From 8eaa98807b9525cb7f022e14cc8cff36f9ee7a06 Mon Sep 17 00:00:00 2001
From: Tom St Denis <tomstdenis@gmail.com>
Date: Mon, 9 Aug 2004 22:15:59 +0000
Subject: [PATCH] added libtommath-0.31

---
 bn.pdf                  |  Bin 320564 -> 320636 bytes
 bn.tex                  |    2 +-
 bn_fast_s_mp_mul_digs.c |    2 +-
 bn_mp_2expt.c           |    2 +-
 bn_mp_clear.c           |    6 +-
 bn_mp_div.c             |    2 +-
 bn_mp_init.c            |   11 +-
 bn_mp_karatsuba_mul.c   |    3 -
 bn_mp_mul.c             |    2 +-
 bn_mp_reduce_is_2k.c    |    5 +-
 bncore.c                |   14 +-
 booker.pl               |    1 +
 changes.txt             |    9 +
 demo/demo.c             |  296 +-
 demo/timing.c           |  291 ++
 etc/makefile            |    3 +-
 etc/makefile.icc        |   67 +
 logs/add.log            |   32 +-
 logs/addsub.png         |  Bin 7084 -> 6254 bytes
 logs/expt.log           |   14 +-
 logs/expt.png           |  Bin 7439 -> 6605 bytes
 logs/expt_2k.log        |   12 +-
 logs/expt_dr.log        |   14 +-
 logs/graphs.dem         |   34 +-
 logs/invmod.log         |   32 -
 logs/invmod.png         |  Bin 5735 -> 4918 bytes
 logs/k7/README          |   13 -
 logs/k7/add.log         |   16 -
 logs/k7/addsub.png      |  Bin 6933 -> 0 bytes
 logs/k7/expt.log        |    7 -
 logs/k7/expt.png        |  Bin 6490 -> 0 bytes
 logs/k7/expt_dr.log     |    7 -
 logs/k7/graphs.dem      |   17 -
 logs/k7/index.html      |   24 -
 logs/k7/invmod.log      |   32 -
 logs/k7/invmod.png      |  Bin 5730 -> 0 bytes
 logs/k7/mult.log        |   17 -
 logs/k7/mult.png        |  Bin 8263 -> 0 bytes
 logs/k7/mult_kara.log   |   17 -
 logs/k7/sqr.log         |   17 -
 logs/k7/sqr_kara.log    |   17 -
 logs/k7/sub.log         |   16 -
 logs/mult.log           |   66 +-
 logs/mult.png           |  Bin 8100 -> 6770 bytes
 logs/mult_kara.log      |   66 +-
 logs/p4/README          |   13 -
 logs/p4/add.log         |   16 -
 logs/p4/addsub.png      |  Bin 6898 -> 0 bytes
 logs/p4/expt.log        |    7 -
 logs/p4/expt.png        |  Bin 6587 -> 0 bytes
 logs/p4/expt_dr.log     |    7 -
 logs/p4/graphs.dem      |   17 -
 logs/p4/index.html      |   24 -
 logs/p4/invmod.log      |   32 -
 logs/p4/invmod.png      |  Bin 5596 -> 0 bytes
 logs/p4/mult.log        |   17 -
 logs/p4/mult.png        |  Bin 8100 -> 0 bytes
 logs/p4/mult_kara.log   |   17 -
 logs/p4/sqr.log         |   17 -
 logs/p4/sqr_kara.log    |   17 -
 logs/p4/sub.log         |   16 -
 logs/sqr.log            |   66 +-
 logs/sqr_kara.log       |   66 +-
 logs/sub.log            |   32 +-
 makefile                |   40 +-
 makefile.bcc            |    3 +-
 makefile.cygwin_dll     |    3 +-
 makefile.icc            |  110 +
 makefile.msvc           |    3 +-
 poster.pdf              |  Bin 40819 -> 40821 bytes
 pre_gen/mpi.c           |   47 +-
 pretty.build            |   66 +
 tommath.pdf             |  Bin 1153705 -> 1154603 bytes
 tommath.src             |  163 +-
 tommath.tex             | 8344 +++++++++++++++++++--------------------
 75 files changed, 5111 insertions(+), 5218 deletions(-)
 create mode 100644 demo/timing.c
 create mode 100644 etc/makefile.icc
 delete mode 100644 logs/k7/README
 delete mode 100644 logs/k7/add.log
 delete mode 100644 logs/k7/addsub.png
 delete mode 100644 logs/k7/expt.log
 delete mode 100644 logs/k7/expt.png
 delete mode 100644 logs/k7/expt_dr.log
 delete mode 100644 logs/k7/graphs.dem
 delete mode 100644 logs/k7/index.html
 delete mode 100644 logs/k7/invmod.log
 delete mode 100644 logs/k7/invmod.png
 delete mode 100644 logs/k7/mult.log
 delete mode 100644 logs/k7/mult.png
 delete mode 100644 logs/k7/mult_kara.log
 delete mode 100644 logs/k7/sqr.log
 delete mode 100644 logs/k7/sqr_kara.log
 delete mode 100644 logs/k7/sub.log
 delete mode 100644 logs/p4/README
 delete mode 100644 logs/p4/add.log
 delete mode 100644 logs/p4/addsub.png
 delete mode 100644 logs/p4/expt.log
 delete mode 100644 logs/p4/expt.png
 delete mode 100644 logs/p4/expt_dr.log
 delete mode 100644 logs/p4/graphs.dem
 delete mode 100644 logs/p4/index.html
 delete mode 100644 logs/p4/invmod.log
 delete mode 100644 logs/p4/invmod.png
 delete mode 100644 logs/p4/mult.log
 delete mode 100644 logs/p4/mult.png
 delete mode 100644 logs/p4/mult_kara.log
 delete mode 100644 logs/p4/sqr.log
 delete mode 100644 logs/p4/sqr_kara.log
 delete mode 100644 logs/p4/sub.log
 create mode 100644 makefile.icc
 create mode 100644 pretty.build

diff --git a/bn.pdf b/bn.pdf
index 52421fd5981b12f134d6f2352d6c7d978f6ffb32..bf80e6a225db3ede3dc544c20cd8b67966853684 100644
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diff --git a/bn.tex b/bn.tex
index a254586..3d1d26d 100644
--- a/bn.tex
+++ b/bn.tex
@@ -49,7 +49,7 @@
 \begin{document}
 \frontmatter
 \pagestyle{empty}
-\title{LibTomMath User Manual \\ v0.30}
+\title{LibTomMath User Manual \\ v0.31}
 \author{Tom St Denis \\ tomstdenis@iahu.ca}
 \maketitle
 This text, the library and the accompanying textbook are all hereby placed in the public domain.  This book has been 
diff --git a/bn_fast_s_mp_mul_digs.c b/bn_fast_s_mp_mul_digs.c
index 75fa706..d268df3 100644
--- a/bn_fast_s_mp_mul_digs.c
+++ b/bn_fast_s_mp_mul_digs.c
@@ -88,7 +88,7 @@ fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
   }
 
   /* setup dest */
-  olduse = c->used;
+  olduse  = c->used;
   c->used = digs;
 
   {
diff --git a/bn_mp_2expt.c b/bn_mp_2expt.c
index 5c4e256..502e85b 100644
--- a/bn_mp_2expt.c
+++ b/bn_mp_2expt.c
@@ -36,7 +36,7 @@ mp_2expt (mp_int * a, int b)
   a->used = b / DIGIT_BIT + 1;
 
   /* put the single bit in its place */
-  a->dp[b / DIGIT_BIT] = 1 << (b % DIGIT_BIT);
+  a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);
 
   return MP_OKAY;
 }
diff --git a/bn_mp_clear.c b/bn_mp_clear.c
index afe9b26..cd439df 100644
--- a/bn_mp_clear.c
+++ b/bn_mp_clear.c
@@ -18,10 +18,14 @@
 void
 mp_clear (mp_int * a)
 {
+  int i;
+
   /* only do anything if a hasn't been freed previously */
   if (a->dp != NULL) {
     /* first zero the digits */
-    memset (a->dp, 0, sizeof (mp_digit) * a->used);
+    for (i = 0; i < a->used; i++) {
+        a->dp[i] = 0;
+    }
 
     /* free ram */
     XFREE(a->dp);
diff --git a/bn_mp_div.c b/bn_mp_div.c
index 652a094..ea2d514 100644
--- a/bn_mp_div.c
+++ b/bn_mp_div.c
@@ -187,7 +187,7 @@ int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
    */
   
   /* get sign before writing to c */
-  x.sign = a->sign;
+  x.sign = x.used == 0 ? MP_ZPOS : a->sign;
 
   if (c != NULL) {
     mp_clamp (&q);
diff --git a/bn_mp_init.c b/bn_mp_init.c
index 5c5c1ad..cac782a 100644
--- a/bn_mp_init.c
+++ b/bn_mp_init.c
@@ -14,15 +14,22 @@
  */
 #include <tommath.h>
 
-/* init a new bigint */
+/* init a new mp_int */
 int mp_init (mp_int * a)
 {
+  int i;
+
   /* allocate memory required and clear it */
-  a->dp = OPT_CAST(mp_digit) XCALLOC (sizeof (mp_digit), MP_PREC);
+  a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC);
   if (a->dp == NULL) {
     return MP_MEM;
   }
 
+  /* set the digits to zero */
+  for (i = 0; i < MP_PREC; i++) {
+      a->dp[i] = 0;
+  }
+
   /* set the used to zero, allocated digits to the default precision
    * and sign to positive */
   a->used  = 0;
diff --git a/bn_mp_karatsuba_mul.c b/bn_mp_karatsuba_mul.c
index 169dacf..105590e 100644
--- a/bn_mp_karatsuba_mul.c
+++ b/bn_mp_karatsuba_mul.c
@@ -76,9 +76,6 @@ int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
     goto X0Y0;
 
   /* now shift the digits */
-  x0.sign = x1.sign = a->sign;
-  y0.sign = y1.sign = b->sign;
-
   x0.used = y0.used = B;
   x1.used = a->used - B;
   y1.used = b->used - B;
diff --git a/bn_mp_mul.c b/bn_mp_mul.c
index 6f3c491..8e11f9f 100644
--- a/bn_mp_mul.c
+++ b/bn_mp_mul.c
@@ -43,6 +43,6 @@ int mp_mul (mp_int * a, mp_int * b, mp_int * c)
       res = s_mp_mul (a, b, c);
     }
   }
-  c->sign = neg;
+  c->sign = (c->used > 0) ? neg : MP_ZPOS;
   return res;
 }
diff --git a/bn_mp_reduce_is_2k.c b/bn_mp_reduce_is_2k.c
index d43b9ff..cc36115 100644
--- a/bn_mp_reduce_is_2k.c
+++ b/bn_mp_reduce_is_2k.c
@@ -17,7 +17,8 @@
 /* determines if mp_reduce_2k can be used */
 int mp_reduce_is_2k(mp_int *a)
 {
-   int ix, iy, iz, iw;
+   int ix, iy, iw;
+   mp_digit iz;
    
    if (a->used == 0) {
       return 0;
@@ -34,7 +35,7 @@ int mp_reduce_is_2k(mp_int *a)
              return 0;
           }
           iz <<= 1;
-          if (iz > (int)MP_MASK) {
+          if (iz > (mp_digit)MP_MASK) {
              ++iw;
              iz = 1;
           }
diff --git a/bncore.c b/bncore.c
index acb78a0..918a99a 100644
--- a/bncore.c
+++ b/bncore.c
@@ -18,14 +18,16 @@
 
  CPU                    /Compiler     /MUL CUTOFF/SQR CUTOFF
 -------------------------------------------------------------
- Intel P4               /GCC v3.2     /        70/       108
- AMD Athlon XP          /GCC v3.2     /       109/       127
-
+ Intel P4 Northwood     /GCC v3.3.3   /        59/        81/profiled build
+ Intel P4 Northwood     /GCC v3.3.3   /        59/        80/profiled_single build
+ Intel P4 Northwood     /ICC v8.0     /        57/        70/profiled build
+ Intel P4 Northwood     /ICC v8.0     /        54/        76/profiled_single build
+ AMD Athlon XP          /GCC v3.2     /       109/       127/
+ 
 */
 
-/* configured for a AMD XP Thoroughbred core with etc/tune.c */
-int     KARATSUBA_MUL_CUTOFF = 109,      /* Min. number of digits before Karatsuba multiplication is used. */
-        KARATSUBA_SQR_CUTOFF = 127,      /* Min. number of digits before Karatsuba squaring is used. */
+int     KARATSUBA_MUL_CUTOFF = 57,      /* Min. number of digits before Karatsuba multiplication is used. */
+        KARATSUBA_SQR_CUTOFF = 70,      /* Min. number of digits before Karatsuba squaring is used. */
         
         TOOM_MUL_CUTOFF      = 350,      /* no optimal values of these are known yet so set em high */
         TOOM_SQR_CUTOFF      = 400; 
diff --git a/booker.pl b/booker.pl
index 4f4231f..de28780 100644
--- a/booker.pl
+++ b/booker.pl
@@ -84,6 +84,7 @@ while (<IN>) {
             $text[$line++] = $_;
             last if ($_ =~ /tommath\.h/);
          }
+         <SRC>;   
       }
       
       $inline = 0;
diff --git a/changes.txt b/changes.txt
index 63e9d61..c90d27a 100644
--- a/changes.txt
+++ b/changes.txt
@@ -1,3 +1,12 @@
+August 9th, 2004
+v0.31  -- "profiled" builds now :-) new timings for Intel Northwoods
+       -- Added "pretty" build target
+       -- Update mp_init() to actually assign 0's instead of relying on calloc()
+       -- "Wolfgang Ehrhardt" <Wolfgang.Ehrhardt@munich.netsurf.de> found a bug in mp_mul() where if
+          you multiply a negative by zero you get negative zero as the result.  Oops.
+       -- J Harper from PeerSec let me toy with his AMD64 and I got 60-bit digits working properly
+          [this also means that I fixed a bug where if sizeof(int) < sizeof(mp_digit) it would bug]
+
 April 11th, 2004
 v0.30  -- Added "mp_toradix_n" which stores upto "n-1" least significant digits of an mp_int
        -- Johan Lindh sent a patch so MSVC wouldn't whine about redefining malloc [in weird dll modes]
diff --git a/demo/demo.c b/demo/demo.c
index 8014ea8..8cbcb8a 100644
--- a/demo/demo.c
+++ b/demo/demo.c
@@ -1,7 +1,5 @@
 #include <time.h>
 
-#define TESTING
-
 #ifdef IOWNANATHLON
 #include <unistd.h>
 #define SLEEP sleep(4)
@@ -11,49 +9,6 @@
 
 #include "tommath.h"
 
-#ifdef TIMER
-ulong64 _tt;
-
-#if defined(__i386__) || defined(_M_IX86) || defined(_M_AMD64)
-/* RDTSC from Scott Duplichan */
-static ulong64 TIMFUNC (void)
-   {
-   #if defined __GNUC__
-      #ifdef __i386__
-         ulong64 a;
-         __asm__ __volatile__ ("rdtsc ":"=A" (a));
-         return a;
-      #else /* gcc-IA64 version */
-         unsigned long result;
-         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
-         while (__builtin_expect ((int) result == -1, 0))
-         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
-         return result;
-      #endif
-
-   // Microsoft and Intel Windows compilers
-   #elif defined _M_IX86
-     __asm rdtsc
-   #elif defined _M_AMD64
-     return __rdtsc ();
-   #elif defined _M_IA64
-     #if defined __INTEL_COMPILER
-       #include <ia64intrin.h>
-     #endif
-      return __getReg (3116);
-   #else
-     #error need rdtsc function for this build
-   #endif
-   }
-#else
-#define TIMFUNC clock
-#endif
-
-ulong64 rdtsc(void) { return TIMFUNC() - _tt; }
-void reset(void) { _tt = TIMFUNC(); }
-
-#endif
-
 void ndraw(mp_int *a, char *name)
 {
    char buf[4096];
@@ -89,10 +44,6 @@ int myrng(unsigned char *dst, int len, void *dat)
 }
 
 
-#define DO2(x) x; x;
-#define DO4(x) DO2(x); DO2(x);
-#define DO8(x) DO4(x); DO4(x);
-#define DO(x)  DO8(x); DO8(x);
 
    char cmd[4096], buf[4096];
 int main(void)
@@ -103,10 +54,6 @@ int main(void)
    unsigned rr;
    int i, n, err, cnt, ix, old_kara_m, old_kara_s;
 
-#ifdef TIMER
-   ulong64 tt, CLK_PER_SEC;
-   FILE *log, *logb, *logc;
-#endif
 
    mp_init(&a);
    mp_init(&b);
@@ -117,11 +64,10 @@ int main(void)
 
    srand(time(NULL));
 
-#ifdef TESTING
   // test mp_get_int
   printf("Testing: mp_get_int\n");
   for(i=0;i<1000;++i) {
-    t = (unsigned long)rand()*rand()+1;
+    t = ((unsigned long)rand()*rand()+1)&0xFFFFFFFF;
     mp_set_int(&a,t);
     if (t!=mp_get_int(&a)) { 
       printf("mp_get_int() bad result!\n");
@@ -141,7 +87,7 @@ int main(void)
 
   // test mp_sqrt
   printf("Testing: mp_sqrt\n");
-  for (i=0;i<10000;++i) { 
+  for (i=0;i<1000;++i) { 
     printf("%6d\r", i); fflush(stdout);
     n = (rand()&15)+1;
     mp_rand(&a,n);
@@ -157,7 +103,7 @@ int main(void)
   }
 
   printf("\nTesting: mp_is_square\n");
-  for (i=0;i<100000;++i) {
+  for (i=0;i<1000;++i) {
     printf("%6d\r", i); fflush(stdout);
 
     /* test mp_is_square false negatives */
@@ -186,11 +132,9 @@ int main(void)
 
   }
   printf("\n\n");
-#endif
 
-#ifdef TESTING 
    /* test for size */
-   for (ix = 16; ix < 512; ix++) {
+   for (ix = 10; ix < 256; ix++) {
        printf("Testing (not safe-prime): %9d bits    \r", ix); fflush(stdout);
        err = mp_prime_random_ex(&a, 8, ix, (rand()&1)?LTM_PRIME_2MSB_OFF:LTM_PRIME_2MSB_ON, myrng, NULL);
        if (err != MP_OKAY) {
@@ -203,7 +147,7 @@ int main(void)
        }
    }
 
-   for (ix = 16; ix < 512; ix++) {
+   for (ix = 16; ix < 256; ix++) {
        printf("Testing (   safe-prime): %9d bits    \r", ix); fflush(stdout);
        err = mp_prime_random_ex(&a, 8, ix, ((rand()&1)?LTM_PRIME_2MSB_OFF:LTM_PRIME_2MSB_ON)|LTM_PRIME_SAFE, myrng, NULL);
        if (err != MP_OKAY) {
@@ -225,9 +169,7 @@ int main(void)
    }
 
    printf("\n\n");
-#endif
 
-#ifdef TESTING
    mp_read_radix(&a, "123456", 10);
    mp_toradix_n(&a, buf, 10, 3);
    printf("a == %s\n", buf);
@@ -235,7 +177,6 @@ int main(void)
    printf("a == %s\n", buf);
    mp_toradix_n(&a, buf, 10, 30);
    printf("a == %s\n", buf);
-#endif
 
 
 #if 0
@@ -248,22 +189,6 @@ int main(void)
    }
 #endif
 
-#if 0
-{
-   mp_word aa, bb;
-
-   for (;;) {
-       aa = abs(rand()) & MP_MASK;
-       bb = abs(rand()) & MP_MASK;
-      if (MULT(aa,bb) != (aa*bb)) {
-             printf("%llu * %llu == %llu or %llu?\n", aa, bb, (ulong64)MULT(aa,bb), (ulong64)(aa*bb));
-             return 0;
-          }
-   }
-}
-#endif
-
-#ifdef TESTING
    /* test mp_cnt_lsb */
    printf("testing mp_cnt_lsb...\n");
    mp_set(&a, 1);
@@ -274,12 +199,10 @@ int main(void)
        }
        mp_mul_2(&a, &a);
    }
-#endif
 
 /* test mp_reduce_2k */
-#ifdef TESTING
    printf("Testing mp_reduce_2k...\n");
-   for (cnt = 3; cnt <= 384; ++cnt) {
+   for (cnt = 3; cnt <= 128; ++cnt) {
        mp_digit tmp;
        mp_2expt(&a, cnt);
        mp_sub_d(&a, 2, &a);  /* a = 2**cnt - 2 */
@@ -289,7 +212,7 @@ int main(void)
        printf("(%d)", mp_reduce_is_2k(&a));
        mp_reduce_2k_setup(&a, &tmp);
        printf("(%d)", tmp);
-       for (ix = 0; ix < 10000; ix++) {
+       for (ix = 0; ix < 1000; ix++) {
            if (!(ix & 127)) {printf("."); fflush(stdout); }
            mp_rand(&b, (cnt/DIGIT_BIT  + 1) * 2);
            mp_copy(&c, &b);
@@ -301,14 +224,11 @@ int main(void)
            }
         }
     }
-#endif
-
 
 /* test mp_div_3  */
-#ifdef TESTING
    printf("Testing mp_div_3...\n");
    mp_set(&d, 3);
-   for (cnt = 0; cnt < 1000000; ) {
+   for (cnt = 0; cnt < 10000; ) {
       mp_digit r1, r2;
 
       if (!(++cnt & 127)) printf("%9d\r", cnt);
@@ -321,12 +241,10 @@ int main(void)
       }
    }
    printf("\n\nPassed div_3 testing\n");
-#endif
 
 /* test the DR reduction */
-#ifdef TESTING
    printf("testing mp_dr_reduce...\n");
-   for (cnt = 2; cnt < 128; cnt++) {
+   for (cnt = 2; cnt < 32; cnt++) {
        printf("%d digit modulus\n", cnt);
        mp_grow(&a, cnt);
        mp_zero(&a);
@@ -334,7 +252,7 @@ int main(void)
            a.dp[ix] = MP_MASK;
        }
        a.used = cnt;
-       mp_prime_next_prime(&a, 3, 0);
+       a.dp[0] = 3;
 
        mp_rand(&b, cnt - 1);
        mp_copy(&b, &c);
@@ -346,206 +264,16 @@ int main(void)
          mp_copy(&b, &c);
 
          mp_mod(&b, &a, &b);
-         mp_dr_reduce(&c, &a, (1<<DIGIT_BIT)-a.dp[0]);
+         mp_dr_reduce(&c, &a, (((mp_digit)1)<<DIGIT_BIT)-a.dp[0]);
 
          if (mp_cmp(&b, &c) != MP_EQ) {
             printf("Failed on trial %lu\n", rr); exit(-1);
 
          }
-      } while (++rr < 100000);
+      } while (++rr < 500);
       printf("Passed DR test for %d digits\n", cnt);
    }
-#endif
 
-#ifdef TIMER
-      /* temp. turn off TOOM */
-      TOOM_MUL_CUTOFF = TOOM_SQR_CUTOFF = 100000;
-
-      reset();
-      sleep(1);
-      CLK_PER_SEC = rdtsc();
-
-      printf("CLK_PER_SEC == %lu\n", CLK_PER_SEC);
-      
-
-      log = fopen("logs/add.log", "w");
-      for (cnt = 8; cnt <= 128; cnt += 8) {
-         SLEEP;
-         mp_rand(&a, cnt);
-         mp_rand(&b, cnt);
-         reset();
-         rr = 0;
-         do {
-            DO(mp_add(&a,&b,&c));
-            rr += 16;
-         } while (rdtsc() < (CLK_PER_SEC * 2));
-         tt = rdtsc();
-         printf("Adding\t\t%4d-bit => %9llu/sec, %9llu ticks\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt, tt);
-         fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, (((ulong64)rr)*CLK_PER_SEC)/tt); fflush(log);
-      }
-      fclose(log);
-
-      log = fopen("logs/sub.log", "w");
-      for (cnt = 8; cnt <= 128; cnt += 8) {
-         SLEEP;
-         mp_rand(&a, cnt);
-         mp_rand(&b, cnt);
-         reset();
-         rr = 0;
-         do {
-            DO(mp_sub(&a,&b,&c));
-            rr += 16;
-         } while (rdtsc() < (CLK_PER_SEC * 2));
-         tt = rdtsc();
-         printf("Subtracting\t\t%4d-bit => %9llu/sec, %9llu ticks\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt, tt);
-         fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, (((ulong64)rr)*CLK_PER_SEC)/tt);  fflush(log);
-      }
-      fclose(log);
-
-   /* do mult/square twice, first without karatsuba and second with */
-mult_test:   
-   old_kara_m = KARATSUBA_MUL_CUTOFF;
-   old_kara_s = KARATSUBA_SQR_CUTOFF;
-   for (ix = 0; ix < 2; ix++) {
-      printf("With%s Karatsuba\n", (ix==0)?"out":"");
-
-      KARATSUBA_MUL_CUTOFF = (ix==0)?9999:old_kara_m;
-      KARATSUBA_SQR_CUTOFF = (ix==0)?9999:old_kara_s;
-
-      log = fopen((ix==0)?"logs/mult.log":"logs/mult_kara.log", "w");
-      for (cnt = 32; cnt <= 288; cnt += 8) {
-         SLEEP;
-         mp_rand(&a, cnt);
-         mp_rand(&b, cnt);
-         reset();
-         rr = 0;
-         do {
-            DO(mp_mul(&a, &b, &c));
-            rr += 16;
-         } while (rdtsc() < (CLK_PER_SEC * 2));
-         tt = rdtsc();
-         printf("Multiplying\t%4d-bit => %9llu/sec, %9llu ticks\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt, tt);
-         fprintf(log, "%d %9llu\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt);  fflush(log);
-      }
-      fclose(log);
-
-      log = fopen((ix==0)?"logs/sqr.log":"logs/sqr_kara.log", "w");
-      for (cnt = 32; cnt <= 288; cnt += 8) {
-         SLEEP;
-         mp_rand(&a, cnt);
-         reset();
-         rr = 0;
-         do {
-            DO(mp_sqr(&a, &b));
-            rr += 16;
-         } while (rdtsc() < (CLK_PER_SEC * 2));
-         tt = rdtsc();
-         printf("Squaring\t%4d-bit => %9llu/sec, %9llu ticks\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt, tt);
-         fprintf(log, "%d %9llu\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt);  fflush(log);
-      }
-      fclose(log);
-
-   }
-expt_test:
-  {
-      char *primes[] = {
-         /* 2K moduli mersenne primes */
-         "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151",
-         "531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127",
-         "10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087",
-         "1475979915214180235084898622737381736312066145333169775147771216478570297878078949377407337049389289382748507531496480477281264838760259191814463365330269540496961201113430156902396093989090226259326935025281409614983499388222831448598601834318536230923772641390209490231836446899608210795482963763094236630945410832793769905399982457186322944729636418890623372171723742105636440368218459649632948538696905872650486914434637457507280441823676813517852099348660847172579408422316678097670224011990280170474894487426924742108823536808485072502240519452587542875349976558572670229633962575212637477897785501552646522609988869914013540483809865681250419497686697771007",
-         "259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071",
-         "190797007524439073807468042969529173669356994749940177394741882673528979787005053706368049835514900244303495954950709725762186311224148828811920216904542206960744666169364221195289538436845390250168663932838805192055137154390912666527533007309292687539092257043362517857366624699975402375462954490293259233303137330643531556539739921926201438606439020075174723029056838272505051571967594608350063404495977660656269020823960825567012344189908927956646011998057988548630107637380993519826582389781888135705408653045219655801758081251164080554609057468028203308718724654081055323215860189611391296030471108443146745671967766308925858547271507311563765171008318248647110097614890313562856541784154881743146033909602737947385055355960331855614540900081456378659068370317267696980001187750995491090350108417050917991562167972281070161305972518044872048331306383715094854938415738549894606070722584737978176686422134354526989443028353644037187375385397838259511833166416134323695660367676897722287918773420968982326089026150031515424165462111337527431154890666327374921446276833564519776797633875503548665093914556482031482248883127023777039667707976559857333357013727342079099064400455741830654320379350833236245819348824064783585692924881021978332974949906122664421376034687815350484991",
-
-         /* DR moduli */
-         "14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368612079",
-         "101745825697019260773923519755878567461315282017759829107608914364075275235254395622580447400994175578963163918967182013639660669771108475957692810857098847138903161308502419410142185759152435680068435915159402496058513611411688900243039",
-         "736335108039604595805923406147184530889923370574768772191969612422073040099331944991573923112581267542507986451953227192970402893063850485730703075899286013451337291468249027691733891486704001513279827771740183629161065194874727962517148100775228363421083691764065477590823919364012917984605619526140821797602431",
-         "38564998830736521417281865696453025806593491967131023221754800625044118265468851210705360385717536794615180260494208076605798671660719333199513807806252394423283413430106003596332513246682903994829528690198205120921557533726473585751382193953592127439965050261476810842071573684505878854588706623484573925925903505747545471088867712185004135201289273405614415899438276535626346098904241020877974002916168099951885406379295536200413493190419727789712076165162175783",
-         "542189391331696172661670440619180536749994166415993334151601745392193484590296600979602378676624808129613777993466242203025054573692562689251250471628358318743978285860720148446448885701001277560572526947619392551574490839286458454994488665744991822837769918095117129546414124448777033941223565831420390846864429504774477949153794689948747680362212954278693335653935890352619041936727463717926744868338358149568368643403037768649616778526013610493696186055899318268339432671541328195724261329606699831016666359440874843103020666106568222401047720269951530296879490444224546654729111504346660859907296364097126834834235287147",
-         "1487259134814709264092032648525971038895865645148901180585340454985524155135260217788758027400478312256339496385275012465661575576202252063145698732079880294664220579764848767704076761853197216563262660046602703973050798218246170835962005598561669706844469447435461092542265792444947706769615695252256130901271870341005768912974433684521436211263358097522726462083917939091760026658925757076733484173202927141441492573799914240222628795405623953109131594523623353044898339481494120112723445689647986475279242446083151413667587008191682564376412347964146113898565886683139407005941383669325997475076910488086663256335689181157957571445067490187939553165903773554290260531009121879044170766615232300936675369451260747671432073394867530820527479172464106442450727640226503746586340279816318821395210726268291535648506190714616083163403189943334431056876038286530365757187367147446004855912033137386225053275419626102417236133948503",
-         "1095121115716677802856811290392395128588168592409109494900178008967955253005183831872715423151551999734857184538199864469605657805519106717529655044054833197687459782636297255219742994736751541815269727940751860670268774903340296040006114013971309257028332849679096824800250742691718610670812374272414086863715763724622797509437062518082383056050144624962776302147890521249477060215148275163688301275847155316042279405557632639366066847442861422164832655874655824221577849928863023018366835675399949740429332468186340518172487073360822220449055340582568461568645259954873303616953776393853174845132081121976327462740354930744487429617202585015510744298530101547706821590188733515880733527449780963163909830077616357506845523215289297624086914545378511082534229620116563260168494523906566709418166011112754529766183554579321224940951177394088465596712620076240067370589036924024728375076210477267488679008016579588696191194060127319035195370137160936882402244399699172017835144537488486396906144217720028992863941288217185353914991583400421682751000603596655790990815525126154394344641336397793791497068253936771017031980867706707490224041075826337383538651825493679503771934836094655802776331664261631740148281763487765852746577808019633679",
-
-         /* generic unrestricted moduli */
-         "17933601194860113372237070562165128350027320072176844226673287945873370751245439587792371960615073855669274087805055507977323024886880985062002853331424203",
-         "2893527720709661239493896562339544088620375736490408468011883030469939904368086092336458298221245707898933583190713188177399401852627749210994595974791782790253946539043962213027074922559572312141181787434278708783207966459019479487",
-         "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319",
-         "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887",
-         "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227",
-         "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207",
-         "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979",
-         NULL
-      };
-   log = fopen("logs/expt.log", "w");
-   logb = fopen("logs/expt_dr.log", "w");
-   logc = fopen("logs/expt_2k.log", "w");
-   for (n = 0; primes[n]; n++) {
-      SLEEP;
-      mp_read_radix(&a, primes[n], 10);
-      mp_zero(&b);
-      for (rr = 0; rr < mp_count_bits(&a); rr++) {
-         mp_mul_2(&b, &b);
-         b.dp[0] |= lbit();
-         b.used  += 1;
-      }
-      mp_sub_d(&a, 1, &c);
-      mp_mod(&b, &c, &b);
-      mp_set(&c, 3);
-      reset();
-      rr = 0;
-      do {
-         DO(mp_exptmod(&c, &b, &a, &d));
-         rr += 16;
-      } while (rdtsc() < (CLK_PER_SEC * 2));
-      tt = rdtsc();
-      mp_sub_d(&a, 1, &e);
-      mp_sub(&e, &b, &b);
-      mp_exptmod(&c, &b, &a, &e);  /* c^(p-1-b) mod a */
-      mp_mulmod(&e, &d, &a, &d);   /* c^b * c^(p-1-b) == c^p-1 == 1 */
-      if (mp_cmp_d(&d, 1)) {
-         printf("Different (%d)!!!\n", mp_count_bits(&a));
-         draw(&d);
-         exit(0);
-      }
-      printf("Exponentiating\t%4d-bit => %9llu/sec, %9llu ticks\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt, tt);
-      fprintf((n < 6) ? logc : (n < 13) ? logb : log, "%d %9llu\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt);
-   }
-   }
-   fclose(log);
-   fclose(logb);
-   fclose(logc);
-
-   log = fopen("logs/invmod.log", "w");
-   for (cnt = 4; cnt <= 128; cnt += 4) {
-      SLEEP;
-      mp_rand(&a, cnt);
-      mp_rand(&b, cnt);
-
-      do {
-         mp_add_d(&b, 1, &b);
-         mp_gcd(&a, &b, &c);
-      } while (mp_cmp_d(&c, 1) != MP_EQ);
-
-      reset();
-      rr = 0;
-      do {
-         DO(mp_invmod(&b, &a, &c));
-         rr += 16;
-      } while (rdtsc() < (CLK_PER_SEC * 2));
-      tt = rdtsc();
-      mp_mulmod(&b, &c, &a, &d);
-      if (mp_cmp_d(&d, 1) != MP_EQ) {
-         printf("Failed to invert\n");
-         return 0;
-      }
-      printf("Inverting mod\t%4d-bit => %9llu/sec, %9llu ticks\n", mp_count_bits(&a), (((ulong64)rr)*CLK_PER_SEC)/tt, tt);
-      fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, (((ulong64)rr)*CLK_PER_SEC)/tt);
-   }
-   fclose(log);
-
-   return 0;
-
-#endif
 
    div2_n = mul2_n = inv_n = expt_n = lcm_n = gcd_n = add_n =
    sub_n = mul_n = div_n = sqr_n = mul2d_n = div2d_n = cnt = add_d_n = sub_d_n= 0;
diff --git a/demo/timing.c b/demo/timing.c
new file mode 100644
index 0000000..30e95ce
--- /dev/null
+++ b/demo/timing.c
@@ -0,0 +1,291 @@
+#include <tommath.h>
+#include <time.h>
+
+ulong64 _tt;
+
+#ifdef IOWNANATHLON
+#include <unistd.h>
+#define SLEEP sleep(4)
+#else
+#define SLEEP
+#endif
+
+
+void ndraw(mp_int *a, char *name)
+{
+   char buf[4096];
+   printf("%s: ", name);
+   mp_toradix(a, buf, 64);
+   printf("%s\n", buf);
+}
+
+static void draw(mp_int *a)
+{
+   ndraw(a, "");
+}
+
+
+unsigned long lfsr = 0xAAAAAAAAUL;
+
+int lbit(void)
+{
+   if (lfsr & 0x80000000UL) {
+      lfsr = ((lfsr << 1) ^ 0x8000001BUL) & 0xFFFFFFFFUL;
+      return 1;
+   } else {
+      lfsr <<= 1;
+      return 0;
+   }
+}
+
+#if defined(__i386__) || defined(_M_IX86) || defined(_M_AMD64)
+/* RDTSC from Scott Duplichan */
+static ulong64 TIMFUNC (void)
+   {
+   #if defined __GNUC__
+      #ifdef __i386__
+         ulong64 a;
+         __asm__ __volatile__ ("rdtsc ":"=A" (a));
+         return a;
+      #else /* gcc-IA64 version */
+         unsigned long result;
+         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
+         while (__builtin_expect ((int) result == -1, 0))
+         __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory");
+         return result;
+      #endif
+
+   // Microsoft and Intel Windows compilers
+   #elif defined _M_IX86
+     __asm rdtsc
+   #elif defined _M_AMD64
+     return __rdtsc ();
+   #elif defined _M_IA64
+     #if defined __INTEL_COMPILER
+       #include <ia64intrin.h>
+     #endif
+      return __getReg (3116);
+   #else
+     #error need rdtsc function for this build
+   #endif
+   }
+#else
+#define TIMFUNC clock
+#endif
+
+#define DO(x) x; x;
+//#define DO4(x) DO2(x); DO2(x);
+//#define DO8(x) DO4(x); DO4(x);
+//#define DO(x)  DO8(x); DO8(x);
+
+int main(void)
+{
+   ulong64 tt, gg, CLK_PER_SEC;
+   FILE *log, *logb, *logc;
+   mp_int a, b, c, d, e, f;
+   int n, cnt, ix, old_kara_m, old_kara_s;
+   unsigned rr;
+
+   mp_init(&a);
+   mp_init(&b);
+   mp_init(&c);
+   mp_init(&d);
+   mp_init(&e);
+   mp_init(&f);
+
+   srand(time(NULL));
+ 
+
+      /* temp. turn off TOOM */
+      TOOM_MUL_CUTOFF = TOOM_SQR_CUTOFF = 100000;
+
+      CLK_PER_SEC = TIMFUNC();
+      sleep(1);
+      CLK_PER_SEC = TIMFUNC() - CLK_PER_SEC;
+
+      printf("CLK_PER_SEC == %llu\n", CLK_PER_SEC);
+      
+      log = fopen("logs/add.log", "w");
+      for (cnt = 8; cnt <= 128; cnt += 8) {
+         SLEEP;
+         mp_rand(&a, cnt);
+         mp_rand(&b, cnt);
+         rr = 0;
+         tt = -1;
+         do {
+            gg = TIMFUNC();
+            DO(mp_add(&a,&b,&c));
+            gg = (TIMFUNC() - gg)>>1;
+            if (tt > gg) tt = gg;
+         } while (++rr < 100000);
+         printf("Adding\t\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt);
+         fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, tt); fflush(log);
+      }
+      fclose(log);
+
+      log = fopen("logs/sub.log", "w");
+      for (cnt = 8; cnt <= 128; cnt += 8) {
+         SLEEP;
+         mp_rand(&a, cnt);
+         mp_rand(&b, cnt);
+         rr = 0;
+         tt = -1;
+         do {
+            gg = TIMFUNC();
+            DO(mp_sub(&a,&b,&c));
+            gg = (TIMFUNC() - gg)>>1;
+            if (tt > gg) tt = gg;
+         } while (++rr < 100000);
+
+         printf("Subtracting\t\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt);
+         fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, tt);  fflush(log);
+      }
+      fclose(log);
+
+   /* do mult/square twice, first without karatsuba and second with */
+   old_kara_m = KARATSUBA_MUL_CUTOFF;
+   old_kara_s = KARATSUBA_SQR_CUTOFF;
+   for (ix = 0; ix < 1; ix++) {
+      printf("With%s Karatsuba\n", (ix==0)?"out":"");
+
+      KARATSUBA_MUL_CUTOFF = (ix==0)?9999:old_kara_m;
+      KARATSUBA_SQR_CUTOFF = (ix==0)?9999:old_kara_s;
+
+      log = fopen((ix==0)?"logs/mult.log":"logs/mult_kara.log", "w");
+      for (cnt = 32; cnt <= 288; cnt += 8) {
+         SLEEP;
+         mp_rand(&a, cnt);
+         mp_rand(&b, cnt);
+         rr = 0;
+         tt = -1;
+         do {
+            gg = TIMFUNC();
+            DO(mp_mul(&a, &b, &c));
+            gg = (TIMFUNC() - gg)>>1;
+            if (tt > gg) tt = gg;
+         } while (++rr < 100);
+         printf("Multiplying\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt);
+         fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt);  fflush(log);
+      }
+      fclose(log);
+
+      log = fopen((ix==0)?"logs/sqr.log":"logs/sqr_kara.log", "w");
+      for (cnt = 32; cnt <= 288; cnt += 8) {
+         SLEEP;
+         mp_rand(&a, cnt);
+         rr = 0;
+         tt = -1;
+         do {
+            gg = TIMFUNC();
+            DO(mp_sqr(&a, &b));
+            gg = (TIMFUNC() - gg)>>1;
+            if (tt > gg) tt = gg;
+         } while (++rr < 100);
+         printf("Squaring\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt);
+         fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt);  fflush(log);
+      }
+      fclose(log);
+
+   }
+
+  {
+      char *primes[] = {
+         /* 2K moduli mersenne primes */
+         "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151",
+         "531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127",
+         "10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087",
+         "1475979915214180235084898622737381736312066145333169775147771216478570297878078949377407337049389289382748507531496480477281264838760259191814463365330269540496961201113430156902396093989090226259326935025281409614983499388222831448598601834318536230923772641390209490231836446899608210795482963763094236630945410832793769905399982457186322944729636418890623372171723742105636440368218459649632948538696905872650486914434637457507280441823676813517852099348660847172579408422316678097670224011990280170474894487426924742108823536808485072502240519452587542875349976558572670229633962575212637477897785501552646522609988869914013540483809865681250419497686697771007",
+         "259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071",
+         "190797007524439073807468042969529173669356994749940177394741882673528979787005053706368049835514900244303495954950709725762186311224148828811920216904542206960744666169364221195289538436845390250168663932838805192055137154390912666527533007309292687539092257043362517857366624699975402375462954490293259233303137330643531556539739921926201438606439020075174723029056838272505051571967594608350063404495977660656269020823960825567012344189908927956646011998057988548630107637380993519826582389781888135705408653045219655801758081251164080554609057468028203308718724654081055323215860189611391296030471108443146745671967766308925858547271507311563765171008318248647110097614890313562856541784154881743146033909602737947385055355960331855614540900081456378659068370317267696980001187750995491090350108417050917991562167972281070161305972518044872048331306383715094854938415738549894606070722584737978176686422134354526989443028353644037187375385397838259511833166416134323695660367676897722287918773420968982326089026150031515424165462111337527431154890666327374921446276833564519776797633875503548665093914556482031482248883127023777039667707976559857333357013727342079099064400455741830654320379350833236245819348824064783585692924881021978332974949906122664421376034687815350484991",
+
+         /* DR moduli */
+         "14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368612079",
+         "101745825697019260773923519755878567461315282017759829107608914364075275235254395622580447400994175578963163918967182013639660669771108475957692810857098847138903161308502419410142185759152435680068435915159402496058513611411688900243039",
+         "736335108039604595805923406147184530889923370574768772191969612422073040099331944991573923112581267542507986451953227192970402893063850485730703075899286013451337291468249027691733891486704001513279827771740183629161065194874727962517148100775228363421083691764065477590823919364012917984605619526140821797602431",
+         "38564998830736521417281865696453025806593491967131023221754800625044118265468851210705360385717536794615180260494208076605798671660719333199513807806252394423283413430106003596332513246682903994829528690198205120921557533726473585751382193953592127439965050261476810842071573684505878854588706623484573925925903505747545471088867712185004135201289273405614415899438276535626346098904241020877974002916168099951885406379295536200413493190419727789712076165162175783",
+         "542189391331696172661670440619180536749994166415993334151601745392193484590296600979602378676624808129613777993466242203025054573692562689251250471628358318743978285860720148446448885701001277560572526947619392551574490839286458454994488665744991822837769918095117129546414124448777033941223565831420390846864429504774477949153794689948747680362212954278693335653935890352619041936727463717926744868338358149568368643403037768649616778526013610493696186055899318268339432671541328195724261329606699831016666359440874843103020666106568222401047720269951530296879490444224546654729111504346660859907296364097126834834235287147",
+         "1487259134814709264092032648525971038895865645148901180585340454985524155135260217788758027400478312256339496385275012465661575576202252063145698732079880294664220579764848767704076761853197216563262660046602703973050798218246170835962005598561669706844469447435461092542265792444947706769615695252256130901271870341005768912974433684521436211263358097522726462083917939091760026658925757076733484173202927141441492573799914240222628795405623953109131594523623353044898339481494120112723445689647986475279242446083151413667587008191682564376412347964146113898565886683139407005941383669325997475076910488086663256335689181157957571445067490187939553165903773554290260531009121879044170766615232300936675369451260747671432073394867530820527479172464106442450727640226503746586340279816318821395210726268291535648506190714616083163403189943334431056876038286530365757187367147446004855912033137386225053275419626102417236133948503",
+         "1095121115716677802856811290392395128588168592409109494900178008967955253005183831872715423151551999734857184538199864469605657805519106717529655044054833197687459782636297255219742994736751541815269727940751860670268774903340296040006114013971309257028332849679096824800250742691718610670812374272414086863715763724622797509437062518082383056050144624962776302147890521249477060215148275163688301275847155316042279405557632639366066847442861422164832655874655824221577849928863023018366835675399949740429332468186340518172487073360822220449055340582568461568645259954873303616953776393853174845132081121976327462740354930744487429617202585015510744298530101547706821590188733515880733527449780963163909830077616357506845523215289297624086914545378511082534229620116563260168494523906566709418166011112754529766183554579321224940951177394088465596712620076240067370589036924024728375076210477267488679008016579588696191194060127319035195370137160936882402244399699172017835144537488486396906144217720028992863941288217185353914991583400421682751000603596655790990815525126154394344641336397793791497068253936771017031980867706707490224041075826337383538651825493679503771934836094655802776331664261631740148281763487765852746577808019633679",
+
+         /* generic unrestricted moduli */
+         "17933601194860113372237070562165128350027320072176844226673287945873370751245439587792371960615073855669274087805055507977323024886880985062002853331424203",
+         "2893527720709661239493896562339544088620375736490408468011883030469939904368086092336458298221245707898933583190713188177399401852627749210994595974791782790253946539043962213027074922559572312141181787434278708783207966459019479487",
+         "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319",
+         "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887",
+         "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227",
+         "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207",
+         "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979",
+         NULL
+      };
+   log = fopen("logs/expt.log", "w");
+   logb = fopen("logs/expt_dr.log", "w");
+   logc = fopen("logs/expt_2k.log", "w");
+   for (n = 0; primes[n]; n++) {
+      SLEEP;
+      mp_read_radix(&a, primes[n], 10);
+      mp_zero(&b);
+      for (rr = 0; rr < (unsigned)mp_count_bits(&a); rr++) {
+         mp_mul_2(&b, &b);
+         b.dp[0] |= lbit();
+         b.used  += 1;
+      }
+      mp_sub_d(&a, 1, &c);
+      mp_mod(&b, &c, &b);
+      mp_set(&c, 3);
+         rr = 0;
+         tt = -1;
+         do {
+            gg = TIMFUNC();
+            DO(mp_exptmod(&c, &b, &a, &d));
+            gg = (TIMFUNC() - gg)>>1;
+            if (tt > gg) tt = gg;
+         } while (++rr < 10);
+      mp_sub_d(&a, 1, &e);
+      mp_sub(&e, &b, &b);
+      mp_exptmod(&c, &b, &a, &e);  /* c^(p-1-b) mod a */
+      mp_mulmod(&e, &d, &a, &d);   /* c^b * c^(p-1-b) == c^p-1 == 1 */
+      if (mp_cmp_d(&d, 1)) {
+         printf("Different (%d)!!!\n", mp_count_bits(&a));
+         draw(&d);
+         exit(0);
+      }
+      printf("Exponentiating\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt);
+      fprintf((n < 6) ? logc : (n < 13) ? logb : log, "%d %9llu\n", mp_count_bits(&a), tt);
+   }
+   }
+   fclose(log);
+   fclose(logb);
+   fclose(logc);
+
+   log = fopen("logs/invmod.log", "w");
+   for (cnt = 4; cnt <= 128; cnt += 4) {
+      SLEEP;
+      mp_rand(&a, cnt);
+      mp_rand(&b, cnt);
+
+      do {
+         mp_add_d(&b, 1, &b);
+         mp_gcd(&a, &b, &c);
+      } while (mp_cmp_d(&c, 1) != MP_EQ);
+
+         rr = 0;
+         tt = -1;
+      do {
+         gg = TIMFUNC();
+         DO(mp_invmod(&b, &a, &c));
+         gg = (TIMFUNC() - gg)>>1;
+         if (tt > gg) tt = gg;
+      } while (++rr < 1000);
+      mp_mulmod(&b, &c, &a, &d);
+      if (mp_cmp_d(&d, 1) != MP_EQ) {
+         printf("Failed to invert\n");
+         return 0;
+      }
+      printf("Inverting mod\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt);
+      fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, tt);
+   }
+   fclose(log);
+
+   return 0;
+}
+
diff --git a/etc/makefile b/etc/makefile
index 98ddb1c..99154d8 100644
--- a/etc/makefile
+++ b/etc/makefile
@@ -46,4 +46,5 @@ mont: mont.o
 
         
 clean:
-	rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat
+	rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat \
+         *.da *.dyn *.dpi *~
diff --git a/etc/makefile.icc b/etc/makefile.icc
new file mode 100644
index 0000000..0a50728
--- /dev/null
+++ b/etc/makefile.icc
@@ -0,0 +1,67 @@
+CC = icc
+
+CFLAGS += -I../
+
+# optimize for SPEED
+#
+# -mcpu= can be pentium, pentiumpro (covers PII through PIII) or pentium4
+# -ax?   specifies make code specifically for ? but compatible with IA-32
+# -x?    specifies compile solely for ? [not specifically IA-32 compatible]
+#
+# where ? is 
+#   K - PIII
+#   W - first P4 [Williamette]
+#   N - P4 Northwood
+#   P - P4 Prescott
+#   B - Blend of P4 and PM [mobile]
+#
+# Default to just generic max opts
+CFLAGS += -O3 -xN -ip
+
+# default lib name (requires install with root)
+# LIBNAME=-ltommath
+
+# libname when you can't install the lib with install
+LIBNAME=../libtommath.a
+
+#provable primes
+pprime: pprime.o
+	$(CC) pprime.o $(LIBNAME) -o pprime
+
+# portable [well requires clock()] tuning app
+tune: tune.o
+	$(CC) tune.o $(LIBNAME) -o tune
+	
+# same app but using RDTSC for higher precision [requires 80586+], coff based gcc installs [e.g. ming, cygwin, djgpp]
+tune86: tune.c
+	nasm -f coff timer.asm
+	$(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o  $(LIBNAME) -o tune86
+	
+# for cygwin
+tune86c: tune.c
+	nasm -f gnuwin32 timer.asm
+	$(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o  $(LIBNAME) -o tune86
+
+#make tune86 for linux or any ELF format
+tune86l: tune.c
+	nasm -f elf -DUSE_ELF timer.asm
+	$(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86l
+        
+# spits out mersenne primes
+mersenne: mersenne.o
+	$(CC) mersenne.o $(LIBNAME) -o mersenne
+
+# fines DR safe primes for the given config
+drprime: drprime.o
+	$(CC) drprime.o $(LIBNAME) -o drprime
+	
+# fines 2k safe primes for the given config
+2kprime: 2kprime.o
+	$(CC) 2kprime.o $(LIBNAME) -o 2kprime
+
+mont: mont.o
+	$(CC) mont.o $(LIBNAME) -o mont
+
+        
+clean:
+	rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat *.il
diff --git a/logs/add.log b/logs/add.log
index e53b415..2ba7207 100644
--- a/logs/add.log
+++ b/logs/add.log
@@ -1,16 +1,16 @@
-224  20297071
-448  15151383
-672  13088682
-896  11111587
-1120   9240621
-1344   8221878
-1568   7227434
-1792   6718051
-2016   6042524
-2240   5685200
-2464   5240465
-2688   4818032
-2912   4412794
-3136   4155883
-3360   3927078
-3584   3722138
+224      1572
+448      1740
+672      1902
+896      2116
+1120      2324
+1344      2484
+1568      2548
+1792      2772
+2016      2958
+2240      3058
+2464      3276
+2688      3436
+2912      3542
+3136      3702
+3360      3926
+3584      4074
diff --git a/logs/addsub.png b/logs/addsub.png
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diff --git a/logs/expt.log b/logs/expt.log
index 2597b48..695c936 100644
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diff --git a/logs/expt_2k.log b/logs/expt_2k.log
index f4c282c..d7c47f3 100644
--- a/logs/expt_2k.log
+++ b/logs/expt_2k.log
@@ -1,6 +1,6 @@
-521       783
-607       585
-1279       138
-2203        39
-3217        15
-4253         6
+521  18847776
+607  24665920
+1279 110036220
+2203 414562036
+3217 1108350966
+4253 2286079370
diff --git a/logs/expt_dr.log b/logs/expt_dr.log
index c552e12..b017e7c 100644
--- a/logs/expt_dr.log
+++ b/logs/expt_dr.log
@@ -1,7 +1,7 @@
-532      1296
-784       551
-1036       283
-1540       109
-2072        52
-3080        18
-4116         7
+532   9656134
+784  23022274
+1036  45227854
+1540 129652848
+2072 280625626
+3080 845619480
+4116 1866206400
diff --git a/logs/graphs.dem b/logs/graphs.dem
index d5c9b8a..dfaf613 100644
--- a/logs/graphs.dem
+++ b/logs/graphs.dem
@@ -1,17 +1,17 @@
-set terminal png
-set size 1.75
-set ylabel "Operations per Second"
-set xlabel "Operand size (bits)"
-
-set output "addsub.png"
-plot 'add.log' smooth bezier title "Addition", 'sub.log' smooth bezier title "Subtraction"
-
-set output "mult.png"
-plot 'sqr.log' smooth bezier title "Squaring (without Karatsuba)", 'sqr_kara.log' smooth bezier title "Squaring (Karatsuba)", 'mult.log' smooth bezier title "Multiplication (without Karatsuba)", 'mult_kara.log' smooth bezier title "Multiplication (Karatsuba)"
-
-set output "expt.png"
-plot 'expt.log' smooth bezier title "Exptmod (Montgomery)", 'expt_dr.log' smooth bezier title "Exptmod (Dimminished Radix)", 'expt_2k.log' smooth bezier title "Exptmod (2k Reduction)"
-
-set output "invmod.png"
-plot 'invmod.log' smooth bezier title "Modular Inverse"
-
+set terminal png
+set size 1.75
+set ylabel "Cycles per Operation"
+set xlabel "Operand size (bits)"
+
+set output "addsub.png"
+plot 'add.log' smooth bezier title "Addition", 'sub.log' smooth bezier title "Subtraction"
+
+set output "mult.png"
+plot 'sqr.log' smooth bezier title "Squaring (without Karatsuba)", 'sqr_kara.log' smooth bezier title "Squaring (Karatsuba)", 'mult.log' smooth bezier title "Multiplication (without Karatsuba)", 'mult_kara.log' smooth bezier title "Multiplication (Karatsuba)"
+
+set output "expt.png"
+plot 'expt.log' smooth bezier title "Exptmod (Montgomery)", 'expt_dr.log' smooth bezier title "Exptmod (Dimminished Radix)", 'expt_2k.log' smooth bezier title "Exptmod (2k Reduction)"
+
+set output "invmod.png"
+plot 'invmod.log' smooth bezier title "Modular Inverse"
+
diff --git a/logs/invmod.log b/logs/invmod.log
index c9294ef..e69de29 100644
--- a/logs/invmod.log
+++ b/logs/invmod.log
@@ -1,32 +0,0 @@
-112     17364
-224      8643
-336      8867
-448      6228
-560      4737
-672      2259
-784      2899
-896      1497
-1008      1238
-1120      1010
-1232       870
-1344      1265
-1456      1102
-1568       981
-1680       539
-1792       484
-1904       722
-2016       392
-2128       604
-2240       551
-2352       511
-2464       469
-2576       263
-2688       247
-2800       227
-2912       354
-3024       336
-3136       312
-3248       296
-3360       166
-3472       155
-3584       248
diff --git a/logs/invmod.png b/logs/invmod.png
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diff --git a/logs/k7/README b/logs/k7/README
deleted file mode 100644
index ea20c81..0000000
--- a/logs/k7/README
+++ /dev/null
@@ -1,13 +0,0 @@
-To use the pretty graphs you have to first build/run the ltmtest from the root directory of the package.  
-Todo this type 
-
-make timing ; ltmtest
-
-in the root.  It will run for a while [about ten minutes on most PCs] and produce a series of .log files in logs/.
-
-After doing that run "gnuplot graphs.dem" to make the PNGs.  If you managed todo that all so far just open index.html to view
-them all :-)
-
-Have fun
-
-Tom
\ No newline at end of file
diff --git a/logs/k7/add.log b/logs/k7/add.log
deleted file mode 100644
index 796ab48..0000000
--- a/logs/k7/add.log
+++ /dev/null
@@ -1,16 +0,0 @@
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diff --git a/logs/k7/addsub.png b/logs/k7/addsub.png
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diff --git a/logs/k7/expt_dr.log b/logs/k7/expt_dr.log
deleted file mode 100644
index 7df658f..0000000
--- a/logs/k7/expt_dr.log
+++ /dev/null
@@ -1,7 +0,0 @@
-532      1088
-784       460
-1036       240
-1540        92
-2072        43
-3080        15
-4116         6
diff --git a/logs/k7/graphs.dem b/logs/k7/graphs.dem
deleted file mode 100644
index c580495..0000000
--- a/logs/k7/graphs.dem
+++ /dev/null
@@ -1,17 +0,0 @@
-set terminal png color
-set size 1.75
-set ylabel "Operations per Second"
-set xlabel "Operand size (bits)"
-
-set output "addsub.png"
-plot 'add.log' smooth bezier title "Addition", 'sub.log' smooth bezier title "Subtraction"
-
-set output "mult.png"
-plot 'sqr.log' smooth bezier title "Squaring (without Karatsuba)", 'sqr_kara.log' smooth bezier title "Squaring (Karatsuba)", 'mult.log' smooth bezier title "Multiplication (without Karatsuba)", 'mult_kara.log' smooth bezier title "Multiplication (Karatsuba)"
-
-set output "expt.png"
-plot 'expt.log' smooth bezier title "Exptmod (Montgomery)", 'expt_dr.log' smooth bezier title "Exptmod (Dimminished Radix)"
-
-set output "invmod.png"
-plot 'invmod.log' smooth bezier title "Modular Inverse"
-
diff --git a/logs/k7/index.html b/logs/k7/index.html
deleted file mode 100644
index 19fe403..0000000
--- a/logs/k7/index.html
+++ /dev/null
@@ -1,24 +0,0 @@
-<html>
-<head>
-<title>LibTomMath Log Plots</title>
-</head>
-<body>
-
-<h1>Addition and Subtraction</h1>
-<center><img src=addsub.png></center>
-<hr>
-
-<h1>Multipliers</h1>
-<center><img src=mult.png></center>
-<hr>
-
-<h1>Exptmod</h1>
-<center><img src=expt.png></center>
-<hr>
-
-<h1>Modular Inverse</h1>
-<center><img src=invmod.png></center>
-<hr>
-
-</body>
-</html>
\ No newline at end of file
diff --git a/logs/k7/invmod.log b/logs/k7/invmod.log
deleted file mode 100644
index d1198fb..0000000
--- a/logs/k7/invmod.log
+++ /dev/null
@@ -1,32 +0,0 @@
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-224      8192
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-448      3560
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-2016       608
-2128       336
-2240       312
-2352       288
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-2800       354
-2912       198
-3024       307
-3136       173
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diff --git a/logs/k7/invmod.png b/logs/k7/invmod.png
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diff --git a/logs/k7/mult.log b/logs/k7/mult.log
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index 4b1bff3..0000000
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diff --git a/logs/k7/mult.png b/logs/k7/mult.png
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deleted file mode 100644
index 53c0864..0000000
--- a/logs/k7/mult_kara.log
+++ /dev/null
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diff --git a/logs/k7/sqr.log b/logs/k7/sqr.log
deleted file mode 100644
index 2fb2e98..0000000
--- a/logs/k7/sqr.log
+++ /dev/null
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deleted file mode 100644
index ba30f9e..0000000
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+++ /dev/null
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diff --git a/logs/k7/sub.log b/logs/k7/sub.log
deleted file mode 100644
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diff --git a/logs/mult.log b/logs/mult.log
index d4f5899..5b2d258 100644
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diff --git a/logs/mult_kara.log b/logs/mult_kara.log
index 6edc439..c69769b 100644
--- a/logs/mult_kara.log
+++ b/logs/mult_kara.log
@@ -1,33 +1,33 @@
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diff --git a/logs/p4/README b/logs/p4/README
deleted file mode 100644
index ea20c81..0000000
--- a/logs/p4/README
+++ /dev/null
@@ -1,13 +0,0 @@
-To use the pretty graphs you have to first build/run the ltmtest from the root directory of the package.  
-Todo this type 
-
-make timing ; ltmtest
-
-in the root.  It will run for a while [about ten minutes on most PCs] and produce a series of .log files in logs/.
-
-After doing that run "gnuplot graphs.dem" to make the PNGs.  If you managed todo that all so far just open index.html to view
-them all :-)
-
-Have fun
-
-Tom
\ No newline at end of file
diff --git a/logs/p4/add.log b/logs/p4/add.log
deleted file mode 100644
index 72b2506..0000000
--- a/logs/p4/add.log
+++ /dev/null
@@ -1,16 +0,0 @@
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diff --git a/logs/p4/addsub.png b/logs/p4/addsub.png
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diff --git a/logs/p4/expt_dr.log b/logs/p4/expt_dr.log
deleted file mode 100644
index 2f5f6a3..0000000
--- a/logs/p4/expt_dr.log
+++ /dev/null
@@ -1,7 +0,0 @@
-532       393
-784       158
-1036        79
-1540        27
-2072        12
-3080         4
-4116         1
diff --git a/logs/p4/graphs.dem b/logs/p4/graphs.dem
deleted file mode 100644
index c580495..0000000
--- a/logs/p4/graphs.dem
+++ /dev/null
@@ -1,17 +0,0 @@
-set terminal png color
-set size 1.75
-set ylabel "Operations per Second"
-set xlabel "Operand size (bits)"
-
-set output "addsub.png"
-plot 'add.log' smooth bezier title "Addition", 'sub.log' smooth bezier title "Subtraction"
-
-set output "mult.png"
-plot 'sqr.log' smooth bezier title "Squaring (without Karatsuba)", 'sqr_kara.log' smooth bezier title "Squaring (Karatsuba)", 'mult.log' smooth bezier title "Multiplication (without Karatsuba)", 'mult_kara.log' smooth bezier title "Multiplication (Karatsuba)"
-
-set output "expt.png"
-plot 'expt.log' smooth bezier title "Exptmod (Montgomery)", 'expt_dr.log' smooth bezier title "Exptmod (Dimminished Radix)"
-
-set output "invmod.png"
-plot 'invmod.log' smooth bezier title "Modular Inverse"
-
diff --git a/logs/p4/index.html b/logs/p4/index.html
deleted file mode 100644
index 19fe403..0000000
--- a/logs/p4/index.html
+++ /dev/null
@@ -1,24 +0,0 @@
-<html>
-<head>
-<title>LibTomMath Log Plots</title>
-</head>
-<body>
-
-<h1>Addition and Subtraction</h1>
-<center><img src=addsub.png></center>
-<hr>
-
-<h1>Multipliers</h1>
-<center><img src=mult.png></center>
-<hr>
-
-<h1>Exptmod</h1>
-<center><img src=expt.png></center>
-<hr>
-
-<h1>Modular Inverse</h1>
-<center><img src=invmod.png></center>
-<hr>
-
-</body>
-</html>
\ No newline at end of file
diff --git a/logs/p4/invmod.log b/logs/p4/invmod.log
deleted file mode 100644
index 096087b..0000000
--- a/logs/p4/invmod.log
+++ /dev/null
@@ -1,32 +0,0 @@
-112     13608
-224      6872
-336      4264
-448      2792
-560      2144
-672      1560
-784      1296
-896      1672
-1008       896
-1120       736
-1232      1024
-1344       888
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-1904       484
-2016       436
-2128       232
-2240       211
-2352       200
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-2576       293
-2688       262
-2800       251
-2912       137
-3024       216
-3136       117
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diff --git a/logs/p4/invmod.png b/logs/p4/invmod.png
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diff --git a/logs/p4/mult_kara.log b/logs/p4/mult_kara.log
deleted file mode 100644
index e1d50a6..0000000
--- a/logs/p4/mult_kara.log
+++ /dev/null
@@ -1,17 +0,0 @@
-896     77752
-1344     35832
-1792     19688
-2240     14704
-2688     10832
-3136      8336
-3584      6600
-4032      5424
-4480      4648
-4928      3976
-5376      3448
-5824      3016
-6272      2664
-6720      2384
-7168      2120
-7616      1912
-8064      1752
diff --git a/logs/p4/sqr.log b/logs/p4/sqr.log
deleted file mode 100644
index b133fb3..0000000
--- a/logs/p4/sqr.log
+++ /dev/null
@@ -1,17 +0,0 @@
-896    128088
-1344     63640
-1792     37968
-2240     25488
-2688     18176
-3136     13672
-3584      4920
-4032      3912
-4480      3160
-4928      2616
-5376      2216
-5824      1896
-6272      1624
-6720      1408
-7168      1240
-7616      1096
-8064       984
diff --git a/logs/p4/sqr_kara.log b/logs/p4/sqr_kara.log
deleted file mode 100644
index 13e4f3e..0000000
--- a/logs/p4/sqr_kara.log
+++ /dev/null
@@ -1,17 +0,0 @@
-896    127456
-1344     63752
-1792     37920
-2240     25440
-2688     18200
-3136     13728
-3584     10968
-4032      9072
-4480      7608
-4928      6440
-5376      5528
-5824      4768
-6272      4328
-6720      3888
-7168      3504
-7616      3176
-8064      2896
diff --git a/logs/p4/sub.log b/logs/p4/sub.log
deleted file mode 100644
index 424de32..0000000
--- a/logs/p4/sub.log
+++ /dev/null
@@ -1,16 +0,0 @@
-224   7355896
-448   6162880
-672   5218984
-896   4622776
-1120   3999320
-1344   3629480
-1568   3290384
-1792   2954752
-2016   2737056
-2240   2563320
-2464   2451928
-2688   2310920
-2912   2139048
-3136   2034080
-3360   1890800
-3584   1808624
diff --git a/logs/sqr.log b/logs/sqr.log
index 81fa612..ec142fe 100644
--- a/logs/sqr.log
+++ b/logs/sqr.log
@@ -1,33 +1,33 @@
-922    471095
-1147    337137
-1366    254327
-1596    199732
-1819    161225
-2044    132852
-2268    111493
-2490     94864
-2715     81745
-2940     71187
-3162     62575
-3387     55418
-3612     14540
-3836     12944
-4060     11627
-4281     10546
-4508      9502
-4730      8688
-4954      7937
-5180      7273
-5402      6701
-5627      6189
-5850      5733
-6076      5310
-6300      4933
-6522      4631
-6748      4313
-6971      4064
-7196      3801
-7420      3576
-7642      3388
-7868      3191
-8092      3020
+924     26026
+1146     37682
+1370     51714
+1595     68130
+1820     86850
+2043    107880
+2267    131236
+2490    156828
+2716    184704
+2940    214934
+3162    247424
+3388    282494
+3608    308390
+3834    345978
+4060    386156
+4282    427648
+4505    471556
+4731    517948
+4954    566396
+5180    618292
+5402    670130
+5628    725674
+5852    783310
+6076    843480
+6300    905136
+6524    969132
+6748   1033680
+6971   1100912
+7195   1170954
+7420   1252576
+7643   1325038
+7867   1413890
+8091   1493140
diff --git a/logs/sqr_kara.log b/logs/sqr_kara.log
index 3b547cf..f75256a 100644
--- a/logs/sqr_kara.log
+++ b/logs/sqr_kara.log
@@ -1,33 +1,33 @@
-922    470930
-1148    337217
-1372    254433
-1596    199827
-1820    161204
-2043    132871
-2267    111522
-2488     94932
-2714     81814
-2939     71231
-3164     62616
-3385     55467
-3611     44426
-3836     40695
-4060     37391
-4283     34371
-4508     31779
-4732     29499
-4956     27426
-5177     25598
-5403     23944
-5628     22416
-5851     21052
-6076     19781
-6299     18588
-6523     17539
-6746     16618
-6972     15705
-7196     13582
-7420     13004
-7643     12496
-7868     11963
-8092     11497
+923    165854
+1146    112539
+1372     80388
+1595     60051
+1820     47498
+2044     38017
+2268     31935
+2492     27373
+2714     23798
+2939     20630
+3164     18198
+3388     16191
+3612     14538
+3836     13038
+4058     11683
+4284     10915
+4508      9998
+4731      9271
+4954      8555
+5180      7910
+5404      7383
+5628      7012
+5852      6527
+6075      6175
+6299      5737
+6524      5398
+6744      5110
+6971      4864
+7196      4567
+7420      4371
+7644      4182
+7868      3981
+8092      3758
diff --git a/logs/sub.log b/logs/sub.log
index f1ade94..97ea200 100644
--- a/logs/sub.log
+++ b/logs/sub.log
@@ -1,16 +1,16 @@
-224  16370431
-448  13327848
-672  11009401
-896   9125342
-1120   7930419
-1344   7114040
-1568   6506998
-1792   5899346
-2016   5435327
-2240   5038931
-2464   4696364
-2688   4425678
-2912   4134476
-3136   3913280
-3360   3692536
-3584   3505219
+224      2012
+448      2208
+672      2366
+896      2532
+1120      2682
+1344      2838
+1568      3016
+1792      3146
+2016      3318
+2240      3538
+2464      3756
+2688      3914
+2912      4060
+3136      4216
+3360      4392
+3584      4550
diff --git a/makefile b/makefile
index 07b7842..95bd003 100644
--- a/makefile
+++ b/makefile
@@ -12,7 +12,10 @@ CFLAGS += -O3 -funroll-loops
 #x86 optimizations [should be valid for any GCC install though]
 CFLAGS  += -fomit-frame-pointer
 
-VERSION=0.30
+#debug
+#CFLAGS += -g3
+
+VERSION=0.31
 
 default: libtommath.a
 
@@ -20,7 +23,7 @@ default: libtommath.a
 LIBNAME=libtommath.a
 HEADERS=tommath.h
 
-#LIBPATH-The directory for libtomcrypt to be installed to.
+#LIBPATH-The directory for libtommath to be installed to.
 #INCPATH-The directory to install the header files for libtommath.
 #DATAPATH-The directory to install the pdf docs.
 DESTDIR=
@@ -58,6 +61,30 @@ libtommath.a:  $(OBJECTS)
 	$(AR) $(ARFLAGS) libtommath.a $(OBJECTS)
 	ranlib libtommath.a
 
+
+#make a profiled library (takes a while!!!)
+#
+# This will build the library with profile generation
+# then run the test demo and rebuild the library.
+# 
+# So far I've seen improvements in the MP math
+profiled:
+	make CFLAGS="$(CFLAGS) -fprofile-arcs -DTESTING" timing
+	./ltmtest
+	rm -f *.a *.o ltmtest
+	make CFLAGS="$(CFLAGS) -fbranch-probabilities"
+
+#make a single object profiled library 
+profiled_single:
+	perl gen.pl
+	$(CC) $(CFLAGS) -fprofile-arcs -DTESTING -c mpi.c -o mpi.o
+	$(CC) $(CFLAGS) -DTESTING -DTIMER demo/timing.c mpi.o -o ltmtest
+	./ltmtest
+	rm -f *.o ltmtest
+	$(CC) $(CFLAGS) -fbranch-probabilities -DTESTING -c mpi.c -o mpi.o
+	$(AR) $(ARFLAGS) libtommath.a mpi.o
+	ranlib libtommath.a	
+
 install: libtommath.a
 	install -d -g root -o root $(DESTDIR)$(LIBPATH)
 	install -d -g root -o root $(DESTDIR)$(INCPATH)
@@ -71,7 +98,7 @@ mtest: test
 	cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest -s
         
 timing: libtommath.a
-	$(CC) $(CFLAGS) -DTIMER demo/demo.c libtommath.a -o ltmtest -s
+	$(CC) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest -s
 
 # makes the LTM book DVI file, requires tetex, perl and makeindex [part of tetex I think]
 docdvi: tommath.src
@@ -106,10 +133,13 @@ mandvi: bn.tex
 manual:	mandvi
 	pdflatex bn >/dev/null
 	rm -f bn.aux bn.dvi bn.log bn.idx bn.lof bn.out bn.toc
-	
+
+pretty: 
+	perl pretty.build
+
 clean:
 	rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \
-        *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c 
+        *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.da *.dyn *.dpi tommath.tex *~ demo/*~ etc/*~
 	cd etc ; make clean
 	cd pics ; make clean
 
diff --git a/makefile.bcc b/makefile.bcc
index 6874d2f..b71f380 100644
--- a/makefile.bcc
+++ b/makefile.bcc
@@ -30,7 +30,8 @@ bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \
 bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \
 bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \
 bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_prime_sizes_tab.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \
-bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj
+bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \
+bn_mp_init_set.obj bn_mp_init_set_int.obj 
 
 TARGET = libtommath.lib
 
diff --git a/makefile.cygwin_dll b/makefile.cygwin_dll
index e5ab814..332a328 100644
--- a/makefile.cygwin_dll
+++ b/makefile.cygwin_dll
@@ -35,7 +35,8 @@ bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \
 bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \
 bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \
 bn_mp_init_multi.o bn_mp_clear_multi.o bn_prime_sizes_tab.o bn_mp_exteuclid.o bn_mp_toradix_n.o \
-bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o
+bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \
+bn_mp_init_set_int.o
 
 # make a Windows DLL via Cygwin
 windll:  $(OBJECTS)
diff --git a/makefile.icc b/makefile.icc
new file mode 100644
index 0000000..e4c0d19
--- /dev/null
+++ b/makefile.icc
@@ -0,0 +1,110 @@
+#Makefile for ICC
+#
+#Tom St Denis
+CC=icc
+
+CFLAGS  +=  -I./
+
+# optimize for SPEED
+#
+# -mcpu= can be pentium, pentiumpro (covers PII through PIII) or pentium4
+# -ax?   specifies make code specifically for ? but compatible with IA-32
+# -x?    specifies compile solely for ? [not specifically IA-32 compatible]
+#
+# where ? is 
+#   K - PIII
+#   W - first P4 [Williamette]
+#   N - P4 Northwood
+#   P - P4 Prescott
+#   B - Blend of P4 and PM [mobile]
+#
+# Default to just generic max opts
+CFLAGS += -O3 -xN
+
+default: libtommath.a
+
+#default files to install
+LIBNAME=libtommath.a
+HEADERS=tommath.h
+
+#LIBPATH-The directory for libtomcrypt to be installed to.
+#INCPATH-The directory to install the header files for libtommath.
+#DATAPATH-The directory to install the pdf docs.
+DESTDIR=
+LIBPATH=/usr/lib
+INCPATH=/usr/include
+DATAPATH=/usr/share/doc/libtommath/pdf
+
+OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \
+bn_mp_clamp.o bn_mp_zero.o  bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \
+bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \
+bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \
+bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \
+bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \
+bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \
+bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \
+bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \
+bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \
+bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \
+bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \
+bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \
+bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o  \
+bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \
+bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \
+bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \
+bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \
+bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \
+bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \
+bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \
+bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \
+bn_mp_init_multi.o bn_mp_clear_multi.o bn_prime_sizes_tab.o bn_mp_exteuclid.o bn_mp_toradix_n.o \
+bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \
+bn_mp_init_set_int.o
+
+libtommath.a:  $(OBJECTS)
+	$(AR) $(ARFLAGS) libtommath.a $(OBJECTS)
+	ranlib libtommath.a
+
+#make a profiled library (takes a while!!!)
+#
+# This will build the library with profile generation
+# then run the test demo and rebuild the library.
+# 
+# So far I've seen improvements in the MP math
+profiled:
+	make -f makefile.icc CFLAGS="$(CFLAGS) -prof_gen -DTESTING" timing
+	./ltmtest
+	rm -f *.a *.o ltmtest
+	make -f makefile.icc CFLAGS="$(CFLAGS) -prof_use"
+
+#make a single object profiled library 
+profiled_single:
+	perl gen.pl
+	$(CC) $(CFLAGS) -prof_gen -DTESTING -c mpi.c -o mpi.o
+	$(CC) $(CFLAGS) -DTESTING -DTIMER demo/demo.c mpi.o -o ltmtest
+	./ltmtest
+	rm -f *.o ltmtest
+	$(CC) $(CFLAGS) -prof_use -ip -DTESTING -c mpi.c -o mpi.o
+	$(AR) $(ARFLAGS) libtommath.a mpi.o
+	ranlib libtommath.a	
+
+install: libtommath.a
+	install -d -g root -o root $(DESTDIR)$(LIBPATH)
+	install -d -g root -o root $(DESTDIR)$(INCPATH)
+	install -g root -o root $(LIBNAME) $(DESTDIR)$(LIBPATH)
+	install -g root -o root $(HEADERS) $(DESTDIR)$(INCPATH)
+
+test: libtommath.a demo/demo.o
+	$(CC) demo/demo.o libtommath.a -o test
+	
+mtest: test	
+	cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest
+        
+timing: libtommath.a
+	$(CC) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest
+
+clean:
+	rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \
+        *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.il etc/*.il *.dyn
+	cd etc ; make clean
+	cd pics ; make clean
diff --git a/makefile.msvc b/makefile.msvc
index beeb77e..7d67442 100644
--- a/makefile.msvc
+++ b/makefile.msvc
@@ -29,7 +29,8 @@ bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \
 bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \
 bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \
 bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_prime_sizes_tab.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \
-bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj
+bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \
+bn_mp_init_set.obj bn_mp_init_set_int.obj
 
 library: $(OBJECTS)
 	lib /out:tommath.lib $(OBJECTS)
diff --git a/poster.pdf b/poster.pdf
index 3731bd2bdaa53bb2052ef8258e5d97368b1b413c..6689f2eb463696b7bcf3c0db068b76926e39a166 100644
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diff --git a/pre_gen/mpi.c b/pre_gen/mpi.c
index 1f9997f..370b34d 100644
--- a/pre_gen/mpi.c
+++ b/pre_gen/mpi.c
@@ -452,7 +452,7 @@ fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
   }
 
   /* setup dest */
-  olduse = c->used;
+  olduse  = c->used;
   c->used = digs;
 
   {
@@ -779,7 +779,7 @@ mp_2expt (mp_int * a, int b)
   a->used = b / DIGIT_BIT + 1;
 
   /* put the single bit in its place */
-  a->dp[b / DIGIT_BIT] = 1 << (b % DIGIT_BIT);
+  a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);
 
   return MP_OKAY;
 }
@@ -1142,10 +1142,14 @@ mp_clamp (mp_int * a)
 void
 mp_clear (mp_int * a)
 {
+  int i;
+
   /* only do anything if a hasn't been freed previously */
   if (a->dp != NULL) {
     /* first zero the digits */
-    memset (a->dp, 0, sizeof (mp_digit) * a->used);
+    for (i = 0; i < a->used; i++) {
+        a->dp[i] = 0;
+    }
 
     /* free ram */
     XFREE(a->dp);
@@ -1677,7 +1681,7 @@ int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
    */
   
   /* get sign before writing to c */
-  x.sign = a->sign;
+  x.sign = x.used == 0 ? MP_ZPOS : a->sign;
 
   if (c != NULL) {
     mp_clamp (&q);
@@ -3083,15 +3087,22 @@ int mp_grow (mp_int * a, int size)
  */
 #include <tommath.h>
 
-/* init a new bigint */
+/* init a new mp_int */
 int mp_init (mp_int * a)
 {
+  int i;
+
   /* allocate memory required and clear it */
-  a->dp = OPT_CAST(mp_digit) XCALLOC (sizeof (mp_digit), MP_PREC);
+  a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC);
   if (a->dp == NULL) {
     return MP_MEM;
   }
 
+  /* set the digits to zero */
+  for (i = 0; i < MP_PREC; i++) {
+      a->dp[i] = 0;
+  }
+
   /* set the used to zero, allocated digits to the default precision
    * and sign to positive */
   a->used  = 0;
@@ -3753,9 +3764,6 @@ int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
     goto X0Y0;
 
   /* now shift the digits */
-  x0.sign = x1.sign = a->sign;
-  y0.sign = y1.sign = b->sign;
-
   x0.used = y0.used = B;
   x1.used = a->used - B;
   y1.used = b->used - B;
@@ -4484,7 +4492,7 @@ int mp_mul (mp_int * a, mp_int * b, mp_int * c)
       res = s_mp_mul (a, b, c);
     }
   }
-  c->sign = neg;
+  c->sign = (c->used > 0) ? neg : MP_ZPOS;
   return res;
 }
 
@@ -6090,7 +6098,8 @@ mp_reduce_2k_setup(mp_int *a, mp_digit *d)
 /* determines if mp_reduce_2k can be used */
 int mp_reduce_is_2k(mp_int *a)
 {
-   int ix, iy, iz, iw;
+   int ix, iy, iw;
+   mp_digit iz;
    
    if (a->used == 0) {
       return 0;
@@ -6107,7 +6116,7 @@ int mp_reduce_is_2k(mp_int *a)
              return 0;
           }
           iz <<= 1;
-          if (iz > (int)MP_MASK) {
+          if (iz > (mp_digit)MP_MASK) {
              ++iw;
              iz = 1;
           }
@@ -8396,14 +8405,16 @@ s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
 
  CPU                    /Compiler     /MUL CUTOFF/SQR CUTOFF
 -------------------------------------------------------------
- Intel P4               /GCC v3.2     /        70/       108
- AMD Athlon XP          /GCC v3.2     /       109/       127
-
+ Intel P4 Northwood     /GCC v3.3.3   /        59/        81/profiled build
+ Intel P4 Northwood     /GCC v3.3.3   /        59/        80/profiled_single build
+ Intel P4 Northwood     /ICC v8.0     /        57/        70/profiled build
+ Intel P4 Northwood     /ICC v8.0     /        54/        76/profiled_single build
+ AMD Athlon XP          /GCC v3.2     /       109/       127/
+ 
 */
 
-/* configured for a AMD XP Thoroughbred core with etc/tune.c */
-int     KARATSUBA_MUL_CUTOFF = 109,      /* Min. number of digits before Karatsuba multiplication is used. */
-        KARATSUBA_SQR_CUTOFF = 127,      /* Min. number of digits before Karatsuba squaring is used. */
+int     KARATSUBA_MUL_CUTOFF = 57,      /* Min. number of digits before Karatsuba multiplication is used. */
+        KARATSUBA_SQR_CUTOFF = 70,      /* Min. number of digits before Karatsuba squaring is used. */
         
         TOOM_MUL_CUTOFF      = 350,      /* no optimal values of these are known yet so set em high */
         TOOM_SQR_CUTOFF      = 400; 
diff --git a/pretty.build b/pretty.build
new file mode 100644
index 0000000..a708b8a
--- /dev/null
+++ b/pretty.build
@@ -0,0 +1,66 @@
+#!/bin/perl -w
+#
+# Cute little builder for perl 
+# Total waste of development time...
+#
+# This will build all the object files and then the archive .a file
+# requires GCC, GNU make and a sense of humour.
+#
+# Tom St Denis
+use strict;
+
+my $count = 0;
+my $starttime = time;
+my $rate  = 0;
+print "Scanning for source files...\n";
+foreach my $filename (glob "*.c") {
+       ++$count;
+}
+print "Source files to build: $count\nBuilding...\n";
+my $i = 0;
+my $lines = 0;
+my $filesbuilt = 0;
+foreach my $filename (glob "*.c") {
+       printf("Building %3.2f%%, ", (++$i/$count)*100.0);
+       if ($i % 4 == 0) { print "/, "; }
+       if ($i % 4 == 1) { print "-, "; }
+       if ($i % 4 == 2) { print "\\, "; }
+       if ($i % 4 == 3) { print "|, "; }
+       if ($rate > 0) {
+           my $tleft = ($count - $i) / $rate;
+           my $tsec  = $tleft%60;
+           my $tmin  = ($tleft/60)%60;
+           my $thour = ($tleft/3600)%60;
+           printf("%2d:%02d:%02d left, ", $thour, $tmin, $tsec);
+       }
+       my $cnt = ($i/$count)*30.0;
+       my $x   = 0;
+       print "[";
+       for (; $x < $cnt; $x++) { print "#"; }
+       for (; $x < 30; $x++)   { print " "; }
+       print "]\r";
+       my $tmp = $filename;
+       $tmp =~ s/\.c/".o"/ge;
+       if (open(SRC, "<$tmp")) {
+          close SRC;
+       } else {
+          !system("make $tmp > /dev/null 2>/dev/null") or die "\nERROR: Failed to make $tmp!!!\n";
+          open( SRC, "<$filename" ) or die "Couldn't open $filename for reading: $!";
+          ++$lines while (<SRC>);
+          close SRC or die "Error closing $filename after reading: $!";
+          ++$filesbuilt;
+       }      
+
+       # update timer 
+       if (time != $starttime) {
+          my $delay = time - $starttime;
+          $rate = $i/$delay;
+       }
+}
+
+# finish building the library 
+printf("\nFinished building source (%d seconds, %3.2f files per second).\n", time - $starttime, $rate);
+print "Compiled approximately $filesbuilt files and $lines lines of code.\n";
+print "Doing final make (building archive...)\n";
+!system("make > /dev/null 2>/dev/null") or die "\nERROR: Failed to perform last make command!!!\n";
+print "done.\n";
\ No newline at end of file
diff --git a/tommath.pdf b/tommath.pdf
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GIT binary patch
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diff --git a/tommath.src b/tommath.src
index 0389831..6ee842d 100644
--- a/tommath.src
+++ b/tommath.src
@@ -258,7 +258,7 @@ floating point is meant to be implemented in hardware the precision of the manti
 a mantissa of much larger precision than hardware alone can efficiently support.  This approach could be useful where 
 scientific applications must minimize the total output error over long calculations.
 
-Another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
+Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
 In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.
 
 \subsection{Benefits of Multiple Precision Arithmetic}
@@ -316,7 +316,7 @@ the reader how the algorithms fit together as well as where to start on various
 
 \section{Discussion and Notation}
 \subsection{Notation}
-A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1} ... x_1 x_0)_{ \beta }$ and represent
+A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
 the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$.  The elements of the array $x$ are said to be the radix $\beta$ digits 
 of the integer.  For example, $x = (1,2,3)_{10}$ would represent the integer 
 $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.  
@@ -339,12 +339,11 @@ algorithms will be used to establish the relevant theory which will subsequently
 precision algorithm to solve the same problem.  
 
 \subsection{Precision Notation}
-For the purposes of this text a single precision variable must be able to represent integers in the range 
-$0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range 
-$0 \le x < q \beta^2$.  The variable $\beta$ represents the radix of a single digit of a multiple precision integer and 
-must be of the form $q^p$ for $q, p \in \Z^+$.  The extra radix-$q$ factor allows additions and subtractions to proceed 
-without truncation of the carry.  Since all modern computers are binary, it is assumed that $q$ is two, for all intents 
-and purposes.
+The variable $\beta$ represents the radix of a single digit of a multiple precision integer and 
+must be of the form $q^p$ for $q, p \in \Z^+$.  A single precision variable must be able to represent integers in 
+the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range 
+$0 \le x < q \beta^2$.  The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the 
+carry.  Since all modern computers are binary, it is assumed that $q$ is two.
 
 \index{mp\_digit} \index{mp\_word}
 Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent 
@@ -376,7 +375,7 @@ the $/$ division symbol is used the intention is to perform an integer division
 $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity.  When an expression is written as a 
 fraction a real value division is implied, for example ${5 \over 2} = 2.5$.  
 
-The norm of a multiple precision integer, for example, $\vert \vert x \vert \vert$ will be used to represent the number of digits in the representation
+The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
 of the integer.  For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.  
 
 \subsection{Work Effort}
@@ -569,7 +568,7 @@ By building outwards from a base foundation instead of using a parallel design m
 highly modular.  Being highly modular is a desirable property of any project as it often means the resulting product
 has a small footprint and updates are easy to perform.  
 
-Usually when I start a project I will begin with the header file.  I define the data types I think I will need and 
+Usually when I start a project I will begin with the header files.  I define the data types I think I will need and 
 prototype the initial functions that are not dependent on other functions (within the library).  After I 
 implement these base functions I prototype more dependent functions and implement them.   The process repeats until
 I implement all of the functions I require.  For example, in the case of LibTomMath I implemented functions such as 
@@ -619,14 +618,26 @@ any such data type but it does provide for making composite data types known as
 used within LibTomMath.
 
 \index{mp\_int}
-\begin{verbatim}
-typedef struct  {
-    int used, alloc, sign;
-    mp_digit *dp;
-} mp_int;
-\end{verbatim}
+\begin{figure}[here]
+\begin{center}
+\begin{small}
+%\begin{verbatim}
+\begin{tabular}{|l|}
+\hline
+typedef struct \{ \\
+\hspace{3mm}int used, alloc, sign;\\
+\hspace{3mm}mp\_digit *dp;\\
+\} \textbf{mp\_int}; \\
+\hline
+\end{tabular}
+%\end{verbatim}
+\end{small}
+\caption{The mp\_int Structure}
+\label{fig:mpint}
+\end{center}
+\end{figure}
 
-The mp\_int structure can be broken down as follows.
+The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.
 
 \begin{enumerate}
 \item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
@@ -701,9 +712,10 @@ fault by dereferencing memory not owned by the application.
 In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for 
 instance) and memory allocation errors.  It will not check that the mp\_int passed to any function is valid nor 
 will it check pointers for validity.  Any function that can cause a runtime error will return an error code as an 
-\textbf{int} data type with one of the following values.
+\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).
 
 \index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
+\begin{figure}[here]
 \begin{center}
 \begin{tabular}{|l|l|}
 \hline \textbf{Value} & \textbf{Meaning} \\
@@ -713,6 +725,9 @@ will it check pointers for validity.  Any function that can cause a runtime erro
 \hline
 \end{tabular}
 \end{center}
+\caption{LibTomMath Error Codes}
+\label{fig:errcodes}
+\end{figure}
 
 When an error is detected within a function it should free any memory it allocated, often during the initialization of
 temporary mp\_ints, and return as soon as possible.  The goal is to leave the system in the same state it was when the 
@@ -748,6 +763,7 @@ to zero.  The \textbf{used} count set to zero and \textbf{sign} set to \textbf{M
 An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the
 structure are set to valid values.  The mp\_init algorithm will perform such an action.
 
+\index{mp\_init}
 \begin{figure}[here]
 \begin{center}
 \begin{tabular}{l}
@@ -770,17 +786,23 @@ structure are set to valid values.  The mp\_init algorithm will perform such an
 \end{figure}
 
 \textbf{Algorithm mp\_init.}
-The \textbf{MP\_PREC} name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} 
-used to dictate the minimum precision of allocated mp\_int integers.  Ideally, it is at least equal to $32$ since for most
-purposes that will be more than enough.
+The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
+manipulte it.  It is assumed that the input may not have had any of its members previously initialized which is certainly
+a valid assumption if the input resides on the stack.  
 
-Memory for the default number of digits is allocated first.  If the allocation fails the algorithm returns immediately
-with the \textbf{MP\_MEM} error code.  If the allocation succeeds the remaining members of the mp\_int structure
-must be initialized to reflect the default initial state.
+Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
+the digits is allocated.  If this fails the function returns before setting any of the other members.  The \textbf{MP\_PREC} 
+name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} 
+used to dictate the minimum precision of newly initialized mp\_int integers.  Ideally, it is at least equal to the smallest
+precision number you'll be working with.
 
-The allocated digits are all set to zero (step three) to ensure they are in a known state.  The \textbf{sign}, \textbf{used}
-and \textbf{alloc} are subsequently initialized to represent the zero integer.  By step seven the algorithm returns a success 
-code and the mp\_int $a$ has been successfully initialized to a valid state representing the integer zero.  
+Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
+heap operations later functions will have to perform in the future.  If \textbf{MP\_PREC} is set correctly the slack 
+memory and the number of heap operations will be trivial.
+
+Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
+\textbf{alloc} members initialized.  This ensures that the mp\_int will always represent the default state of zero regardless
+of the original condition of the input.
 
 \textbf{Remark.}
 This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
@@ -796,19 +818,21 @@ One immediate observation of this initializtion function is that it does not ret
 is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack.  The 
 call to mp\_init() is used only to initialize the members of the structure to a known default state.  
 
-Before any of the other members of the structure are initialized memory from the application heap is allocated with
-the calloc() function (line @22,calloc@).  The size of the allocated memory is large enough to hold \textbf{MP\_PREC} 
-mp\_digit variables.  The calloc() function is used instead\footnote{calloc() will allocate memory in the same
-manner as malloc() except that it also sets the contents to zero upon successfully allocating the memory.} of malloc() 
-since digits have to be set to zero for the function to finish correctly.  The \textbf{OPT\_CAST} token is a macro 
-definition which will turn into a cast from void * to mp\_digit * for C++ compilers.  It is not required for C compilers.
+Here we see (line @23,XMALLOC@) the memory allocation is performed first.  This allows us to exit cleanly and quickly
+if there is an error.  If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
+was a memory error.  The function XMALLOC is what actually allocates the memory.  Technically XMALLOC is not a function
+but a macro defined in ``tommath.h``.  By default, XMALLOC will evaluate to malloc() which is the C library's built--in
+memory allocation routine.
 
-After the memory has been successfully allocated the remainder of the members are initialized 
+In order to assure the mp\_int is in a known state the digits must be set to zero.  On most platforms this could have been
+accomplished by using calloc() instead of malloc().  However,  to correctly initialize a integer type to a given value in a 
+portable fashion you have to actually assign the value.  The for loop (line @28,for@) performs this required
+operation.
+
+After the memory has been successfully initialized the remainder of the members are initialized 
 (lines @29,used@ through @31,sign@) to their respective default states.  At this point the algorithm has succeeded and
-a success code is returned to the calling function.
-
-If this function returns \textbf{MP\_OKAY} it is safe to assume the mp\_int structure has been properly initialized and
-is safe to use with other functions within the library.  
+a success code is returned to the calling function.  If this function returns \textbf{MP\_OKAY} it is safe to assume the 
+mp\_int structure has been properly initialized and is safe to use with other functions within the library.  
 
 \subsection{Clearing an mp\_int}
 When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be 
@@ -819,7 +843,7 @@ returned to the application's memory pool with the mp\_clear algorithm.
 \begin{tabular}{l}
 \hline Algorithm \textbf{mp\_clear}. \\
 \textbf{Input}.   An mp\_int $a$ \\
-\textbf{Output}.  The memory for $a$ is freed for reuse.  \\
+\textbf{Output}.  The memory for $a$ shall be deallocated.  \\
 \hline \\
 1.  If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
 2.  for $n$ from 0 to $a.used - 1$ do \\
@@ -836,32 +860,31 @@ returned to the application's memory pool with the mp\_clear algorithm.
 \end{figure}
 
 \textbf{Algorithm mp\_clear.}
-This algorithm releases the memory allocated for an mp\_int back into the memory pool for reuse.  It is designed
-such that a given mp\_int structure can be cleared multiple times between initializations without attempting to 
-free the memory twice\footnote{In ISO C for example, calling free() twice on the same memory block causes undefinied
-behaviour.}.  
+This algorithm accomplishes two goals.  First, it clears the digits and the other mp\_int members.  This ensures that 
+if a developer accidentally re-uses a cleared structure it is less likely to cause problems.  The second goal
+is to free the allocated memory.
 
-The first step determines if the mp\_int structure has been marked as free already.  If it has, the algorithm returns
-success immediately as no further actions are required.  Otherwise, the algorithm will proceed to put the structure 
-in a known empty and otherwise invalid state.  First the digits of the mp\_int are set to zero.  The memory that has been allocated for the 
-digits is then freed.  The \textbf{used} and \textbf{alloc} counts are both set to zero and the \textbf{sign} set to 
-\textbf{MP\_ZPOS}.  This known fixed state for cleared mp\_int structures will make debuging easier for the end 
-developer.  That is, if they spot (via their debugger) an mp\_int they are using that is in this state it will be 
-obvious that they erroneously and prematurely cleared the mp\_int structure.
+The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
+algorithm will not try to free the memory multiple times.  Cleared mp\_ints are detectable by having a pre-defined invalid 
+digit pointer \textbf{dp} setting.
 
-Note that once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
+Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
 with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.
 
 EXAM,bn_mp_clear.c
 
-The ``if'' statement (line @21,a->dp != NULL@) prevents the heap from being corrupted if a user double-frees an 
-mp\_int.  This is because once the memory is freed the pointer is set to \textbf{NULL} (line @30,NULL@).  
+The algorithm only operates on the mp\_int if it hasn't been previously cleared.  The if statement (line @23,a->dp != NULL@)
+checks to see if the \textbf{dp} member is not \textbf{NULL}.  If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
+\textbf{NULL} in which case the if statement will evaluate to true.
 
-Without the check, code that accidentally calls mp\_clear twice for a given mp\_int structure would try to free the memory 
-allocated for the digits twice.  This may cause some C libraries to signal a fault.  By setting the pointer to 
-\textbf{NULL} it helps debug code that may inadvertently free the mp\_int before it is truly not needed, because attempts 
-to reference digits should fail immediately. The allocated digits are set to zero before being freed (line @24,memset@).  
-This is ideal for cryptographic situations where the integer that the mp\_int represents might need to be kept a secret.
+The digits of the mp\_int are cleared by the for loop (line @25,for@) which assigns a zero to every digit.  Similar to mp\_init()
+the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.  
+
+The digits are deallocated off the heap via the XFREE macro.  Similar to XMALLOC the XFREE macro actually evaluates to
+a standard C library function.  In this case the free() function.  Since free() only deallocates the memory the pointer
+still has to be reset to \textbf{NULL} manually (line @33,NULL@).  
+
+Now that the digits have been cleared and deallocated the other members are set to their final values (lines @34,= 0@ and @35,ZPOS@).
 
 \section{Maintenance Algorithms}
 
@@ -889,7 +912,7 @@ must be re-sized appropriately to accomodate the result.  The mp\_grow algorithm
 1.  if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
 2.  $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
 3.  $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
-4.  Re-Allocate the array of digits $a$ to size $v$ \\
+4.  Re-allocate the array of digits $a$ to size $v$ \\
 5.  If the allocation failed then return(\textit{MP\_MEM}). \\
 6.  for n from a.alloc to $v - 1$ do  \\
 \hspace{+3mm}6.1  $a_n \leftarrow 0$ \\
@@ -914,15 +937,19 @@ assumed to contain undefined values they are initially set to zero.
 
 EXAM,bn_mp_grow.c
 
-The first step is to see if we actually need to perform a re-allocation at all (line @24,a->alloc < size@).  If a reallocation
-must occur the digit count is padded upwards to help prevent many trivial reallocations (line @28,size@).  Next the reallocation is performed
-and the return of realloc() is stored in a temporary pointer named $tmp$ (line @36,realloc@).  The return is stored in a temporary
-instead of $a.dp$ to prevent the code from losing the original pointer in case the reallocation fails.  Had the return been stored 
-in $a.dp$ instead there would be no way to reclaim the heap originally used.
+A quick optimization is to first determine if a memory re-allocation is required at all.  The if statement (line @23,if@) checks
+if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count.  If the count is not larger than \textbf{alloc}
+the function skips the re-allocation part thus saving time.
 
-If the reallocation fails the function will return \textbf{MP\_MEM} (line @39,return@), otherwise, the value of $tmp$ is assigned
-to the pointer $a.dp$ and the function continues.  A simple for loop from line @48,a->alloc@ to line @50,}@ will zero all digits 
-that were above the old \textbf{alloc} limit to make sure the integer is in a known state.
+When a re-allocation is performed it is turned into an optimal request to save time in the future.  The requested digit count is
+padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line @25, size@).  The XREALLOC function is used
+to re-allocate the memory.  As per the other functions XREALLOC is actually a macro which evaluates to realloc by default.  The realloc
+function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
+the re-allocation.  All	that is left is to clear the newly allocated digits and return.
+
+Note that the re-allocation result is actually stored in a temporary pointer $tmp$.  This is to allow this function to return
+an error with a valid pointer.  Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$.  That would
+result in a memory leak if XREALLOC ever failed.  
 
 \subsection{Initializing Variable Precision mp\_ints}
 Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size 
@@ -970,7 +997,7 @@ The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmen
 mp\_int is placed in a default state representing the integer zero.  Otherwise, the error code \textbf{MP\_MEM} will be 
 returned (line @27,return@).  
 
-The digits are allocated and set to zero at the same time with the calloc() function (line @25,calloc@).  The 
+The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@).  The 
 \textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set 
 to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@).  If the function 
 returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the 
diff --git a/tommath.tex b/tommath.tex
index 629edba..3fbe907 100644
--- a/tommath.tex
+++ b/tommath.tex
@@ -258,7 +258,7 @@ floating point is meant to be implemented in hardware the precision of the manti
 a mantissa of much larger precision than hardware alone can efficiently support.  This approach could be useful where 
 scientific applications must minimize the total output error over long calculations.
 
-Another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
+Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
 In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.
 
 \subsection{Benefits of Multiple Precision Arithmetic}
@@ -316,7 +316,7 @@ the reader how the algorithms fit together as well as where to start on various
 
 \section{Discussion and Notation}
 \subsection{Notation}
-A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1} ... x_1 x_0)_{ \beta }$ and represent
+A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
 the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$.  The elements of the array $x$ are said to be the radix $\beta$ digits 
 of the integer.  For example, $x = (1,2,3)_{10}$ would represent the integer 
 $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.  
@@ -339,12 +339,11 @@ algorithms will be used to establish the relevant theory which will subsequently
 precision algorithm to solve the same problem.  
 
 \subsection{Precision Notation}
-For the purposes of this text a single precision variable must be able to represent integers in the range 
-$0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range 
-$0 \le x < q \beta^2$.  The variable $\beta$ represents the radix of a single digit of a multiple precision integer and 
-must be of the form $q^p$ for $q, p \in \Z^+$.  The extra radix-$q$ factor allows additions and subtractions to proceed 
-without truncation of the carry.  Since all modern computers are binary, it is assumed that $q$ is two, for all intents 
-and purposes.
+The variable $\beta$ represents the radix of a single digit of a multiple precision integer and 
+must be of the form $q^p$ for $q, p \in \Z^+$.  A single precision variable must be able to represent integers in 
+the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range 
+$0 \le x < q \beta^2$.  The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the 
+carry.  Since all modern computers are binary, it is assumed that $q$ is two.
 
 \index{mp\_digit} \index{mp\_word}
 Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent 
@@ -376,7 +375,7 @@ the $/$ division symbol is used the intention is to perform an integer division
 $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity.  When an expression is written as a 
 fraction a real value division is implied, for example ${5 \over 2} = 2.5$.  
 
-The norm of a multiple precision integer, for example, $\vert \vert x \vert \vert$ will be used to represent the number of digits in the representation
+The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
 of the integer.  For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.  
 
 \subsection{Work Effort}
@@ -569,7 +568,7 @@ By building outwards from a base foundation instead of using a parallel design m
 highly modular.  Being highly modular is a desirable property of any project as it often means the resulting product
 has a small footprint and updates are easy to perform.  
 
-Usually when I start a project I will begin with the header file.  I define the data types I think I will need and 
+Usually when I start a project I will begin with the header files.  I define the data types I think I will need and 
 prototype the initial functions that are not dependent on other functions (within the library).  After I 
 implement these base functions I prototype more dependent functions and implement them.   The process repeats until
 I implement all of the functions I require.  For example, in the case of LibTomMath I implemented functions such as 
@@ -625,14 +624,26 @@ any such data type but it does provide for making composite data types known as
 used within LibTomMath.
 
 \index{mp\_int}
-\begin{verbatim}
-typedef struct  {
-    int used, alloc, sign;
-    mp_digit *dp;
-} mp_int;
-\end{verbatim}
+\begin{figure}[here]
+\begin{center}
+\begin{small}
+%\begin{verbatim}
+\begin{tabular}{|l|}
+\hline
+typedef struct \{ \\
+\hspace{3mm}int used, alloc, sign;\\
+\hspace{3mm}mp\_digit *dp;\\
+\} \textbf{mp\_int}; \\
+\hline
+\end{tabular}
+%\end{verbatim}
+\end{small}
+\caption{The mp\_int Structure}
+\label{fig:mpint}
+\end{center}
+\end{figure}
 
-The mp\_int structure can be broken down as follows.
+The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.
 
 \begin{enumerate}
 \item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
@@ -707,9 +718,10 @@ fault by dereferencing memory not owned by the application.
 In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for 
 instance) and memory allocation errors.  It will not check that the mp\_int passed to any function is valid nor 
 will it check pointers for validity.  Any function that can cause a runtime error will return an error code as an 
-\textbf{int} data type with one of the following values.
+\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).
 
 \index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
+\begin{figure}[here]
 \begin{center}
 \begin{tabular}{|l|l|}
 \hline \textbf{Value} & \textbf{Meaning} \\
@@ -719,6 +731,9 @@ will it check pointers for validity.  Any function that can cause a runtime erro
 \hline
 \end{tabular}
 \end{center}
+\caption{LibTomMath Error Codes}
+\label{fig:errcodes}
+\end{figure}
 
 When an error is detected within a function it should free any memory it allocated, often during the initialization of
 temporary mp\_ints, and return as soon as possible.  The goal is to leave the system in the same state it was when the 
@@ -754,6 +769,7 @@ to zero.  The \textbf{used} count set to zero and \textbf{sign} set to \textbf{M
 An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the
 structure are set to valid values.  The mp\_init algorithm will perform such an action.
 
+\index{mp\_init}
 \begin{figure}[here]
 \begin{center}
 \begin{tabular}{l}
@@ -776,17 +792,23 @@ structure are set to valid values.  The mp\_init algorithm will perform such an
 \end{figure}
 
 \textbf{Algorithm mp\_init.}
-The \textbf{MP\_PREC} name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} 
-used to dictate the minimum precision of allocated mp\_int integers.  Ideally, it is at least equal to $32$ since for most
-purposes that will be more than enough.
+The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
+manipulte it.  It is assumed that the input may not have had any of its members previously initialized which is certainly
+a valid assumption if the input resides on the stack.  
 
-Memory for the default number of digits is allocated first.  If the allocation fails the algorithm returns immediately
-with the \textbf{MP\_MEM} error code.  If the allocation succeeds the remaining members of the mp\_int structure
-must be initialized to reflect the default initial state.
+Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
+the digits is allocated.  If this fails the function returns before setting any of the other members.  The \textbf{MP\_PREC} 
+name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} 
+used to dictate the minimum precision of newly initialized mp\_int integers.  Ideally, it is at least equal to the smallest
+precision number you'll be working with.
 
-The allocated digits are all set to zero (step three) to ensure they are in a known state.  The \textbf{sign}, \textbf{used}
-and \textbf{alloc} are subsequently initialized to represent the zero integer.  By step seven the algorithm returns a success 
-code and the mp\_int $a$ has been successfully initialized to a valid state representing the integer zero.  
+Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
+heap operations later functions will have to perform in the future.  If \textbf{MP\_PREC} is set correctly the slack 
+memory and the number of heap operations will be trivial.
+
+Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
+\textbf{alloc} members initialized.  This ensures that the mp\_int will always represent the default state of zero regardless
+of the original condition of the input.
 
 \textbf{Remark.}
 This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
@@ -800,24 +822,30 @@ decrementally.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_init.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* init a new bigint */
-018   int mp_init (mp_int * a)
-019   \{
-020     /* allocate memory required and clear it */
-021     a->dp = OPT_CAST(mp_digit) XCALLOC (sizeof (mp_digit), MP_PREC);
-022     if (a->dp == NULL) \{
-023       return MP_MEM;
-024     \}
-025   
-026     /* set the used to zero, allocated digits to the default precision
-027      * and sign to positive */
-028     a->used  = 0;
-029     a->alloc = MP_PREC;
-030     a->sign  = MP_ZPOS;
+016   /* init a new mp_int */
+017   int mp_init (mp_int * a)
+018   \{
+019     int i;
+020   
+021     /* allocate memory required and clear it */
+022     a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC);
+023     if (a->dp == NULL) \{
+024       return MP_MEM;
+025     \}
+026   
+027     /* set the digits to zero */
+028     for (i = 0; i < MP_PREC; i++) \{
+029         a->dp[i] = 0;
+030     \}
 031   
-032     return MP_OKAY;
-033   \}
+032     /* set the used to zero, allocated digits to the default precision
+033      * and sign to positive */
+034     a->used  = 0;
+035     a->alloc = MP_PREC;
+036     a->sign  = MP_ZPOS;
+037   
+038     return MP_OKAY;
+039   \}
 \end{alltt}
 \end{small}
 
@@ -825,19 +853,21 @@ One immediate observation of this initializtion function is that it does not ret
 is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack.  The 
 call to mp\_init() is used only to initialize the members of the structure to a known default state.  
 
-Before any of the other members of the structure are initialized memory from the application heap is allocated with
-the calloc() function (line @22,calloc@).  The size of the allocated memory is large enough to hold \textbf{MP\_PREC} 
-mp\_digit variables.  The calloc() function is used instead\footnote{calloc() will allocate memory in the same
-manner as malloc() except that it also sets the contents to zero upon successfully allocating the memory.} of malloc() 
-since digits have to be set to zero for the function to finish correctly.  The \textbf{OPT\_CAST} token is a macro 
-definition which will turn into a cast from void * to mp\_digit * for C++ compilers.  It is not required for C compilers.
+Here we see (line 22) the memory allocation is performed first.  This allows us to exit cleanly and quickly
+if there is an error.  If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
+was a memory error.  The function XMALLOC is what actually allocates the memory.  Technically XMALLOC is not a function
+but a macro defined in ``tommath.h``.  By default, XMALLOC will evaluate to malloc() which is the C library's built--in
+memory allocation routine.
 
-After the memory has been successfully allocated the remainder of the members are initialized 
-(lines 28 through 30) to their respective default states.  At this point the algorithm has succeeded and
-a success code is returned to the calling function.
+In order to assure the mp\_int is in a known state the digits must be set to zero.  On most platforms this could have been
+accomplished by using calloc() instead of malloc().  However,  to correctly initialize a integer type to a given value in a 
+portable fashion you have to actually assign the value.  The for loop (line 28) performs this required
+operation.
 
-If this function returns \textbf{MP\_OKAY} it is safe to assume the mp\_int structure has been properly initialized and
-is safe to use with other functions within the library.  
+After the memory has been successfully initialized the remainder of the members are initialized 
+(lines 32 through 33) to their respective default states.  At this point the algorithm has succeeded and
+a success code is returned to the calling function.  If this function returns \textbf{MP\_OKAY} it is safe to assume the 
+mp\_int structure has been properly initialized and is safe to use with other functions within the library.  
 
 \subsection{Clearing an mp\_int}
 When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be 
@@ -848,7 +878,7 @@ returned to the application's memory pool with the mp\_clear algorithm.
 \begin{tabular}{l}
 \hline Algorithm \textbf{mp\_clear}. \\
 \textbf{Input}.   An mp\_int $a$ \\
-\textbf{Output}.  The memory for $a$ is freed for reuse.  \\
+\textbf{Output}.  The memory for $a$ shall be deallocated.  \\
 \hline \\
 1.  If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
 2.  for $n$ from 0 to $a.used - 1$ do \\
@@ -865,56 +895,58 @@ returned to the application's memory pool with the mp\_clear algorithm.
 \end{figure}
 
 \textbf{Algorithm mp\_clear.}
-This algorithm releases the memory allocated for an mp\_int back into the memory pool for reuse.  It is designed
-such that a given mp\_int structure can be cleared multiple times between initializations without attempting to 
-free the memory twice\footnote{In ISO C for example, calling free() twice on the same memory block causes undefinied
-behaviour.}.  
+This algorithm accomplishes two goals.  First, it clears the digits and the other mp\_int members.  This ensures that 
+if a developer accidentally re-uses a cleared structure it is less likely to cause problems.  The second goal
+is to free the allocated memory.
 
-The first step determines if the mp\_int structure has been marked as free already.  If it has, the algorithm returns
-success immediately as no further actions are required.  Otherwise, the algorithm will proceed to put the structure 
-in a known empty and otherwise invalid state.  First the digits of the mp\_int are set to zero.  The memory that has been allocated for the 
-digits is then freed.  The \textbf{used} and \textbf{alloc} counts are both set to zero and the \textbf{sign} set to 
-\textbf{MP\_ZPOS}.  This known fixed state for cleared mp\_int structures will make debuging easier for the end 
-developer.  That is, if they spot (via their debugger) an mp\_int they are using that is in this state it will be 
-obvious that they erroneously and prematurely cleared the mp\_int structure.
+The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
+algorithm will not try to free the memory multiple times.  Cleared mp\_ints are detectable by having a pre-defined invalid 
+digit pointer \textbf{dp} setting.
 
-Note that once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
+Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
 with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.
 
 \vspace{+3mm}\begin{small}
 \hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* clear one (frees)  */
-018   void
-019   mp_clear (mp_int * a)
-020   \{
-021     /* only do anything if a hasn't been freed previously */
-022     if (a->dp != NULL) \{
-023       /* first zero the digits */
-024       memset (a->dp, 0, sizeof (mp_digit) * a->used);
-025   
-026       /* free ram */
-027       XFREE(a->dp);
+016   /* clear one (frees)  */
+017   void
+018   mp_clear (mp_int * a)
+019   \{
+020     int i;
+021   
+022     /* only do anything if a hasn't been freed previously */
+023     if (a->dp != NULL) \{
+024       /* first zero the digits */
+025       for (i = 0; i < a->used; i++) \{
+026           a->dp[i] = 0;
+027       \}
 028   
-029       /* reset members to make debugging easier */
-030       a->dp    = NULL;
-031       a->alloc = a->used = 0;
-032       a->sign  = MP_ZPOS;
-033     \}
-034   \}
+029       /* free ram */
+030       XFREE(a->dp);
+031   
+032       /* reset members to make debugging easier */
+033       a->dp    = NULL;
+034       a->alloc = a->used = 0;
+035       a->sign  = MP_ZPOS;
+036     \}
+037   \}
 \end{alltt}
 \end{small}
 
-The ``if'' statement (line 22) prevents the heap from being corrupted if a user double-frees an 
-mp\_int.  This is because once the memory is freed the pointer is set to \textbf{NULL} (line 30).  
+The algorithm only operates on the mp\_int if it hasn't been previously cleared.  The if statement (line 23)
+checks to see if the \textbf{dp} member is not \textbf{NULL}.  If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
+\textbf{NULL} in which case the if statement will evaluate to true.
 
-Without the check, code that accidentally calls mp\_clear twice for a given mp\_int structure would try to free the memory 
-allocated for the digits twice.  This may cause some C libraries to signal a fault.  By setting the pointer to 
-\textbf{NULL} it helps debug code that may inadvertently free the mp\_int before it is truly not needed, because attempts 
-to reference digits should fail immediately. The allocated digits are set to zero before being freed (line 24).  
-This is ideal for cryptographic situations where the integer that the mp\_int represents might need to be kept a secret.
+The digits of the mp\_int are cleared by the for loop (line 25) which assigns a zero to every digit.  Similar to mp\_init()
+the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.  
+
+The digits are deallocated off the heap via the XFREE macro.  Similar to XMALLOC the XFREE macro actually evaluates to
+a standard C library function.  In this case the free() function.  Since free() only deallocates the memory the pointer
+still has to be reset to \textbf{NULL} manually (line 33).  
+
+Now that the digits have been cleared and deallocated the other members are set to their final values (lines 34 and 35).
 
 \section{Maintenance Algorithms}
 
@@ -942,7 +974,7 @@ must be re-sized appropriately to accomodate the result.  The mp\_grow algorithm
 1.  if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
 2.  $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
 3.  $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
-4.  Re-Allocate the array of digits $a$ to size $v$ \\
+4.  Re-allocate the array of digits $a$ to size $v$ \\
 5.  If the allocation failed then return(\textit{MP\_MEM}). \\
 6.  for n from a.alloc to $v - 1$ do  \\
 \hspace{+3mm}6.1  $a_n \leftarrow 0$ \\
@@ -969,54 +1001,57 @@ assumed to contain undefined values they are initially set to zero.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* grow as required */
-018   int mp_grow (mp_int * a, int size)
-019   \{
-020     int     i;
-021     mp_digit *tmp;
-022   
-023     /* if the alloc size is smaller alloc more ram */
-024     if (a->alloc < size) \{
-025       /* ensure there are always at least MP_PREC digits extra on top */
-026       size += (MP_PREC * 2) - (size % MP_PREC);
-027   
-028       /* reallocate the array a->dp
-029        *
-030        * We store the return in a temporary variable
-031        * in case the operation failed we don't want
-032        * to overwrite the dp member of a.
-033        */
-034       tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size);
-035       if (tmp == NULL) \{
-036         /* reallocation failed but "a" is still valid [can be freed] */
-037         return MP_MEM;
-038       \}
-039   
-040       /* reallocation succeeded so set a->dp */
-041       a->dp = tmp;
-042   
-043       /* zero excess digits */
-044       i        = a->alloc;
-045       a->alloc = size;
-046       for (; i < a->alloc; i++) \{
-047         a->dp[i] = 0;
-048       \}
-049     \}
-050     return MP_OKAY;
-051   \}
+016   /* grow as required */
+017   int mp_grow (mp_int * a, int size)
+018   \{
+019     int     i;
+020     mp_digit *tmp;
+021   
+022     /* if the alloc size is smaller alloc more ram */
+023     if (a->alloc < size) \{
+024       /* ensure there are always at least MP_PREC digits extra on top */
+025       size += (MP_PREC * 2) - (size % MP_PREC);
+026   
+027       /* reallocate the array a->dp
+028        *
+029        * We store the return in a temporary variable
+030        * in case the operation failed we don't want
+031        * to overwrite the dp member of a.
+032        */
+033       tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size);
+034       if (tmp == NULL) \{
+035         /* reallocation failed but "a" is still valid [can be freed] */
+036         return MP_MEM;
+037       \}
+038   
+039       /* reallocation succeeded so set a->dp */
+040       a->dp = tmp;
+041   
+042       /* zero excess digits */
+043       i        = a->alloc;
+044       a->alloc = size;
+045       for (; i < a->alloc; i++) \{
+046         a->dp[i] = 0;
+047       \}
+048     \}
+049     return MP_OKAY;
+050   \}
 \end{alltt}
 \end{small}
 
-The first step is to see if we actually need to perform a re-allocation at all (line 24).  If a reallocation
-must occur the digit count is padded upwards to help prevent many trivial reallocations (line 26).  Next the reallocation is performed
-and the return of realloc() is stored in a temporary pointer named $tmp$ (line 36).  The return is stored in a temporary
-instead of $a.dp$ to prevent the code from losing the original pointer in case the reallocation fails.  Had the return been stored 
-in $a.dp$ instead there would be no way to reclaim the heap originally used.
+A quick optimization is to first determine if a memory re-allocation is required at all.  The if statement (line 23) checks
+if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count.  If the count is not larger than \textbf{alloc}
+the function skips the re-allocation part thus saving time.
 
-If the reallocation fails the function will return \textbf{MP\_MEM} (line 37), otherwise, the value of $tmp$ is assigned
-to the pointer $a.dp$ and the function continues.  A simple for loop from line 46 to line 51 will zero all digits 
-that were above the old \textbf{alloc} limit to make sure the integer is in a known state.
+When a re-allocation is performed it is turned into an optimal request to save time in the future.  The requested digit count is
+padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 25).  The XREALLOC function is used
+to re-allocate the memory.  As per the other functions XREALLOC is actually a macro which evaluates to realloc by default.  The realloc
+function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
+the re-allocation.  All	that is left is to clear the newly allocated digits and return.
+
+Note that the re-allocation result is actually stored in a temporary pointer $tmp$.  This is to allow this function to return
+an error with a valid pointer.  Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$.  That would
+result in a memory leak if XREALLOC ever failed.  
 
 \subsection{Initializing Variable Precision mp\_ints}
 Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size 
@@ -1061,35 +1096,34 @@ correct no further memory re-allocations are required to work with the mp\_int.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* init an mp_init for a given size */
-018   int mp_init_size (mp_int * a, int size)
-019   \{
-020     /* pad size so there are always extra digits */
-021     size += (MP_PREC * 2) - (size % MP_PREC);    
-022     
-023     /* alloc mem */
-024     a->dp = OPT_CAST(mp_digit) XCALLOC (sizeof (mp_digit), size);
-025     if (a->dp == NULL) \{
-026       return MP_MEM;
-027     \}
-028     a->used  = 0;
-029     a->alloc = size;
-030     a->sign  = MP_ZPOS;
-031   
-032     return MP_OKAY;
-033   \}
+016   /* init an mp_init for a given size */
+017   int mp_init_size (mp_int * a, int size)
+018   \{
+019     /* pad size so there are always extra digits */
+020     size += (MP_PREC * 2) - (size % MP_PREC);    
+021     
+022     /* alloc mem */
+023     a->dp = OPT_CAST(mp_digit) XCALLOC (sizeof (mp_digit), size);
+024     if (a->dp == NULL) \{
+025       return MP_MEM;
+026     \}
+027     a->used  = 0;
+028     a->alloc = size;
+029     a->sign  = MP_ZPOS;
+030   
+031     return MP_OKAY;
+032   \}
 \end{alltt}
 \end{small}
 
-The number of digits $b$ requested is padded (line 21) by first augmenting it to the next multiple of 
+The number of digits $b$ requested is padded (line 20) by first augmenting it to the next multiple of 
 \textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result.  If the memory can be successfully allocated the 
 mp\_int is placed in a default state representing the integer zero.  Otherwise, the error code \textbf{MP\_MEM} will be 
-returned (line 26).  
+returned (line 25).  
 
-The digits are allocated and set to zero at the same time with the calloc() function (line @25,calloc@).  The 
+The digits are allocated and set to zero at the same time with the calloc() function (line 23).  The 
 \textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set 
-to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 28, 29 and 30).  If the function 
+to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 27, 28 and 29).  If the function 
 returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the 
 functions to work with.
 
@@ -1127,44 +1161,43 @@ initialization which allows for quick recovery from runtime errors.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c
 \vspace{-3mm}
 \begin{alltt}
-016   #include <stdarg.h>
-017   
-018   int mp_init_multi(mp_int *mp, ...) 
-019   \{
-020       mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
-021       int n = 0;                 /* Number of ok inits */
-022       mp_int* cur_arg = mp;
-023       va_list args;
-024   
-025       va_start(args, mp);        /* init args to next argument from caller */
-026       while (cur_arg != NULL) \{
-027           if (mp_init(cur_arg) != MP_OKAY) \{
-028               /* Oops - error! Back-track and mp_clear what we already
-029                  succeeded in init-ing, then return error.
-030               */
-031               va_list clean_args;
-032               
-033               /* end the current list */
-034               va_end(args);
-035               
-036               /* now start cleaning up */            
-037               cur_arg = mp;
-038               va_start(clean_args, mp);
-039               while (n--) \{
-040                   mp_clear(cur_arg);
-041                   cur_arg = va_arg(clean_args, mp_int*);
-042               \}
-043               va_end(clean_args);
-044               res = MP_MEM;
-045               break;
-046           \}
-047           n++;
-048           cur_arg = va_arg(args, mp_int*);
-049       \}
-050       va_end(args);
-051       return res;                /* Assumed ok, if error flagged above. */
-052   \}
-053   
+016   
+017   int mp_init_multi(mp_int *mp, ...) 
+018   \{
+019       mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
+020       int n = 0;                 /* Number of ok inits */
+021       mp_int* cur_arg = mp;
+022       va_list args;
+023   
+024       va_start(args, mp);        /* init args to next argument from caller */
+025       while (cur_arg != NULL) \{
+026           if (mp_init(cur_arg) != MP_OKAY) \{
+027               /* Oops - error! Back-track and mp_clear what we already
+028                  succeeded in init-ing, then return error.
+029               */
+030               va_list clean_args;
+031               
+032               /* end the current list */
+033               va_end(args);
+034               
+035               /* now start cleaning up */            
+036               cur_arg = mp;
+037               va_start(clean_args, mp);
+038               while (n--) \{
+039                   mp_clear(cur_arg);
+040                   cur_arg = va_arg(clean_args, mp_int*);
+041               \}
+042               va_end(clean_args);
+043               res = MP_MEM;
+044               break;
+045           \}
+046           n++;
+047           cur_arg = va_arg(args, mp_int*);
+048       \}
+049       va_end(args);
+050       return res;                /* Assumed ok, if error flagged above. */
+051   \}
+052   
 \end{alltt}
 \end{small}
 
@@ -1174,8 +1207,8 @@ structures in an actual C array they are simply passed as arguments to the funct
 appended on the right.  
 
 The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function.  A count
-$n$ of succesfully initialized mp\_int structures is maintained (line 47) such that if a failure does occur,
-the algorithm can backtrack and free the previously initialized structures (lines 27 to 46).  
+$n$ of succesfully initialized mp\_int structures is maintained (line 46) such that if a failure does occur,
+the algorithm can backtrack and free the previously initialized structures (lines 26 to 45).  
 
 
 \subsection{Clamping Excess Digits}
@@ -1226,36 +1259,35 @@ when all of the digits are zero to ensure that the mp\_int is valid at all times
 \hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* trim unused digits 
-018    *
-019    * This is used to ensure that leading zero digits are
-020    * trimed and the leading "used" digit will be non-zero
-021    * Typically very fast.  Also fixes the sign if there
-022    * are no more leading digits
-023    */
-024   void
-025   mp_clamp (mp_int * a)
-026   \{
-027     /* decrease used while the most significant digit is
-028      * zero.
-029      */
-030     while (a->used > 0 && a->dp[a->used - 1] == 0) \{
-031       --(a->used);
-032     \}
-033   
-034     /* reset the sign flag if used == 0 */
-035     if (a->used == 0) \{
-036       a->sign = MP_ZPOS;
-037     \}
-038   \}
+016   /* trim unused digits 
+017    *
+018    * This is used to ensure that leading zero digits are
+019    * trimed and the leading "used" digit will be non-zero
+020    * Typically very fast.  Also fixes the sign if there
+021    * are no more leading digits
+022    */
+023   void
+024   mp_clamp (mp_int * a)
+025   \{
+026     /* decrease used while the most significant digit is
+027      * zero.
+028      */
+029     while (a->used > 0 && a->dp[a->used - 1] == 0) \{
+030       --(a->used);
+031     \}
+032   
+033     /* reset the sign flag if used == 0 */
+034     if (a->used == 0) \{
+035       a->sign = MP_ZPOS;
+036     \}
+037   \}
 \end{alltt}
 \end{small}
 
-Note on line 27 how to test for the \textbf{used} count is made on the left of the \&\& operator.  In the C programming
+Note on line 26 how to test for the \textbf{used} count is made on the left of the \&\& operator.  In the C programming
 language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails.  This is 
 important since if the \textbf{used} is zero the test on the right would fetch below the array.  That is obviously 
-undesirable.  The parenthesis on line 30 is used to make sure the \textbf{used} count is decremented and not
+undesirable.  The parenthesis on line 29 is used to make sure the \textbf{used} count is decremented and not
 the pointer ``a''.  
 
 \section*{Exercises}
@@ -1338,68 +1370,67 @@ implement the pseudo-code.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* copy, b = a */
-018   int
-019   mp_copy (mp_int * a, mp_int * b)
-020   \{
-021     int     res, n;
-022   
-023     /* if dst == src do nothing */
-024     if (a == b) \{
-025       return MP_OKAY;
-026     \}
-027   
-028     /* grow dest */
-029     if (b->alloc < a->used) \{
-030        if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
-031           return res;
-032        \}
-033     \}
-034   
-035     /* zero b and copy the parameters over */
-036     \{
-037       register mp_digit *tmpa, *tmpb;
-038   
-039       /* pointer aliases */
-040   
-041       /* source */
-042       tmpa = a->dp;
-043   
-044       /* destination */
-045       tmpb = b->dp;
-046   
-047       /* copy all the digits */
-048       for (n = 0; n < a->used; n++) \{
-049         *tmpb++ = *tmpa++;
-050       \}
-051   
-052       /* clear high digits */
-053       for (; n < b->used; n++) \{
-054         *tmpb++ = 0;
-055       \}
-056     \}
-057   
-058     /* copy used count and sign */
-059     b->used = a->used;
-060     b->sign = a->sign;
-061     return MP_OKAY;
-062   \}
+016   /* copy, b = a */
+017   int
+018   mp_copy (mp_int * a, mp_int * b)
+019   \{
+020     int     res, n;
+021   
+022     /* if dst == src do nothing */
+023     if (a == b) \{
+024       return MP_OKAY;
+025     \}
+026   
+027     /* grow dest */
+028     if (b->alloc < a->used) \{
+029        if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
+030           return res;
+031        \}
+032     \}
+033   
+034     /* zero b and copy the parameters over */
+035     \{
+036       register mp_digit *tmpa, *tmpb;
+037   
+038       /* pointer aliases */
+039   
+040       /* source */
+041       tmpa = a->dp;
+042   
+043       /* destination */
+044       tmpb = b->dp;
+045   
+046       /* copy all the digits */
+047       for (n = 0; n < a->used; n++) \{
+048         *tmpb++ = *tmpa++;
+049       \}
+050   
+051       /* clear high digits */
+052       for (; n < b->used; n++) \{
+053         *tmpb++ = 0;
+054       \}
+055     \}
+056   
+057     /* copy used count and sign */
+058     b->used = a->used;
+059     b->sign = a->sign;
+060     return MP_OKAY;
+061   \}
 \end{alltt}
 \end{small}
 
 Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output
 mp\_int structures passed to a function are one and the same.  For this case it is optimal to return immediately without 
-copying digits (line 24).  
+copying digits (line 23).  
 
 The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$.  If $b.alloc$ is less than
-$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 29 to 33).  In order to
+$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 28 to 32).  In order to
 simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits
-of the mp\_ints $a$ and $b$ respectively.  These aliases (lines 42 and 45) allow the compiler to access the digits without first dereferencing the
+of the mp\_ints $a$ and $b$ respectively.  These aliases (lines 41 and 44) allow the compiler to access the digits without first dereferencing the
 mp\_int pointers and then subsequently the pointer to the digits.  
 
-After the aliases are established the digits from $a$ are copied into $b$ (lines 48 to 50) and then the excess 
-digits of $b$ are set to zero (lines 53 to 55).  Both ``for'' loops make use of the pointer aliases and in 
+After the aliases are established the digits from $a$ are copied into $b$ (lines 47 to 49) and then the excess 
+digits of $b$ are set to zero (lines 52 to 55).  Both ``for'' loops make use of the pointer aliases and in 
 fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits.  This optimization 
 allows the alias to stay in a machine register fairly easy between the two loops.
 
@@ -1487,17 +1518,16 @@ such this algorithm will perform two operations in one step.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* creates "a" then copies b into it */
-018   int mp_init_copy (mp_int * a, mp_int * b)
-019   \{
-020     int     res;
-021   
-022     if ((res = mp_init (a)) != MP_OKAY) \{
-023       return res;
-024     \}
-025     return mp_copy (b, a);
-026   \}
+016   /* creates "a" then copies b into it */
+017   int mp_init_copy (mp_int * a, mp_int * b)
+018   \{
+019     int     res;
+020   
+021     if ((res = mp_init (a)) != MP_OKAY) \{
+022       return res;
+023     \}
+024     return mp_copy (b, a);
+025   \}
 \end{alltt}
 \end{small}
 
@@ -1533,15 +1563,14 @@ This algorithm simply resets a mp\_int to the default state.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* set to zero */
-018   void
-019   mp_zero (mp_int * a)
-020   \{
-021     a->sign = MP_ZPOS;
-022     a->used = 0;
-023     memset (a->dp, 0, sizeof (mp_digit) * a->alloc);
-024   \}
+016   /* set to zero */
+017   void
+018   mp_zero (mp_int * a)
+019   \{
+020     a->sign = MP_ZPOS;
+021     a->used = 0;
+022     memset (a->dp, 0, sizeof (mp_digit) * a->alloc);
+023   \}
 \end{alltt}
 \end{small}
 
@@ -1580,28 +1609,27 @@ logic to handle it.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* b = |a| 
-018    *
-019    * Simple function copies the input and fixes the sign to positive
-020    */
-021   int
-022   mp_abs (mp_int * a, mp_int * b)
-023   \{
-024     int     res;
-025   
-026     /* copy a to b */
-027     if (a != b) \{
-028        if ((res = mp_copy (a, b)) != MP_OKAY) \{
-029          return res;
-030        \}
-031     \}
-032   
-033     /* force the sign of b to positive */
-034     b->sign = MP_ZPOS;
-035   
-036     return MP_OKAY;
-037   \}
+016   /* b = |a| 
+017    *
+018    * Simple function copies the input and fixes the sign to positive
+019    */
+020   int
+021   mp_abs (mp_int * a, mp_int * b)
+022   \{
+023     int     res;
+024   
+025     /* copy a to b */
+026     if (a != b) \{
+027        if ((res = mp_copy (a, b)) != MP_OKAY) \{
+028          return res;
+029        \}
+030     \}
+031   
+032     /* force the sign of b to positive */
+033     b->sign = MP_ZPOS;
+034   
+035     return MP_OKAY;
+036   \}
 \end{alltt}
 \end{small}
 
@@ -1640,19 +1668,18 @@ zero as negative.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* b = -a */
-018   int mp_neg (mp_int * a, mp_int * b)
-019   \{
-020     int     res;
-021     if ((res = mp_copy (a, b)) != MP_OKAY) \{
-022       return res;
-023     \}
-024     if (mp_iszero(b) != MP_YES) \{
-025        b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS;
-026     \}
-027     return MP_OKAY;
-028   \}
+016   /* b = -a */
+017   int mp_neg (mp_int * a, mp_int * b)
+018   \{
+019     int     res;
+020     if ((res = mp_copy (a, b)) != MP_OKAY) \{
+021       return res;
+022     \}
+023     if (mp_iszero(b) != MP_YES) \{
+024        b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS;
+025     \}
+026     return MP_OKAY;
+027   \}
 \end{alltt}
 \end{small}
 
@@ -1687,21 +1714,20 @@ single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adj
 \hspace{-5.1mm}{\bf File}: bn\_mp\_set.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* set to a digit */
-018   void mp_set (mp_int * a, mp_digit b)
-019   \{
-020     mp_zero (a);
-021     a->dp[0] = b & MP_MASK;
-022     a->used  = (a->dp[0] != 0) ? 1 : 0;
-023   \}
+016   /* set to a digit */
+017   void mp_set (mp_int * a, mp_digit b)
+018   \{
+019     mp_zero (a);
+020     a->dp[0] = b & MP_MASK;
+021     a->used  = (a->dp[0] != 0) ? 1 : 0;
+022   \}
 \end{alltt}
 \end{small}
 
-Line 20 calls mp\_zero() to clear the mp\_int and reset the sign.  Line 21 copies the digit 
+Line 19 calls mp\_zero() to clear the mp\_int and reset the sign.  Line 20 copies the digit 
 into the least significant location.  Note the usage of a new constant \textbf{MP\_MASK}.  This constant is used to quickly
 reduce an integer modulo $\beta$.  Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with 
-$MP\_MASK = 2^k - 1$ to perform the reduction.  Finally line 22 will set the \textbf{used} member with respect to the 
+$MP\_MASK = 2^k - 1$ to perform the reduction.  Finally line 21 will set the \textbf{used} member with respect to the 
 digit actually set. This function will always make the integer positive.
 
 One important limitation of this function is that it will only set one digit.  The size of a digit is not fixed, meaning source that uses 
@@ -1744,40 +1770,39 @@ Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorith
 \hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* set a 32-bit const */
-018   int mp_set_int (mp_int * a, unsigned long b)
-019   \{
-020     int     x, res;
-021   
-022     mp_zero (a);
-023     
-024     /* set four bits at a time */
-025     for (x = 0; x < 8; x++) \{
-026       /* shift the number up four bits */
-027       if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{
-028         return res;
-029       \}
-030   
-031       /* OR in the top four bits of the source */
-032       a->dp[0] |= (b >> 28) & 15;
-033   
-034       /* shift the source up to the next four bits */
-035       b <<= 4;
-036   
-037       /* ensure that digits are not clamped off */
-038       a->used += 1;
-039     \}
-040     mp_clamp (a);
-041     return MP_OKAY;
-042   \}
+016   /* set a 32-bit const */
+017   int mp_set_int (mp_int * a, unsigned long b)
+018   \{
+019     int     x, res;
+020   
+021     mp_zero (a);
+022     
+023     /* set four bits at a time */
+024     for (x = 0; x < 8; x++) \{
+025       /* shift the number up four bits */
+026       if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{
+027         return res;
+028       \}
+029   
+030       /* OR in the top four bits of the source */
+031       a->dp[0] |= (b >> 28) & 15;
+032   
+033       /* shift the source up to the next four bits */
+034       b <<= 4;
+035   
+036       /* ensure that digits are not clamped off */
+037       a->used += 1;
+038     \}
+039     mp_clamp (a);
+040     return MP_OKAY;
+041   \}
 \end{alltt}
 \end{small}
 
 This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes.  The weird
-addition on line 38 ensures that the newly added in bits are added to the number of digits.  While it may not 
-seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 27 
-as well as the  call to mp\_clamp() on line 40.  Both functions will clamp excess leading digits which keeps 
+addition on line 37 ensures that the newly added in bits are added to the number of digits.  While it may not 
+seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 26 
+as well as the  call to mp\_clamp() on line 39.  Both functions will clamp excess leading digits which keeps 
 the number of used digits low.
 
 \section{Comparisons}
@@ -1838,44 +1863,43 @@ the zero'th digit.  If after all of the digits have been compared, no difference
 \hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* compare maginitude of two ints (unsigned) */
-018   int mp_cmp_mag (mp_int * a, mp_int * b)
-019   \{
-020     int     n;
-021     mp_digit *tmpa, *tmpb;
-022   
-023     /* compare based on # of non-zero digits */
-024     if (a->used > b->used) \{
-025       return MP_GT;
-026     \}
-027     
-028     if (a->used < b->used) \{
-029       return MP_LT;
-030     \}
-031   
-032     /* alias for a */
-033     tmpa = a->dp + (a->used - 1);
-034   
-035     /* alias for b */
-036     tmpb = b->dp + (a->used - 1);
-037   
-038     /* compare based on digits  */
-039     for (n = 0; n < a->used; ++n, --tmpa, --tmpb) \{
-040       if (*tmpa > *tmpb) \{
-041         return MP_GT;
-042       \}
-043   
-044       if (*tmpa < *tmpb) \{
-045         return MP_LT;
-046       \}
-047     \}
-048     return MP_EQ;
-049   \}
+016   /* compare maginitude of two ints (unsigned) */
+017   int mp_cmp_mag (mp_int * a, mp_int * b)
+018   \{
+019     int     n;
+020     mp_digit *tmpa, *tmpb;
+021   
+022     /* compare based on # of non-zero digits */
+023     if (a->used > b->used) \{
+024       return MP_GT;
+025     \}
+026     
+027     if (a->used < b->used) \{
+028       return MP_LT;
+029     \}
+030   
+031     /* alias for a */
+032     tmpa = a->dp + (a->used - 1);
+033   
+034     /* alias for b */
+035     tmpb = b->dp + (a->used - 1);
+036   
+037     /* compare based on digits  */
+038     for (n = 0; n < a->used; ++n, --tmpa, --tmpb) \{
+039       if (*tmpa > *tmpb) \{
+040         return MP_GT;
+041       \}
+042   
+043       if (*tmpa < *tmpb) \{
+044         return MP_LT;
+045       \}
+046     \}
+047     return MP_EQ;
+048   \}
 \end{alltt}
 \end{small}
 
-The two if statements on lines 24 and 28 compare the number of digits in the two inputs.  These two are performed before all of the digits
+The two if statements on lines 23 and 27 compare the number of digits in the two inputs.  These two are performed before all of the digits
 are compared since it is a very cheap test to perform and can potentially save considerable time.  The implementation given is also not valid 
 without those two statements.  $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the 
 array of digits.
@@ -1913,34 +1937,33 @@ $\vert a \vert < \vert b \vert$.  Step number four will compare the two when the
 \hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* compare two ints (signed)*/
-018   int
-019   mp_cmp (mp_int * a, mp_int * b)
-020   \{
-021     /* compare based on sign */
-022     if (a->sign != b->sign) \{
-023        if (a->sign == MP_NEG) \{
-024           return MP_LT;
-025        \} else \{
-026           return MP_GT;
-027        \}
-028     \}
-029     
-030     /* compare digits */
-031     if (a->sign == MP_NEG) \{
-032        /* if negative compare opposite direction */
-033        return mp_cmp_mag(b, a);
-034     \} else \{
-035        return mp_cmp_mag(a, b);
-036     \}
-037   \}
+016   /* compare two ints (signed)*/
+017   int
+018   mp_cmp (mp_int * a, mp_int * b)
+019   \{
+020     /* compare based on sign */
+021     if (a->sign != b->sign) \{
+022        if (a->sign == MP_NEG) \{
+023           return MP_LT;
+024        \} else \{
+025           return MP_GT;
+026        \}
+027     \}
+028     
+029     /* compare digits */
+030     if (a->sign == MP_NEG) \{
+031        /* if negative compare opposite direction */
+032        return mp_cmp_mag(b, a);
+033     \} else \{
+034        return mp_cmp_mag(a, b);
+035     \}
+036   \}
 \end{alltt}
 \end{small}
 
-The two if statements on lines 22 and 23 perform the initial sign comparison.  If the signs are not the equal then which ever
-has the positive sign is larger.   At line 31, the inputs are compared based on magnitudes.  If the signs were both negative then 
-the unsigned comparison is performed in the opposite direction (\textit{line 33}).  Otherwise, the signs are assumed to 
+The two if statements on lines 22 and 30 perform the initial sign comparison.  If the signs are not the equal then which ever
+has the positive sign is larger.   At line 30, the inputs are compared based on magnitudes.  If the signs were both negative then 
+the unsigned comparison is performed in the opposite direction (\textit{line 32}).  Otherwise, the signs are assumed to 
 be both positive and a forward direction unsigned comparison is performed.
 
 \section*{Exercises}
@@ -2064,110 +2087,109 @@ The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are
 \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* low level addition, based on HAC pp.594, Algorithm 14.7 */
-018   int
-019   s_mp_add (mp_int * a, mp_int * b, mp_int * c)
-020   \{
-021     mp_int *x;
-022     int     olduse, res, min, max;
-023   
-024     /* find sizes, we let |a| <= |b| which means we have to sort
-025      * them.  "x" will point to the input with the most digits
-026      */
-027     if (a->used > b->used) \{
-028       min = b->used;
-029       max = a->used;
-030       x = a;
-031     \} else \{
-032       min = a->used;
-033       max = b->used;
-034       x = b;
-035     \}
-036   
-037     /* init result */
-038     if (c->alloc < max + 1) \{
-039       if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{
-040         return res;
-041       \}
-042     \}
-043   
-044     /* get old used digit count and set new one */
-045     olduse = c->used;
-046     c->used = max + 1;
-047   
-048     \{
-049       register mp_digit u, *tmpa, *tmpb, *tmpc;
-050       register int i;
-051   
-052       /* alias for digit pointers */
-053   
-054       /* first input */
-055       tmpa = a->dp;
-056   
-057       /* second input */
-058       tmpb = b->dp;
-059   
-060       /* destination */
-061       tmpc = c->dp;
-062   
-063       /* zero the carry */
-064       u = 0;
-065       for (i = 0; i < min; i++) \{
-066         /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */
-067         *tmpc = *tmpa++ + *tmpb++ + u;
-068   
-069         /* U = carry bit of T[i] */
-070         u = *tmpc >> ((mp_digit)DIGIT_BIT);
-071   
-072         /* take away carry bit from T[i] */
-073         *tmpc++ &= MP_MASK;
-074       \}
-075   
-076       /* now copy higher words if any, that is in A+B 
-077        * if A or B has more digits add those in 
-078        */
-079       if (min != max) \{
-080         for (; i < max; i++) \{
-081           /* T[i] = X[i] + U */
-082           *tmpc = x->dp[i] + u;
-083   
-084           /* U = carry bit of T[i] */
-085           u = *tmpc >> ((mp_digit)DIGIT_BIT);
-086   
-087           /* take away carry bit from T[i] */
-088           *tmpc++ &= MP_MASK;
-089         \}
-090       \}
-091   
-092       /* add carry */
-093       *tmpc++ = u;
-094   
-095       /* clear digits above oldused */
-096       for (i = c->used; i < olduse; i++) \{
-097         *tmpc++ = 0;
-098       \}
-099     \}
-100   
-101     mp_clamp (c);
-102     return MP_OKAY;
-103   \}
+016   /* low level addition, based on HAC pp.594, Algorithm 14.7 */
+017   int
+018   s_mp_add (mp_int * a, mp_int * b, mp_int * c)
+019   \{
+020     mp_int *x;
+021     int     olduse, res, min, max;
+022   
+023     /* find sizes, we let |a| <= |b| which means we have to sort
+024      * them.  "x" will point to the input with the most digits
+025      */
+026     if (a->used > b->used) \{
+027       min = b->used;
+028       max = a->used;
+029       x = a;
+030     \} else \{
+031       min = a->used;
+032       max = b->used;
+033       x = b;
+034     \}
+035   
+036     /* init result */
+037     if (c->alloc < max + 1) \{
+038       if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{
+039         return res;
+040       \}
+041     \}
+042   
+043     /* get old used digit count and set new one */
+044     olduse = c->used;
+045     c->used = max + 1;
+046   
+047     \{
+048       register mp_digit u, *tmpa, *tmpb, *tmpc;
+049       register int i;
+050   
+051       /* alias for digit pointers */
+052   
+053       /* first input */
+054       tmpa = a->dp;
+055   
+056       /* second input */
+057       tmpb = b->dp;
+058   
+059       /* destination */
+060       tmpc = c->dp;
+061   
+062       /* zero the carry */
+063       u = 0;
+064       for (i = 0; i < min; i++) \{
+065         /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */
+066         *tmpc = *tmpa++ + *tmpb++ + u;
+067   
+068         /* U = carry bit of T[i] */
+069         u = *tmpc >> ((mp_digit)DIGIT_BIT);
+070   
+071         /* take away carry bit from T[i] */
+072         *tmpc++ &= MP_MASK;
+073       \}
+074   
+075       /* now copy higher words if any, that is in A+B 
+076        * if A or B has more digits add those in 
+077        */
+078       if (min != max) \{
+079         for (; i < max; i++) \{
+080           /* T[i] = X[i] + U */
+081           *tmpc = x->dp[i] + u;
+082   
+083           /* U = carry bit of T[i] */
+084           u = *tmpc >> ((mp_digit)DIGIT_BIT);
+085   
+086           /* take away carry bit from T[i] */
+087           *tmpc++ &= MP_MASK;
+088         \}
+089       \}
+090   
+091       /* add carry */
+092       *tmpc++ = u;
+093   
+094       /* clear digits above oldused */
+095       for (i = c->used; i < olduse; i++) \{
+096         *tmpc++ = 0;
+097       \}
+098     \}
+099   
+100     mp_clamp (c);
+101     return MP_OKAY;
+102   \}
 \end{alltt}
 \end{small}
 
-Lines 27 to 35 perform the initial sorting of the inputs and determine the $min$ and $max$ variables.  Note that $x$ is a pointer to a 
-mp\_int assigned to the largest input, in effect it is a local alias.  Lines 37 to 42 ensure that the destination is grown to 
+Lines 26 to 34 perform the initial sorting of the inputs and determine the $min$ and $max$ variables.  Note that $x$ is a pointer to a 
+mp\_int assigned to the largest input, in effect it is a local alias.  Lines 36 to 41 ensure that the destination is grown to 
 accomodate the result of the addition. 
 
 Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style.  The three aliases that are on 
-lines 55, 58 and 61 represent the two inputs and destination variables respectively.  These aliases are used to ensure the
+lines 54, 57 and 60 represent the two inputs and destination variables respectively.  These aliases are used to ensure the
 compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.
 
-The initial carry $u$ is cleared on line 64, note that $u$ is of type mp\_digit which ensures type compatibility within the 
-implementation.  The initial addition loop begins on line 65 and ends on line 74.  Similarly the conditional addition loop
-begins on line 80 and ends on line 90.  The addition is finished with the final carry being stored in $tmpc$ on line 93.  
-Note the ``++'' operator on the same line.  After line 93 $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$.  This is useful
-for the next loop on lines 96 to 99 which set any old upper digits to zero.
+The initial carry $u$ is cleared on line 63, note that $u$ is of type mp\_digit which ensures type compatibility within the 
+implementation.  The initial addition loop begins on line 64 and ends on line 73.  Similarly the conditional addition loop
+begins on line 79 and ends on line 89.  The addition is finished with the final carry being stored in $tmpc$ on line 96.  
+Note the ``++'' operator on the same line.  After line 96 $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$.  This is useful
+for the next loop on lines 95 to 98 which set any old upper digits to zero.
 
 \subsection{Low Level Subtraction}
 The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm.  The principle difference is that the
@@ -2251,91 +2273,90 @@ If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and cop
 \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
-018   int
-019   s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
-020   \{
-021     int     olduse, res, min, max;
-022   
-023     /* find sizes */
-024     min = b->used;
-025     max = a->used;
-026   
-027     /* init result */
-028     if (c->alloc < max) \{
-029       if ((res = mp_grow (c, max)) != MP_OKAY) \{
-030         return res;
-031       \}
-032     \}
-033     olduse = c->used;
-034     c->used = max;
-035   
-036     \{
-037       register mp_digit u, *tmpa, *tmpb, *tmpc;
-038       register int i;
-039   
-040       /* alias for digit pointers */
-041       tmpa = a->dp;
-042       tmpb = b->dp;
-043       tmpc = c->dp;
-044   
-045       /* set carry to zero */
-046       u = 0;
-047       for (i = 0; i < min; i++) \{
-048         /* T[i] = A[i] - B[i] - U */
-049         *tmpc = *tmpa++ - *tmpb++ - u;
-050   
-051         /* U = carry bit of T[i]
-052          * Note this saves performing an AND operation since
-053          * if a carry does occur it will propagate all the way to the
-054          * MSB.  As a result a single shift is enough to get the carry
-055          */
-056         u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
-057   
-058         /* Clear carry from T[i] */
-059         *tmpc++ &= MP_MASK;
-060       \}
-061   
-062       /* now copy higher words if any, e.g. if A has more digits than B  */
-063       for (; i < max; i++) \{
-064         /* T[i] = A[i] - U */
-065         *tmpc = *tmpa++ - u;
-066   
-067         /* U = carry bit of T[i] */
-068         u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
-069   
-070         /* Clear carry from T[i] */
-071         *tmpc++ &= MP_MASK;
-072       \}
-073   
-074       /* clear digits above used (since we may not have grown result above) */
+016   /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
+017   int
+018   s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
+019   \{
+020     int     olduse, res, min, max;
+021   
+022     /* find sizes */
+023     min = b->used;
+024     max = a->used;
+025   
+026     /* init result */
+027     if (c->alloc < max) \{
+028       if ((res = mp_grow (c, max)) != MP_OKAY) \{
+029         return res;
+030       \}
+031     \}
+032     olduse = c->used;
+033     c->used = max;
+034   
+035     \{
+036       register mp_digit u, *tmpa, *tmpb, *tmpc;
+037       register int i;
+038   
+039       /* alias for digit pointers */
+040       tmpa = a->dp;
+041       tmpb = b->dp;
+042       tmpc = c->dp;
+043   
+044       /* set carry to zero */
+045       u = 0;
+046       for (i = 0; i < min; i++) \{
+047         /* T[i] = A[i] - B[i] - U */
+048         *tmpc = *tmpa++ - *tmpb++ - u;
+049   
+050         /* U = carry bit of T[i]
+051          * Note this saves performing an AND operation since
+052          * if a carry does occur it will propagate all the way to the
+053          * MSB.  As a result a single shift is enough to get the carry
+054          */
+055         u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
+056   
+057         /* Clear carry from T[i] */
+058         *tmpc++ &= MP_MASK;
+059       \}
+060   
+061       /* now copy higher words if any, e.g. if A has more digits than B  */
+062       for (; i < max; i++) \{
+063         /* T[i] = A[i] - U */
+064         *tmpc = *tmpa++ - u;
+065   
+066         /* U = carry bit of T[i] */
+067         u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
+068   
+069         /* Clear carry from T[i] */
+070         *tmpc++ &= MP_MASK;
+071       \}
+072   
+073       /* clear digits above used (since we may not have grown result above) */
       
-075       for (i = c->used; i < olduse; i++) \{
-076         *tmpc++ = 0;
-077       \}
-078     \}
-079   
-080     mp_clamp (c);
-081     return MP_OKAY;
-082   \}
-083   
+074       for (i = c->used; i < olduse; i++) \{
+075         *tmpc++ = 0;
+076       \}
+077     \}
+078   
+079     mp_clamp (c);
+080     return MP_OKAY;
+081   \}
+082   
 \end{alltt}
 \end{small}
 
-Line 24 and 25 perform the initial hardcoded sorting of the inputs.  In reality the $min$ and $max$ variables are only aliases and are only 
-used to make the source code easier to read.  Again the pointer alias optimization is used within this algorithm.  Lines 41, 42 and 43 initialize the aliases for 
+Line 23 and 24 perform the initial hardcoded sorting of the inputs.  In reality the $min$ and $max$ variables are only aliases and are only 
+used to make the source code easier to read.  Again the pointer alias optimization is used within this algorithm.  Lines 40, 41 and 42 initialize the aliases for 
 $a$, $b$ and $c$ respectively.
 
-The first subtraction loop occurs on lines 46 through 60.  The theory behind the subtraction loop is exactly the same as that for
+The first subtraction loop occurs on lines 45 through 59.  The theory behind the subtraction loop is exactly the same as that for
 the addition loop.  As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry 
-(\textit{see line 56}).  The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND 
+(\textit{see line 55}).  The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND 
 the least significant bit.  The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry
 occurs from subtraction.  This carry extraction requires two relatively cheap operations to extract the carry.  The other method is to simply 
 shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation.  This optimization only works on
 twos compliment machines which is a safe assumption to make.
 
-If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines 63 through 72}) is required to propagate the carry through
+If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines 62 through 71}) is required to propagate the carry through
 $a$ and copy the result to $c$.  
 
 \subsection{High Level Addition}
@@ -2419,38 +2440,37 @@ within algorithm s\_mp\_add will force $-0$ to become $0$.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_add.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* high level addition (handles signs) */
-018   int mp_add (mp_int * a, mp_int * b, mp_int * c)
-019   \{
-020     int     sa, sb, res;
-021   
-022     /* get sign of both inputs */
-023     sa = a->sign;
-024     sb = b->sign;
-025   
-026     /* handle two cases, not four */
-027     if (sa == sb) \{
-028       /* both positive or both negative */
-029       /* add their magnitudes, copy the sign */
-030       c->sign = sa;
-031       res = s_mp_add (a, b, c);
-032     \} else \{
-033       /* one positive, the other negative */
-034       /* subtract the one with the greater magnitude from */
-035       /* the one of the lesser magnitude.  The result gets */
-036       /* the sign of the one with the greater magnitude. */
-037       if (mp_cmp_mag (a, b) == MP_LT) \{
-038         c->sign = sb;
-039         res = s_mp_sub (b, a, c);
-040       \} else \{
-041         c->sign = sa;
-042         res = s_mp_sub (a, b, c);
-043       \}
-044     \}
-045     return res;
-046   \}
-047   
+016   /* high level addition (handles signs) */
+017   int mp_add (mp_int * a, mp_int * b, mp_int * c)
+018   \{
+019     int     sa, sb, res;
+020   
+021     /* get sign of both inputs */
+022     sa = a->sign;
+023     sb = b->sign;
+024   
+025     /* handle two cases, not four */
+026     if (sa == sb) \{
+027       /* both positive or both negative */
+028       /* add their magnitudes, copy the sign */
+029       c->sign = sa;
+030       res = s_mp_add (a, b, c);
+031     \} else \{
+032       /* one positive, the other negative */
+033       /* subtract the one with the greater magnitude from */
+034       /* the one of the lesser magnitude.  The result gets */
+035       /* the sign of the one with the greater magnitude. */
+036       if (mp_cmp_mag (a, b) == MP_LT) \{
+037         c->sign = sb;
+038         res = s_mp_sub (b, a, c);
+039       \} else \{
+040         c->sign = sa;
+041         res = s_mp_sub (a, b, c);
+042       \}
+043     \}
+044     return res;
+045   \}
+046   
 \end{alltt}
 \end{small}
 
@@ -2524,49 +2544,48 @@ algorithm from producing $-a - -a = -0$ as a result.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* high level subtraction (handles signs) */
-018   int
-019   mp_sub (mp_int * a, mp_int * b, mp_int * c)
-020   \{
-021     int     sa, sb, res;
-022   
-023     sa = a->sign;
-024     sb = b->sign;
-025   
-026     if (sa != sb) \{
-027       /* subtract a negative from a positive, OR */
-028       /* subtract a positive from a negative. */
-029       /* In either case, ADD their magnitudes, */
-030       /* and use the sign of the first number. */
-031       c->sign = sa;
-032       res = s_mp_add (a, b, c);
-033     \} else \{
-034       /* subtract a positive from a positive, OR */
-035       /* subtract a negative from a negative. */
-036       /* First, take the difference between their */
-037       /* magnitudes, then... */
-038       if (mp_cmp_mag (a, b) != MP_LT) \{
-039         /* Copy the sign from the first */
-040         c->sign = sa;
-041         /* The first has a larger or equal magnitude */
-042         res = s_mp_sub (a, b, c);
-043       \} else \{
-044         /* The result has the *opposite* sign from */
-045         /* the first number. */
-046         c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS;
-047         /* The second has a larger magnitude */
-048         res = s_mp_sub (b, a, c);
-049       \}
-050     \}
-051     return res;
-052   \}
-053   
+016   /* high level subtraction (handles signs) */
+017   int
+018   mp_sub (mp_int * a, mp_int * b, mp_int * c)
+019   \{
+020     int     sa, sb, res;
+021   
+022     sa = a->sign;
+023     sb = b->sign;
+024   
+025     if (sa != sb) \{
+026       /* subtract a negative from a positive, OR */
+027       /* subtract a positive from a negative. */
+028       /* In either case, ADD their magnitudes, */
+029       /* and use the sign of the first number. */
+030       c->sign = sa;
+031       res = s_mp_add (a, b, c);
+032     \} else \{
+033       /* subtract a positive from a positive, OR */
+034       /* subtract a negative from a negative. */
+035       /* First, take the difference between their */
+036       /* magnitudes, then... */
+037       if (mp_cmp_mag (a, b) != MP_LT) \{
+038         /* Copy the sign from the first */
+039         c->sign = sa;
+040         /* The first has a larger or equal magnitude */
+041         res = s_mp_sub (a, b, c);
+042       \} else \{
+043         /* The result has the *opposite* sign from */
+044         /* the first number. */
+045         c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS;
+046         /* The second has a larger magnitude */
+047         res = s_mp_sub (b, a, c);
+048       \}
+049     \}
+050     return res;
+051   \}
+052   
 \end{alltt}
 \end{small}
 
 Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
-and forward it to the end of the function.  On line 38 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a 
+and forward it to the end of the function.  On line 37 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a 
 ``greater than or equal to'' comparison.  
 
 \section{Bit and Digit Shifting}
@@ -2634,72 +2653,71 @@ Step 8 clears any leading digits of $b$ in case it originally had a larger magni
 \hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* b = a*2 */
-018   int mp_mul_2(mp_int * a, mp_int * b)
-019   \{
-020     int     x, res, oldused;
-021   
-022     /* grow to accomodate result */
-023     if (b->alloc < a->used + 1) \{
-024       if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{
-025         return res;
-026       \}
-027     \}
-028   
-029     oldused = b->used;
-030     b->used = a->used;
-031   
-032     \{
-033       register mp_digit r, rr, *tmpa, *tmpb;
-034   
-035       /* alias for source */
-036       tmpa = a->dp;
-037       
-038       /* alias for dest */
-039       tmpb = b->dp;
-040   
-041       /* carry */
-042       r = 0;
-043       for (x = 0; x < a->used; x++) \{
-044       
-045         /* get what will be the *next* carry bit from the 
-046          * MSB of the current digit 
-047          */
-048         rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));
-049         
-050         /* now shift up this digit, add in the carry [from the previous] */
-051         *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;
-052         
-053         /* copy the carry that would be from the source 
-054          * digit into the next iteration 
-055          */
-056         r = rr;
-057       \}
-058   
-059       /* new leading digit? */
-060       if (r != 0) \{
-061         /* add a MSB which is always 1 at this point */
-062         *tmpb = 1;
-063         ++(b->used);
-064       \}
-065   
-066       /* now zero any excess digits on the destination 
-067        * that we didn't write to 
-068        */
-069       tmpb = b->dp + b->used;
-070       for (x = b->used; x < oldused; x++) \{
-071         *tmpb++ = 0;
-072       \}
-073     \}
-074     b->sign = a->sign;
-075     return MP_OKAY;
-076   \}
+016   /* b = a*2 */
+017   int mp_mul_2(mp_int * a, mp_int * b)
+018   \{
+019     int     x, res, oldused;
+020   
+021     /* grow to accomodate result */
+022     if (b->alloc < a->used + 1) \{
+023       if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{
+024         return res;
+025       \}
+026     \}
+027   
+028     oldused = b->used;
+029     b->used = a->used;
+030   
+031     \{
+032       register mp_digit r, rr, *tmpa, *tmpb;
+033   
+034       /* alias for source */
+035       tmpa = a->dp;
+036       
+037       /* alias for dest */
+038       tmpb = b->dp;
+039   
+040       /* carry */
+041       r = 0;
+042       for (x = 0; x < a->used; x++) \{
+043       
+044         /* get what will be the *next* carry bit from the 
+045          * MSB of the current digit 
+046          */
+047         rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));
+048         
+049         /* now shift up this digit, add in the carry [from the previous] */
+050         *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;
+051         
+052         /* copy the carry that would be from the source 
+053          * digit into the next iteration 
+054          */
+055         r = rr;
+056       \}
+057   
+058       /* new leading digit? */
+059       if (r != 0) \{
+060         /* add a MSB which is always 1 at this point */
+061         *tmpb = 1;
+062         ++(b->used);
+063       \}
+064   
+065       /* now zero any excess digits on the destination 
+066        * that we didn't write to 
+067        */
+068       tmpb = b->dp + b->used;
+069       for (x = b->used; x < oldused; x++) \{
+070         *tmpb++ = 0;
+071       \}
+072     \}
+073     b->sign = a->sign;
+074     return MP_OKAY;
+075   \}
 \end{alltt}
 \end{small}
 
 This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input.  The only noteworthy difference
-is the use of the logical shift operator on line 51 to perform a single precision doubling.  
+is the use of the logical shift operator on line 50 to perform a single precision doubling.  
 
 \subsection{Division by Two}
 A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.
@@ -2747,53 +2765,52 @@ least significant bit not the most significant bit.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* b = a/2 */
-018   int mp_div_2(mp_int * a, mp_int * b)
-019   \{
-020     int     x, res, oldused;
-021   
-022     /* copy */
-023     if (b->alloc < a->used) \{
-024       if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
-025         return res;
-026       \}
-027     \}
-028   
-029     oldused = b->used;
-030     b->used = a->used;
-031     \{
-032       register mp_digit r, rr, *tmpa, *tmpb;
-033   
-034       /* source alias */
-035       tmpa = a->dp + b->used - 1;
-036   
-037       /* dest alias */
-038       tmpb = b->dp + b->used - 1;
-039   
-040       /* carry */
-041       r = 0;
-042       for (x = b->used - 1; x >= 0; x--) \{
-043         /* get the carry for the next iteration */
-044         rr = *tmpa & 1;
-045   
-046         /* shift the current digit, add in carry and store */
-047         *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1));
-048   
-049         /* forward carry to next iteration */
-050         r = rr;
-051       \}
-052   
-053       /* zero excess digits */
-054       tmpb = b->dp + b->used;
-055       for (x = b->used; x < oldused; x++) \{
-056         *tmpb++ = 0;
-057       \}
-058     \}
-059     b->sign = a->sign;
-060     mp_clamp (b);
-061     return MP_OKAY;
-062   \}
+016   /* b = a/2 */
+017   int mp_div_2(mp_int * a, mp_int * b)
+018   \{
+019     int     x, res, oldused;
+020   
+021     /* copy */
+022     if (b->alloc < a->used) \{
+023       if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
+024         return res;
+025       \}
+026     \}
+027   
+028     oldused = b->used;
+029     b->used = a->used;
+030     \{
+031       register mp_digit r, rr, *tmpa, *tmpb;
+032   
+033       /* source alias */
+034       tmpa = a->dp + b->used - 1;
+035   
+036       /* dest alias */
+037       tmpb = b->dp + b->used - 1;
+038   
+039       /* carry */
+040       r = 0;
+041       for (x = b->used - 1; x >= 0; x--) \{
+042         /* get the carry for the next iteration */
+043         rr = *tmpa & 1;
+044   
+045         /* shift the current digit, add in carry and store */
+046         *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1));
+047   
+048         /* forward carry to next iteration */
+049         r = rr;
+050       \}
+051   
+052       /* zero excess digits */
+053       tmpb = b->dp + b->used;
+054       for (x = b->used; x < oldused; x++) \{
+055         *tmpb++ = 0;
+056       \}
+057     \}
+058     b->sign = a->sign;
+059     mp_clamp (b);
+060     return MP_OKAY;
+061   \}
 \end{alltt}
 \end{small}
 
@@ -2867,58 +2884,57 @@ step 8 sets the lower $b$ digits to zero.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* shift left a certain amount of digits */
-018   int mp_lshd (mp_int * a, int b)
-019   \{
-020     int     x, res;
-021   
-022     /* if its less than zero return */
-023     if (b <= 0) \{
-024       return MP_OKAY;
-025     \}
-026   
-027     /* grow to fit the new digits */
-028     if (a->alloc < a->used + b) \{
-029        if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{
-030          return res;
-031        \}
-032     \}
-033   
-034     \{
-035       register mp_digit *top, *bottom;
-036   
-037       /* increment the used by the shift amount then copy upwards */
-038       a->used += b;
-039   
-040       /* top */
-041       top = a->dp + a->used - 1;
-042   
-043       /* base */
-044       bottom = a->dp + a->used - 1 - b;
-045   
-046       /* much like mp_rshd this is implemented using a sliding window
-047        * except the window goes the otherway around.  Copying from
-048        * the bottom to the top.  see bn_mp_rshd.c for more info.
-049        */
-050       for (x = a->used - 1; x >= b; x--) \{
-051         *top-- = *bottom--;
-052       \}
-053   
-054       /* zero the lower digits */
-055       top = a->dp;
-056       for (x = 0; x < b; x++) \{
-057         *top++ = 0;
-058       \}
-059     \}
-060     return MP_OKAY;
-061   \}
+016   /* shift left a certain amount of digits */
+017   int mp_lshd (mp_int * a, int b)
+018   \{
+019     int     x, res;
+020   
+021     /* if its less than zero return */
+022     if (b <= 0) \{
+023       return MP_OKAY;
+024     \}
+025   
+026     /* grow to fit the new digits */
+027     if (a->alloc < a->used + b) \{
+028        if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{
+029          return res;
+030        \}
+031     \}
+032   
+033     \{
+034       register mp_digit *top, *bottom;
+035   
+036       /* increment the used by the shift amount then copy upwards */
+037       a->used += b;
+038   
+039       /* top */
+040       top = a->dp + a->used - 1;
+041   
+042       /* base */
+043       bottom = a->dp + a->used - 1 - b;
+044   
+045       /* much like mp_rshd this is implemented using a sliding window
+046        * except the window goes the otherway around.  Copying from
+047        * the bottom to the top.  see bn_mp_rshd.c for more info.
+048        */
+049       for (x = a->used - 1; x >= b; x--) \{
+050         *top-- = *bottom--;
+051       \}
+052   
+053       /* zero the lower digits */
+054       top = a->dp;
+055       for (x = 0; x < b; x++) \{
+056         *top++ = 0;
+057       \}
+058     \}
+059     return MP_OKAY;
+060   \}
 \end{alltt}
 \end{small}
 
-The if statement on line 23 ensures that the $b$ variable is greater than zero.  The \textbf{used} count is incremented by $b$ before
-the copy loop begins.  This elminates the need for an additional variable in the for loop.  The variable $top$ on line 41 is an alias
-for the leading digit while $bottom$ on line 44 is an alias for the trailing edge.  The aliases form a window of exactly $b$ digits
+The if statement on line 22 ensures that the $b$ variable is greater than zero.  The \textbf{used} count is incremented by $b$ before
+the copy loop begins.  This elminates the need for an additional variable in the for loop.  The variable $top$ on line 40 is an alias
+for the leading digit while $bottom$ on line 47 is an alias for the trailing edge.  The aliases form a window of exactly $b$ digits
 over the input.  
 
 \subsection{Division by $x$}
@@ -2971,57 +2987,56 @@ Once the window copy is complete the upper digits must be zeroed and the \textbf
 \hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* shift right a certain amount of digits */
-018   void mp_rshd (mp_int * a, int b)
-019   \{
-020     int     x;
-021   
-022     /* if b <= 0 then ignore it */
-023     if (b <= 0) \{
-024       return;
-025     \}
-026   
-027     /* if b > used then simply zero it and return */
-028     if (a->used <= b) \{
-029       mp_zero (a);
-030       return;
-031     \}
-032   
-033     \{
-034       register mp_digit *bottom, *top;
-035   
-036       /* shift the digits down */
-037   
-038       /* bottom */
-039       bottom = a->dp;
-040   
-041       /* top [offset into digits] */
-042       top = a->dp + b;
-043   
-044       /* this is implemented as a sliding window where 
-045        * the window is b-digits long and digits from 
-046        * the top of the window are copied to the bottom
-047        *
-048        * e.g.
-049   
-050        b-2 | b-1 | b0 | b1 | b2 | ... | bb |   ---->
-051                    /\symbol{92}                   |      ---->
-052                     \symbol{92}-------------------/      ---->
-053        */
-054       for (x = 0; x < (a->used - b); x++) \{
-055         *bottom++ = *top++;
-056       \}
-057   
-058       /* zero the top digits */
-059       for (; x < a->used; x++) \{
-060         *bottom++ = 0;
-061       \}
-062     \}
-063     
-064     /* remove excess digits */
-065     a->used -= b;
-066   \}
+016   /* shift right a certain amount of digits */
+017   void mp_rshd (mp_int * a, int b)
+018   \{
+019     int     x;
+020   
+021     /* if b <= 0 then ignore it */
+022     if (b <= 0) \{
+023       return;
+024     \}
+025   
+026     /* if b > used then simply zero it and return */
+027     if (a->used <= b) \{
+028       mp_zero (a);
+029       return;
+030     \}
+031   
+032     \{
+033       register mp_digit *bottom, *top;
+034   
+035       /* shift the digits down */
+036   
+037       /* bottom */
+038       bottom = a->dp;
+039   
+040       /* top [offset into digits] */
+041       top = a->dp + b;
+042   
+043       /* this is implemented as a sliding window where 
+044        * the window is b-digits long and digits from 
+045        * the top of the window are copied to the bottom
+046        *
+047        * e.g.
+048   
+049        b-2 | b-1 | b0 | b1 | b2 | ... | bb |   ---->
+050                    /\symbol{92}                   |      ---->
+051                     \symbol{92}-------------------/      ---->
+052        */
+053       for (x = 0; x < (a->used - b); x++) \{
+054         *bottom++ = *top++;
+055       \}
+056   
+057       /* zero the top digits */
+058       for (; x < a->used; x++) \{
+059         *bottom++ = 0;
+060       \}
+061     \}
+062     
+063     /* remove excess digits */
+064     a->used -= b;
+065   \}
 \end{alltt}
 \end{small}
 
@@ -3090,70 +3105,69 @@ complete.  It is possible to optimize this algorithm down to a $O(n)$ algorithm
 \hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* shift left by a certain bit count */
-018   int mp_mul_2d (mp_int * a, int b, mp_int * c)
-019   \{
-020     mp_digit d;
-021     int      res;
-022   
-023     /* copy */
-024     if (a != c) \{
-025        if ((res = mp_copy (a, c)) != MP_OKAY) \{
-026          return res;
-027        \}
-028     \}
-029   
-030     if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) \{
-031        if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) \{
-032          return res;
-033        \}
-034     \}
-035   
-036     /* shift by as many digits in the bit count */
-037     if (b >= (int)DIGIT_BIT) \{
-038       if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{
-039         return res;
-040       \}
-041     \}
-042   
-043     /* shift any bit count < DIGIT_BIT */
-044     d = (mp_digit) (b % DIGIT_BIT);
-045     if (d != 0) \{
-046       register mp_digit *tmpc, shift, mask, r, rr;
-047       register int x;
-048   
-049       /* bitmask for carries */
-050       mask = (((mp_digit)1) << d) - 1;
-051   
-052       /* shift for msbs */
-053       shift = DIGIT_BIT - d;
-054   
-055       /* alias */
-056       tmpc = c->dp;
-057   
-058       /* carry */
-059       r    = 0;
-060       for (x = 0; x < c->used; x++) \{
-061         /* get the higher bits of the current word */
-062         rr = (*tmpc >> shift) & mask;
-063   
-064         /* shift the current word and OR in the carry */
-065         *tmpc = ((*tmpc << d) | r) & MP_MASK;
-066         ++tmpc;
-067   
-068         /* set the carry to the carry bits of the current word */
-069         r = rr;
-070       \}
-071       
-072       /* set final carry */
-073       if (r != 0) \{
-074          c->dp[(c->used)++] = r;
-075       \}
-076     \}
-077     mp_clamp (c);
-078     return MP_OKAY;
-079   \}
+016   /* shift left by a certain bit count */
+017   int mp_mul_2d (mp_int * a, int b, mp_int * c)
+018   \{
+019     mp_digit d;
+020     int      res;
+021   
+022     /* copy */
+023     if (a != c) \{
+024        if ((res = mp_copy (a, c)) != MP_OKAY) \{
+025          return res;
+026        \}
+027     \}
+028   
+029     if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) \{
+030        if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) \{
+031          return res;
+032        \}
+033     \}
+034   
+035     /* shift by as many digits in the bit count */
+036     if (b >= (int)DIGIT_BIT) \{
+037       if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{
+038         return res;
+039       \}
+040     \}
+041   
+042     /* shift any bit count < DIGIT_BIT */
+043     d = (mp_digit) (b % DIGIT_BIT);
+044     if (d != 0) \{
+045       register mp_digit *tmpc, shift, mask, r, rr;
+046       register int x;
+047   
+048       /* bitmask for carries */
+049       mask = (((mp_digit)1) << d) - 1;
+050   
+051       /* shift for msbs */
+052       shift = DIGIT_BIT - d;
+053   
+054       /* alias */
+055       tmpc = c->dp;
+056   
+057       /* carry */
+058       r    = 0;
+059       for (x = 0; x < c->used; x++) \{
+060         /* get the higher bits of the current word */
+061         rr = (*tmpc >> shift) & mask;
+062   
+063         /* shift the current word and OR in the carry */
+064         *tmpc = ((*tmpc << d) | r) & MP_MASK;
+065         ++tmpc;
+066   
+067         /* set the carry to the carry bits of the current word */
+068         r = rr;
+069       \}
+070       
+071       /* set final carry */
+072       if (r != 0) \{
+073          c->dp[(c->used)++] = r;
+074       \}
+075     \}
+076     mp_clamp (c);
+077     return MP_OKAY;
+078   \}
 \end{alltt}
 \end{small}
 
@@ -3203,84 +3217,83 @@ by using algorithm mp\_mod\_2d.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* shift right by a certain bit count (store quotient in c, optional remaind
+016   /* shift right by a certain bit count (store quotient in c, optional remaind
       er in d) */
-018   int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d)
-019   \{
-020     mp_digit D, r, rr;
-021     int     x, res;
-022     mp_int  t;
+017   int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d)
+018   \{
+019     mp_digit D, r, rr;
+020     int     x, res;
+021     mp_int  t;
+022   
 023   
-024   
-025     /* if the shift count is <= 0 then we do no work */
-026     if (b <= 0) \{
-027       res = mp_copy (a, c);
-028       if (d != NULL) \{
-029         mp_zero (d);
-030       \}
-031       return res;
-032     \}
-033   
-034     if ((res = mp_init (&t)) != MP_OKAY) \{
-035       return res;
-036     \}
-037   
-038     /* get the remainder */
-039     if (d != NULL) \{
-040       if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{
-041         mp_clear (&t);
-042         return res;
-043       \}
-044     \}
-045   
-046     /* copy */
-047     if ((res = mp_copy (a, c)) != MP_OKAY) \{
-048       mp_clear (&t);
-049       return res;
-050     \}
-051   
-052     /* shift by as many digits in the bit count */
-053     if (b >= (int)DIGIT_BIT) \{
-054       mp_rshd (c, b / DIGIT_BIT);
-055     \}
-056   
-057     /* shift any bit count < DIGIT_BIT */
-058     D = (mp_digit) (b % DIGIT_BIT);
-059     if (D != 0) \{
-060       register mp_digit *tmpc, mask, shift;
-061   
-062       /* mask */
-063       mask = (((mp_digit)1) << D) - 1;
-064   
-065       /* shift for lsb */
-066       shift = DIGIT_BIT - D;
-067   
-068       /* alias */
-069       tmpc = c->dp + (c->used - 1);
-070   
-071       /* carry */
-072       r = 0;
-073       for (x = c->used - 1; x >= 0; x--) \{
-074         /* get the lower  bits of this word in a temp */
-075         rr = *tmpc & mask;
-076   
-077         /* shift the current word and mix in the carry bits from the previous 
+024     /* if the shift count is <= 0 then we do no work */
+025     if (b <= 0) \{
+026       res = mp_copy (a, c);
+027       if (d != NULL) \{
+028         mp_zero (d);
+029       \}
+030       return res;
+031     \}
+032   
+033     if ((res = mp_init (&t)) != MP_OKAY) \{
+034       return res;
+035     \}
+036   
+037     /* get the remainder */
+038     if (d != NULL) \{
+039       if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{
+040         mp_clear (&t);
+041         return res;
+042       \}
+043     \}
+044   
+045     /* copy */
+046     if ((res = mp_copy (a, c)) != MP_OKAY) \{
+047       mp_clear (&t);
+048       return res;
+049     \}
+050   
+051     /* shift by as many digits in the bit count */
+052     if (b >= (int)DIGIT_BIT) \{
+053       mp_rshd (c, b / DIGIT_BIT);
+054     \}
+055   
+056     /* shift any bit count < DIGIT_BIT */
+057     D = (mp_digit) (b % DIGIT_BIT);
+058     if (D != 0) \{
+059       register mp_digit *tmpc, mask, shift;
+060   
+061       /* mask */
+062       mask = (((mp_digit)1) << D) - 1;
+063   
+064       /* shift for lsb */
+065       shift = DIGIT_BIT - D;
+066   
+067       /* alias */
+068       tmpc = c->dp + (c->used - 1);
+069   
+070       /* carry */
+071       r = 0;
+072       for (x = c->used - 1; x >= 0; x--) \{
+073         /* get the lower  bits of this word in a temp */
+074         rr = *tmpc & mask;
+075   
+076         /* shift the current word and mix in the carry bits from the previous 
       word */
-078         *tmpc = (*tmpc >> D) | (r << shift);
-079         --tmpc;
-080   
-081         /* set the carry to the carry bits of the current word found above */
-082         r = rr;
-083       \}
-084     \}
-085     mp_clamp (c);
-086     if (d != NULL) \{
-087       mp_exch (&t, d);
-088     \}
-089     mp_clear (&t);
-090     return MP_OKAY;
-091   \}
+077         *tmpc = (*tmpc >> D) | (r << shift);
+078         --tmpc;
+079   
+080         /* set the carry to the carry bits of the current word found above */
+081         r = rr;
+082       \}
+083     \}
+084     mp_clamp (c);
+085     if (d != NULL) \{
+086       mp_exch (&t, d);
+087     \}
+088     mp_clear (&t);
+089     return MP_OKAY;
+090   \}
 \end{alltt}
 \end{small}
 
@@ -3334,42 +3347,41 @@ is copied to $b$, leading digits are removed and the remaining leading digit is
 \hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* calc a value mod 2**b */
-018   int
-019   mp_mod_2d (mp_int * a, int b, mp_int * c)
-020   \{
-021     int     x, res;
-022   
-023     /* if b is <= 0 then zero the int */
-024     if (b <= 0) \{
-025       mp_zero (c);
-026       return MP_OKAY;
-027     \}
-028   
-029     /* if the modulus is larger than the value than return */
-030     if (b > (int) (a->used * DIGIT_BIT)) \{
-031       res = mp_copy (a, c);
-032       return res;
-033     \}
-034   
-035     /* copy */
-036     if ((res = mp_copy (a, c)) != MP_OKAY) \{
-037       return res;
-038     \}
-039   
-040     /* zero digits above the last digit of the modulus */
-041     for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x+
+016   /* calc a value mod 2**b */
+017   int
+018   mp_mod_2d (mp_int * a, int b, mp_int * c)
+019   \{
+020     int     x, res;
+021   
+022     /* if b is <= 0 then zero the int */
+023     if (b <= 0) \{
+024       mp_zero (c);
+025       return MP_OKAY;
+026     \}
+027   
+028     /* if the modulus is larger than the value than return */
+029     if (b > (int) (a->used * DIGIT_BIT)) \{
+030       res = mp_copy (a, c);
+031       return res;
+032     \}
+033   
+034     /* copy */
+035     if ((res = mp_copy (a, c)) != MP_OKAY) \{
+036       return res;
+037     \}
+038   
+039     /* zero digits above the last digit of the modulus */
+040     for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x+
       +) \{
-042       c->dp[x] = 0;
-043     \}
-044     /* clear the digit that is not completely outside/inside the modulus */
-045     c->dp[b / DIGIT_BIT] &=
-046       (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi
+041       c->dp[x] = 0;
+042     \}
+043     /* clear the digit that is not completely outside/inside the modulus */
+044     c->dp[b / DIGIT_BIT] &=
+045       (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi
       t) 1));
-047     mp_clamp (c);
-048     return MP_OKAY;
-049   \}
+046     mp_clamp (c);
+047     return MP_OKAY;
+048   \}
 \end{alltt}
 \end{small}
 
@@ -3533,86 +3545,85 @@ exceed the precision requested.
 \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* multiplies |a| * |b| and only computes upto digs digits of result
-018    * HAC pp. 595, Algorithm 14.12  Modified so you can control how 
-019    * many digits of output are created.
-020    */
-021   int
-022   s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
-023   \{
-024     mp_int  t;
-025     int     res, pa, pb, ix, iy;
-026     mp_digit u;
-027     mp_word r;
-028     mp_digit tmpx, *tmpt, *tmpy;
-029   
-030     /* can we use the fast multiplier? */
-031     if (((digs) < MP_WARRAY) &&
-032         MIN (a->used, b->used) < 
-033             (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
-034       return fast_s_mp_mul_digs (a, b, c, digs);
-035     \}
-036   
-037     if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{
-038       return res;
-039     \}
-040     t.used = digs;
-041   
-042     /* compute the digits of the product directly */
-043     pa = a->used;
-044     for (ix = 0; ix < pa; ix++) \{
-045       /* set the carry to zero */
-046       u = 0;
-047   
-048       /* limit ourselves to making digs digits of output */
-049       pb = MIN (b->used, digs - ix);
-050   
-051       /* setup some aliases */
-052       /* copy of the digit from a used within the nested loop */
-053       tmpx = a->dp[ix];
-054       
-055       /* an alias for the destination shifted ix places */
-056       tmpt = t.dp + ix;
-057       
-058       /* an alias for the digits of b */
-059       tmpy = b->dp;
-060   
-061       /* compute the columns of the output and propagate the carry */
-062       for (iy = 0; iy < pb; iy++) \{
-063         /* compute the column as a mp_word */
-064         r       = ((mp_word)*tmpt) +
-065                   ((mp_word)tmpx) * ((mp_word)*tmpy++) +
-066                   ((mp_word) u);
-067   
-068         /* the new column is the lower part of the result */
-069         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
-070   
-071         /* get the carry word from the result */
-072         u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
-073       \}
-074       /* set carry if it is placed below digs */
-075       if (ix + iy < digs) \{
-076         *tmpt = u;
-077       \}
-078     \}
-079   
-080     mp_clamp (&t);
-081     mp_exch (&t, c);
-082   
-083     mp_clear (&t);
-084     return MP_OKAY;
-085   \}
+016   /* multiplies |a| * |b| and only computes upto digs digits of result
+017    * HAC pp. 595, Algorithm 14.12  Modified so you can control how 
+018    * many digits of output are created.
+019    */
+020   int
+021   s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
+022   \{
+023     mp_int  t;
+024     int     res, pa, pb, ix, iy;
+025     mp_digit u;
+026     mp_word r;
+027     mp_digit tmpx, *tmpt, *tmpy;
+028   
+029     /* can we use the fast multiplier? */
+030     if (((digs) < MP_WARRAY) &&
+031         MIN (a->used, b->used) < 
+032             (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
+033       return fast_s_mp_mul_digs (a, b, c, digs);
+034     \}
+035   
+036     if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{
+037       return res;
+038     \}
+039     t.used = digs;
+040   
+041     /* compute the digits of the product directly */
+042     pa = a->used;
+043     for (ix = 0; ix < pa; ix++) \{
+044       /* set the carry to zero */
+045       u = 0;
+046   
+047       /* limit ourselves to making digs digits of output */
+048       pb = MIN (b->used, digs - ix);
+049   
+050       /* setup some aliases */
+051       /* copy of the digit from a used within the nested loop */
+052       tmpx = a->dp[ix];
+053       
+054       /* an alias for the destination shifted ix places */
+055       tmpt = t.dp + ix;
+056       
+057       /* an alias for the digits of b */
+058       tmpy = b->dp;
+059   
+060       /* compute the columns of the output and propagate the carry */
+061       for (iy = 0; iy < pb; iy++) \{
+062         /* compute the column as a mp_word */
+063         r       = ((mp_word)*tmpt) +
+064                   ((mp_word)tmpx) * ((mp_word)*tmpy++) +
+065                   ((mp_word) u);
+066   
+067         /* the new column is the lower part of the result */
+068         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
+069   
+070         /* get the carry word from the result */
+071         u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
+072       \}
+073       /* set carry if it is placed below digs */
+074       if (ix + iy < digs) \{
+075         *tmpt = u;
+076       \}
+077     \}
+078   
+079     mp_clamp (&t);
+080     mp_exch (&t, c);
+081   
+082     mp_clear (&t);
+083     return MP_OKAY;
+084   \}
 \end{alltt}
 \end{small}
 
-Lines 31 to 35 determine if the Comba method can be used first.  The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and
+Lines 30 to 34 determine if the Comba method can be used first.  The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and
 the number of digits of output is less than \textbf{MP\_WARRAY}.  This new constant is used to control 
 the stack usage in the Comba routines.  By default it is set to $\delta$ but can be reduced when memory is at a premium.
 
-Of particular importance is the calculation of the $ix+iy$'th column on lines 64, 65 and 66.  Note how all of the
+Of particular importance is the calculation of the $ix+iy$'th column on lines 64, 65 and 65.  Note how all of the
 variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$.  That is to ensure that double precision operations 
-are used instead of single precision.  The multiplication on line 65 makes use of a specific GCC optimizer behaviour.  On the outset it looks like 
+are used instead of single precision.  The multiplication on line 64 makes use of a specific GCC optimizer behaviour.  On the outset it looks like 
 the compiler will have to use a double precision multiplication to produce the result required.  Such an operation would be horribly slow on most 
 processors and drag this to a crawl.  However, GCC is smart enough to realize that double wide output single precision multipliers can be used.  For 
 example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result.  
@@ -3786,129 +3797,128 @@ and addition operations in the nested loop in parallel.
 \hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* Fast (comba) multiplier
-018    *
-019    * This is the fast column-array [comba] multiplier.  It is 
-020    * designed to compute the columns of the product first 
-021    * then handle the carries afterwards.  This has the effect 
-022    * of making the nested loops that compute the columns very
-023    * simple and schedulable on super-scalar processors.
-024    *
-025    * This has been modified to produce a variable number of 
-026    * digits of output so if say only a half-product is required 
-027    * you don't have to compute the upper half (a feature 
-028    * required for fast Barrett reduction).
-029    *
-030    * Based on Algorithm 14.12 on pp.595 of HAC.
-031    *
-032    */
-033   int
-034   fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
-035   \{
-036     int     olduse, res, pa, ix;
-037     mp_word W[MP_WARRAY];
-038   
-039     /* grow the destination as required */
-040     if (c->alloc < digs) \{
-041       if ((res = mp_grow (c, digs)) != MP_OKAY) \{
-042         return res;
-043       \}
-044     \}
-045   
-046     /* clear temp buf (the columns) */
-047     memset (W, 0, sizeof (mp_word) * digs);
-048   
-049     /* calculate the columns */
-050     pa = a->used;
-051     for (ix = 0; ix < pa; ix++) \{
-052       /* this multiplier has been modified to allow you to 
-053        * control how many digits of output are produced.  
-054        * So at most we want to make upto "digs" digits of output.
-055        *
-056        * this adds products to distinct columns (at ix+iy) of W
-057        * note that each step through the loop is not dependent on
-058        * the previous which means the compiler can easily unroll
-059        * the loop without scheduling problems
-060        */
-061       \{
-062         register mp_digit tmpx, *tmpy;
-063         register mp_word *_W;
-064         register int iy, pb;
-065   
-066         /* alias for the the word on the left e.g. A[ix] * A[iy] */
-067         tmpx = a->dp[ix];
-068   
-069         /* alias for the right side */
-070         tmpy = b->dp;
-071   
-072         /* alias for the columns, each step through the loop adds a new
-073            term to each column
-074          */
-075         _W = W + ix;
-076   
-077         /* the number of digits is limited by their placement.  E.g.
-078            we avoid multiplying digits that will end up above the # of
-079            digits of precision requested
-080          */
-081         pb = MIN (b->used, digs - ix);
-082   
-083         for (iy = 0; iy < pb; iy++) \{
-084           *_W++ += ((mp_word)tmpx) * ((mp_word)*tmpy++);
-085         \}
-086       \}
-087   
-088     \}
-089   
-090     /* setup dest */
-091     olduse = c->used;
-092     c->used = digs;
-093   
-094     \{
-095       register mp_digit *tmpc;
-096   
-097       /* At this point W[] contains the sums of each column.  To get the
-098        * correct result we must take the extra bits from each column and
-099        * carry them down
-100        *
-101        * Note that while this adds extra code to the multiplier it 
-102        * saves time since the carry propagation is removed from the 
-103        * above nested loop.This has the effect of reducing the work 
-104        * from N*(N+N*c)==N**2 + c*N**2 to N**2 + N*c where c is the 
-105        * cost of the shifting.  On very small numbers this is slower 
-106        * but on most cryptographic size numbers it is faster.
-107        *
-108        * In this particular implementation we feed the carries from
-109        * behind which means when the loop terminates we still have one
-110        * last digit to copy
-111        */
-112       tmpc = c->dp;
-113       for (ix = 1; ix < digs; ix++) \{
-114         /* forward the carry from the previous temp */
-115         W[ix] += (W[ix - 1] >> ((mp_word) DIGIT_BIT));
-116   
-117         /* now extract the previous digit [below the carry] */
-118         *tmpc++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
-119       \}
-120       /* fetch the last digit */
-121       *tmpc++ = (mp_digit) (W[digs - 1] & ((mp_word) MP_MASK));
-122   
-123       /* clear unused digits [that existed in the old copy of c] */
-124       for (; ix < olduse; ix++) \{
-125         *tmpc++ = 0;
-126       \}
-127     \}
-128     mp_clamp (c);
-129     return MP_OKAY;
-130   \}
+016   /* Fast (comba) multiplier
+017    *
+018    * This is the fast column-array [comba] multiplier.  It is 
+019    * designed to compute the columns of the product first 
+020    * then handle the carries afterwards.  This has the effect 
+021    * of making the nested loops that compute the columns very
+022    * simple and schedulable on super-scalar processors.
+023    *
+024    * This has been modified to produce a variable number of 
+025    * digits of output so if say only a half-product is required 
+026    * you don't have to compute the upper half (a feature 
+027    * required for fast Barrett reduction).
+028    *
+029    * Based on Algorithm 14.12 on pp.595 of HAC.
+030    *
+031    */
+032   int
+033   fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
+034   \{
+035     int     olduse, res, pa, ix;
+036     mp_word W[MP_WARRAY];
+037   
+038     /* grow the destination as required */
+039     if (c->alloc < digs) \{
+040       if ((res = mp_grow (c, digs)) != MP_OKAY) \{
+041         return res;
+042       \}
+043     \}
+044   
+045     /* clear temp buf (the columns) */
+046     memset (W, 0, sizeof (mp_word) * digs);
+047   
+048     /* calculate the columns */
+049     pa = a->used;
+050     for (ix = 0; ix < pa; ix++) \{
+051       /* this multiplier has been modified to allow you to 
+052        * control how many digits of output are produced.  
+053        * So at most we want to make upto "digs" digits of output.
+054        *
+055        * this adds products to distinct columns (at ix+iy) of W
+056        * note that each step through the loop is not dependent on
+057        * the previous which means the compiler can easily unroll
+058        * the loop without scheduling problems
+059        */
+060       \{
+061         register mp_digit tmpx, *tmpy;
+062         register mp_word *_W;
+063         register int iy, pb;
+064   
+065         /* alias for the the word on the left e.g. A[ix] * A[iy] */
+066         tmpx = a->dp[ix];
+067   
+068         /* alias for the right side */
+069         tmpy = b->dp;
+070   
+071         /* alias for the columns, each step through the loop adds a new
+072            term to each column
+073          */
+074         _W = W + ix;
+075   
+076         /* the number of digits is limited by their placement.  E.g.
+077            we avoid multiplying digits that will end up above the # of
+078            digits of precision requested
+079          */
+080         pb = MIN (b->used, digs - ix);
+081   
+082         for (iy = 0; iy < pb; iy++) \{
+083           *_W++ += ((mp_word)tmpx) * ((mp_word)*tmpy++);
+084         \}
+085       \}
+086   
+087     \}
+088   
+089     /* setup dest */
+090     olduse  = c->used;
+091     c->used = digs;
+092   
+093     \{
+094       register mp_digit *tmpc;
+095   
+096       /* At this point W[] contains the sums of each column.  To get the
+097        * correct result we must take the extra bits from each column and
+098        * carry them down
+099        *
+100        * Note that while this adds extra code to the multiplier it 
+101        * saves time since the carry propagation is removed from the 
+102        * above nested loop.This has the effect of reducing the work 
+103        * from N*(N+N*c)==N**2 + c*N**2 to N**2 + N*c where c is the 
+104        * cost of the shifting.  On very small numbers this is slower 
+105        * but on most cryptographic size numbers it is faster.
+106        *
+107        * In this particular implementation we feed the carries from
+108        * behind which means when the loop terminates we still have one
+109        * last digit to copy
+110        */
+111       tmpc = c->dp;
+112       for (ix = 1; ix < digs; ix++) \{
+113         /* forward the carry from the previous temp */
+114         W[ix] += (W[ix - 1] >> ((mp_word) DIGIT_BIT));
+115   
+116         /* now extract the previous digit [below the carry] */
+117         *tmpc++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
+118       \}
+119       /* fetch the last digit */
+120       *tmpc++ = (mp_digit) (W[digs - 1] & ((mp_word) MP_MASK));
+121   
+122       /* clear unused digits [that existed in the old copy of c] */
+123       for (; ix < olduse; ix++) \{
+124         *tmpc++ = 0;
+125       \}
+126     \}
+127     mp_clamp (c);
+128     return MP_OKAY;
+129   \}
 \end{alltt}
 \end{small}
 
-The memset on line 47 clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication
-implementation a series of aliases (\textit{lines 67, 70 and 75}) are used to simplify the inner $O(n^2)$ loop.  
+The memset on line 46 clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication
+implementation a series of aliases (\textit{lines 66, 69 and 74}) are used to simplify the inner $O(n^2)$ loop.  
 In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass.  
 
-The inner loop on lines 83, 84 and 85 is where the algorithm will spend the majority of the time, which is why it has been 
+The inner loop on lines 82, 83 and 85 is where the algorithm will spend the majority of the time, which is why it has been 
 stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}.  On x86 processors the multiplication and additions amount to at the 
 very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three 
 (\textit{one load, one store, one multiply-add}).   For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop 
@@ -4105,161 +4115,157 @@ The remaining steps 13 through 18 compute the Karatsuba polynomial through a var
 \hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* c = |a| * |b| using Karatsuba Multiplication using 
-018    * three half size multiplications
-019    *
-020    * Let B represent the radix [e.g. 2**DIGIT_BIT] and 
-021    * let n represent half of the number of digits in 
-022    * the min(a,b)
-023    *
-024    * a = a1 * B**n + a0
-025    * b = b1 * B**n + b0
-026    *
-027    * Then, a * b => 
-028      a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0
-029    *
-030    * Note that a1b1 and a0b0 are used twice and only need to be 
-031    * computed once.  So in total three half size (half # of 
-032    * digit) multiplications are performed, a0b0, a1b1 and 
-033    * (a1-b1)(a0-b0)
-034    *
-035    * Note that a multiplication of half the digits requires
-036    * 1/4th the number of single precision multiplications so in 
-037    * total after one call 25% of the single precision multiplications 
-038    * are saved.  Note also that the call to mp_mul can end up back 
-039    * in this function if the a0, a1, b0, or b1 are above the threshold.  
-040    * This is known as divide-and-conquer and leads to the famous 
-041    * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than 
-042    * the standard O(N**2) that the baseline/comba methods use.  
-043    * Generally though the overhead of this method doesn't pay off 
-044    * until a certain size (N ~ 80) is reached.
-045    */
-046   int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
-047   \{
-048     mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
-049     int     B, err;
-050   
-051     /* default the return code to an error */
-052     err = MP_MEM;
-053   
-054     /* min # of digits */
-055     B = MIN (a->used, b->used);
-056   
-057     /* now divide in two */
-058     B = B >> 1;
-059   
-060     /* init copy all the temps */
-061     if (mp_init_size (&x0, B) != MP_OKAY)
-062       goto ERR;
-063     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
-064       goto X0;
-065     if (mp_init_size (&y0, B) != MP_OKAY)
-066       goto X1;
-067     if (mp_init_size (&y1, b->used - B) != MP_OKAY)
-068       goto Y0;
-069   
-070     /* init temps */
-071     if (mp_init_size (&t1, B * 2) != MP_OKAY)
-072       goto Y1;
-073     if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
-074       goto T1;
-075     if (mp_init_size (&x1y1, B * 2) != MP_OKAY)
-076       goto X0Y0;
-077   
-078     /* now shift the digits */
-079     x0.sign = x1.sign = a->sign;
-080     y0.sign = y1.sign = b->sign;
+016   /* c = |a| * |b| using Karatsuba Multiplication using 
+017    * three half size multiplications
+018    *
+019    * Let B represent the radix [e.g. 2**DIGIT_BIT] and 
+020    * let n represent half of the number of digits in 
+021    * the min(a,b)
+022    *
+023    * a = a1 * B**n + a0
+024    * b = b1 * B**n + b0
+025    *
+026    * Then, a * b => 
+027      a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0
+028    *
+029    * Note that a1b1 and a0b0 are used twice and only need to be 
+030    * computed once.  So in total three half size (half # of 
+031    * digit) multiplications are performed, a0b0, a1b1 and 
+032    * (a1-b1)(a0-b0)
+033    *
+034    * Note that a multiplication of half the digits requires
+035    * 1/4th the number of single precision multiplications so in 
+036    * total after one call 25% of the single precision multiplications 
+037    * are saved.  Note also that the call to mp_mul can end up back 
+038    * in this function if the a0, a1, b0, or b1 are above the threshold.  
+039    * This is known as divide-and-conquer and leads to the famous 
+040    * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than 
+041    * the standard O(N**2) that the baseline/comba methods use.  
+042    * Generally though the overhead of this method doesn't pay off 
+043    * until a certain size (N ~ 80) is reached.
+044    */
+045   int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
+046   \{
+047     mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
+048     int     B, err;
+049   
+050     /* default the return code to an error */
+051     err = MP_MEM;
+052   
+053     /* min # of digits */
+054     B = MIN (a->used, b->used);
+055   
+056     /* now divide in two */
+057     B = B >> 1;
+058   
+059     /* init copy all the temps */
+060     if (mp_init_size (&x0, B) != MP_OKAY)
+061       goto ERR;
+062     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
+063       goto X0;
+064     if (mp_init_size (&y0, B) != MP_OKAY)
+065       goto X1;
+066     if (mp_init_size (&y1, b->used - B) != MP_OKAY)
+067       goto Y0;
+068   
+069     /* init temps */
+070     if (mp_init_size (&t1, B * 2) != MP_OKAY)
+071       goto Y1;
+072     if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
+073       goto T1;
+074     if (mp_init_size (&x1y1, B * 2) != MP_OKAY)
+075       goto X0Y0;
+076   
+077     /* now shift the digits */
+078     x0.used = y0.used = B;
+079     x1.used = a->used - B;
+080     y1.used = b->used - B;
 081   
-082     x0.used = y0.used = B;
-083     x1.used = a->used - B;
-084     y1.used = b->used - B;
+082     \{
+083       register int x;
+084       register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
 085   
-086     \{
-087       register int x;
-088       register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
-089   
-090       /* we copy the digits directly instead of using higher level functions
-091        * since we also need to shift the digits
-092        */
-093       tmpa = a->dp;
-094       tmpb = b->dp;
-095   
-096       tmpx = x0.dp;
-097       tmpy = y0.dp;
-098       for (x = 0; x < B; x++) \{
-099         *tmpx++ = *tmpa++;
-100         *tmpy++ = *tmpb++;
-101       \}
-102   
-103       tmpx = x1.dp;
-104       for (x = B; x < a->used; x++) \{
-105         *tmpx++ = *tmpa++;
-106       \}
-107   
-108       tmpy = y1.dp;
-109       for (x = B; x < b->used; x++) \{
-110         *tmpy++ = *tmpb++;
-111       \}
-112     \}
-113   
-114     /* only need to clamp the lower words since by definition the 
-115      * upper words x1/y1 must have a known number of digits
-116      */
-117     mp_clamp (&x0);
-118     mp_clamp (&y0);
-119   
-120     /* now calc the products x0y0 and x1y1 */
-121     /* after this x0 is no longer required, free temp [x0==t2]! */
-122     if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)  
-123       goto X1Y1;          /* x0y0 = x0*y0 */
-124     if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
-125       goto X1Y1;          /* x1y1 = x1*y1 */
-126   
-127     /* now calc x1-x0 and y1-y0 */
-128     if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
-129       goto X1Y1;          /* t1 = x1 - x0 */
-130     if (mp_sub (&y1, &y0, &x0) != MP_OKAY)
-131       goto X1Y1;          /* t2 = y1 - y0 */
-132     if (mp_mul (&t1, &x0, &t1) != MP_OKAY)
-133       goto X1Y1;          /* t1 = (x1 - x0) * (y1 - y0) */
-134   
-135     /* add x0y0 */
-136     if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY)
-137       goto X1Y1;          /* t2 = x0y0 + x1y1 */
-138     if (mp_sub (&x0, &t1, &t1) != MP_OKAY)
-139       goto X1Y1;          /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */
-140   
-141     /* shift by B */
-142     if (mp_lshd (&t1, B) != MP_OKAY)
-143       goto X1Y1;          /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
-144     if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
-145       goto X1Y1;          /* x1y1 = x1y1 << 2*B */
-146   
-147     if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
-148       goto X1Y1;          /* t1 = x0y0 + t1 */
-149     if (mp_add (&t1, &x1y1, c) != MP_OKAY)
-150       goto X1Y1;          /* t1 = x0y0 + t1 + x1y1 */
-151   
-152     /* Algorithm succeeded set the return code to MP_OKAY */
-153     err = MP_OKAY;
-154   
-155   X1Y1:mp_clear (&x1y1);
-156   X0Y0:mp_clear (&x0y0);
-157   T1:mp_clear (&t1);
-158   Y1:mp_clear (&y1);
-159   Y0:mp_clear (&y0);
-160   X1:mp_clear (&x1);
-161   X0:mp_clear (&x0);
-162   ERR:
-163     return err;
-164   \}
+086       /* we copy the digits directly instead of using higher level functions
+087        * since we also need to shift the digits
+088        */
+089       tmpa = a->dp;
+090       tmpb = b->dp;
+091   
+092       tmpx = x0.dp;
+093       tmpy = y0.dp;
+094       for (x = 0; x < B; x++) \{
+095         *tmpx++ = *tmpa++;
+096         *tmpy++ = *tmpb++;
+097       \}
+098   
+099       tmpx = x1.dp;
+100       for (x = B; x < a->used; x++) \{
+101         *tmpx++ = *tmpa++;
+102       \}
+103   
+104       tmpy = y1.dp;
+105       for (x = B; x < b->used; x++) \{
+106         *tmpy++ = *tmpb++;
+107       \}
+108     \}
+109   
+110     /* only need to clamp the lower words since by definition the 
+111      * upper words x1/y1 must have a known number of digits
+112      */
+113     mp_clamp (&x0);
+114     mp_clamp (&y0);
+115   
+116     /* now calc the products x0y0 and x1y1 */
+117     /* after this x0 is no longer required, free temp [x0==t2]! */
+118     if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)  
+119       goto X1Y1;          /* x0y0 = x0*y0 */
+120     if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
+121       goto X1Y1;          /* x1y1 = x1*y1 */
+122   
+123     /* now calc x1-x0 and y1-y0 */
+124     if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
+125       goto X1Y1;          /* t1 = x1 - x0 */
+126     if (mp_sub (&y1, &y0, &x0) != MP_OKAY)
+127       goto X1Y1;          /* t2 = y1 - y0 */
+128     if (mp_mul (&t1, &x0, &t1) != MP_OKAY)
+129       goto X1Y1;          /* t1 = (x1 - x0) * (y1 - y0) */
+130   
+131     /* add x0y0 */
+132     if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY)
+133       goto X1Y1;          /* t2 = x0y0 + x1y1 */
+134     if (mp_sub (&x0, &t1, &t1) != MP_OKAY)
+135       goto X1Y1;          /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */
+136   
+137     /* shift by B */
+138     if (mp_lshd (&t1, B) != MP_OKAY)
+139       goto X1Y1;          /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
+140     if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
+141       goto X1Y1;          /* x1y1 = x1y1 << 2*B */
+142   
+143     if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
+144       goto X1Y1;          /* t1 = x0y0 + t1 */
+145     if (mp_add (&t1, &x1y1, c) != MP_OKAY)
+146       goto X1Y1;          /* t1 = x0y0 + t1 + x1y1 */
+147   
+148     /* Algorithm succeeded set the return code to MP_OKAY */
+149     err = MP_OKAY;
+150   
+151   X1Y1:mp_clear (&x1y1);
+152   X0Y0:mp_clear (&x0y0);
+153   T1:mp_clear (&t1);
+154   Y1:mp_clear (&y1);
+155   Y0:mp_clear (&y0);
+156   X1:mp_clear (&x1);
+157   X0:mp_clear (&x0);
+158   ERR:
+159     return err;
+160   \}
 \end{alltt}
 \end{small}
 
 The new coding element in this routine, not  seen in previous routines, is the usage of goto statements.  The conventional
 wisdom is that goto statements should be avoided.  This is generally true, however when every single function call can fail, it makes sense
-to handle error recovery with a single piece of code.  Lines 61 to 75 handle initializing all of the temporary variables 
+to handle error recovery with a single piece of code.  Lines 62 to 74 handle initializing all of the temporary variables 
 required.  Note how each of the if statements goes to a different label in case of failure.  This allows the routine to correctly free only
 the temporaries that have been successfully allocated so far.
 
@@ -4269,13 +4275,13 @@ number of digits for the next section of code.
 
 The first algebraic portion of the algorithm is to split the two inputs into their halves.  However, instead of using mp\_mod\_2d and mp\_rshd
 to extract the halves, the respective code has been placed inline within the body of the function.  To initialize the halves, the \textbf{used} and 
-\textbf{sign} members are copied first.  The first for loop on line 98 copies the lower halves.  Since they are both the same magnitude it 
-is simpler to calculate both lower halves in a single loop.  The for loop on lines 104 and 109 calculate the upper halves $x1$ and 
+\textbf{sign} members are copied first.  The first for loop on line 100 copies the lower halves.  Since they are both the same magnitude it 
+is simpler to calculate both lower halves in a single loop.  The for loop on lines 105 and 105 calculate the upper halves $x1$ and 
 $y1$ respectively.
 
 By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.
 
-When line 153 is reached, the algorithm has completed succesfully.  The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
+When line 149 is reached, the algorithm has completed succesfully.  The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
 the same code that handles errors can be used to clear the temporary variables and return.  
 
 \subsection{Toom-Cook $3$-Way Multiplication}
@@ -4393,263 +4399,262 @@ result $a \cdot b$ is produced.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* multiplication using the Toom-Cook 3-way algorithm */
-018   int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
-019   \{
-020       mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
-021       int res, B;
-022           
-023       /* init temps */
-024       if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, 
-025                                &a0, &a1, &a2, &b0, &b1, 
-026                                &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{
-027          return res;
-028       \}
-029       
-030       /* B */
-031       B = MIN(a->used, b->used) / 3;
-032       
-033       /* a = a2 * B**2 + a1 * B + a0 */
-034       if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{
-035          goto ERR;
-036       \}
-037   
-038       if ((res = mp_copy(a, &a1)) != MP_OKAY) \{
-039          goto ERR;
-040       \}
-041       mp_rshd(&a1, B);
-042       mp_mod_2d(&a1, DIGIT_BIT * B, &a1);
-043   
-044       if ((res = mp_copy(a, &a2)) != MP_OKAY) \{
-045          goto ERR;
-046       \}
-047       mp_rshd(&a2, B*2);
-048       
-049       /* b = b2 * B**2 + b1 * B + b0 */
-050       if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{
-051          goto ERR;
-052       \}
-053   
-054       if ((res = mp_copy(b, &b1)) != MP_OKAY) \{
-055          goto ERR;
-056       \}
-057       mp_rshd(&b1, B);
-058       mp_mod_2d(&b1, DIGIT_BIT * B, &b1);
-059   
-060       if ((res = mp_copy(b, &b2)) != MP_OKAY) \{
-061          goto ERR;
-062       \}
-063       mp_rshd(&b2, B*2);
-064       
-065       /* w0 = a0*b0 */
-066       if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{
-067          goto ERR;
-068       \}
-069       
-070       /* w4 = a2 * b2 */
-071       if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{
-072          goto ERR;
-073       \}
-074       
-075       /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
-076       if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{
-077          goto ERR;
-078       \}
-079       if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
-080          goto ERR;
-081       \}
-082       if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
-083          goto ERR;
-084       \}
-085       if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{
-086          goto ERR;
-087       \}
-088       
-089       if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{
-090          goto ERR;
-091       \}
-092       if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
-093          goto ERR;
-094       \}
-095       if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
-096          goto ERR;
-097       \}
-098       if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{
-099          goto ERR;
-100       \}
-101       
-102       if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{
-103          goto ERR;
-104       \}
-105       
-106       /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
-107       if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{
-108          goto ERR;
-109       \}
-110       if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
-111          goto ERR;
-112       \}
-113       if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
-114          goto ERR;
-115       \}
-116       if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
-117          goto ERR;
-118       \}
-119       
-120       if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{
-121          goto ERR;
-122       \}
-123       if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
-124          goto ERR;
-125       \}
-126       if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
-127          goto ERR;
-128       \}
-129       if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
-130          goto ERR;
-131       \}
-132       
-133       if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{
-134          goto ERR;
-135       \}
-136       
-137   
-138       /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
-139       if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{
-140          goto ERR;
-141       \}
-142       if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
-143          goto ERR;
-144       \}
-145       if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{
-146          goto ERR;
-147       \}
-148       if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
-149          goto ERR;
-150       \}
-151       if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{
-152          goto ERR;
-153       \}
-154       
-155       /* now solve the matrix 
-156       
-157          0  0  0  0  1
-158          1  2  4  8  16
-159          1  1  1  1  1
-160          16 8  4  2  1
-161          1  0  0  0  0
-162          
-163          using 12 subtractions, 4 shifts, 
-164                 2 small divisions and 1 small multiplication 
-165        */
-166        
-167        /* r1 - r4 */
-168        if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{
-169           goto ERR;
-170        \}
-171        /* r3 - r0 */
-172        if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{
-173           goto ERR;
-174        \}
-175        /* r1/2 */
-176        if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{
-177           goto ERR;
-178        \}
-179        /* r3/2 */
-180        if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{
-181           goto ERR;
-182        \}
-183        /* r2 - r0 - r4 */
-184        if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{
-185           goto ERR;
-186        \}
-187        if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{
-188           goto ERR;
-189        \}
-190        /* r1 - r2 */
-191        if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
-192           goto ERR;
-193        \}
-194        /* r3 - r2 */
-195        if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
-196           goto ERR;
-197        \}
-198        /* r1 - 8r0 */
-199        if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{
-200           goto ERR;
-201        \}
-202        if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{
-203           goto ERR;
-204        \}
-205        /* r3 - 8r4 */
-206        if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{
-207           goto ERR;
-208        \}
-209        if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{
-210           goto ERR;
-211        \}
-212        /* 3r2 - r1 - r3 */
-213        if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{
-214           goto ERR;
-215        \}
-216        if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{
-217           goto ERR;
-218        \}
-219        if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{
-220           goto ERR;
-221        \}
-222        /* r1 - r2 */
-223        if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
-224           goto ERR;
-225        \}
-226        /* r3 - r2 */
-227        if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
-228           goto ERR;
-229        \}
-230        /* r1/3 */
-231        if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{
-232           goto ERR;
-233        \}
-234        /* r3/3 */
-235        if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{
-236           goto ERR;
-237        \}
-238        
-239        /* at this point shift W[n] by B*n */
-240        if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{
-241           goto ERR;
-242        \}
-243        if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{
-244           goto ERR;
-245        \}
-246        if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{
-247           goto ERR;
-248        \}
-249        if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{
-250           goto ERR;
-251        \}     
-252        
-253        if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{
-254           goto ERR;
-255        \}
-256        if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{
-257           goto ERR;
-258        \}
-259        if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{
-260           goto ERR;
-261        \}
-262        if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{
-263           goto ERR;
-264        \}     
-265        
-266   ERR:
-267        mp_clear_multi(&w0, &w1, &w2, &w3, &w4, 
-268                       &a0, &a1, &a2, &b0, &b1, 
-269                       &b2, &tmp1, &tmp2, NULL);
-270        return res;
-271   \}     
-272        
+016   /* multiplication using the Toom-Cook 3-way algorithm */
+017   int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
+018   \{
+019       mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
+020       int res, B;
+021           
+022       /* init temps */
+023       if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, 
+024                                &a0, &a1, &a2, &b0, &b1, 
+025                                &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{
+026          return res;
+027       \}
+028       
+029       /* B */
+030       B = MIN(a->used, b->used) / 3;
+031       
+032       /* a = a2 * B**2 + a1 * B + a0 */
+033       if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{
+034          goto ERR;
+035       \}
+036   
+037       if ((res = mp_copy(a, &a1)) != MP_OKAY) \{
+038          goto ERR;
+039       \}
+040       mp_rshd(&a1, B);
+041       mp_mod_2d(&a1, DIGIT_BIT * B, &a1);
+042   
+043       if ((res = mp_copy(a, &a2)) != MP_OKAY) \{
+044          goto ERR;
+045       \}
+046       mp_rshd(&a2, B*2);
+047       
+048       /* b = b2 * B**2 + b1 * B + b0 */
+049       if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{
+050          goto ERR;
+051       \}
+052   
+053       if ((res = mp_copy(b, &b1)) != MP_OKAY) \{
+054          goto ERR;
+055       \}
+056       mp_rshd(&b1, B);
+057       mp_mod_2d(&b1, DIGIT_BIT * B, &b1);
+058   
+059       if ((res = mp_copy(b, &b2)) != MP_OKAY) \{
+060          goto ERR;
+061       \}
+062       mp_rshd(&b2, B*2);
+063       
+064       /* w0 = a0*b0 */
+065       if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{
+066          goto ERR;
+067       \}
+068       
+069       /* w4 = a2 * b2 */
+070       if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{
+071          goto ERR;
+072       \}
+073       
+074       /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
+075       if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{
+076          goto ERR;
+077       \}
+078       if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
+079          goto ERR;
+080       \}
+081       if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
+082          goto ERR;
+083       \}
+084       if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{
+085          goto ERR;
+086       \}
+087       
+088       if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{
+089          goto ERR;
+090       \}
+091       if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
+092          goto ERR;
+093       \}
+094       if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
+095          goto ERR;
+096       \}
+097       if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{
+098          goto ERR;
+099       \}
+100       
+101       if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{
+102          goto ERR;
+103       \}
+104       
+105       /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
+106       if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{
+107          goto ERR;
+108       \}
+109       if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
+110          goto ERR;
+111       \}
+112       if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
+113          goto ERR;
+114       \}
+115       if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
+116          goto ERR;
+117       \}
+118       
+119       if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{
+120          goto ERR;
+121       \}
+122       if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
+123          goto ERR;
+124       \}
+125       if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
+126          goto ERR;
+127       \}
+128       if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
+129          goto ERR;
+130       \}
+131       
+132       if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{
+133          goto ERR;
+134       \}
+135       
+136   
+137       /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
+138       if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{
+139          goto ERR;
+140       \}
+141       if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
+142          goto ERR;
+143       \}
+144       if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{
+145          goto ERR;
+146       \}
+147       if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
+148          goto ERR;
+149       \}
+150       if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{
+151          goto ERR;
+152       \}
+153       
+154       /* now solve the matrix 
+155       
+156          0  0  0  0  1
+157          1  2  4  8  16
+158          1  1  1  1  1
+159          16 8  4  2  1
+160          1  0  0  0  0
+161          
+162          using 12 subtractions, 4 shifts, 
+163                 2 small divisions and 1 small multiplication 
+164        */
+165        
+166        /* r1 - r4 */
+167        if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{
+168           goto ERR;
+169        \}
+170        /* r3 - r0 */
+171        if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{
+172           goto ERR;
+173        \}
+174        /* r1/2 */
+175        if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{
+176           goto ERR;
+177        \}
+178        /* r3/2 */
+179        if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{
+180           goto ERR;
+181        \}
+182        /* r2 - r0 - r4 */
+183        if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{
+184           goto ERR;
+185        \}
+186        if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{
+187           goto ERR;
+188        \}
+189        /* r1 - r2 */
+190        if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
+191           goto ERR;
+192        \}
+193        /* r3 - r2 */
+194        if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
+195           goto ERR;
+196        \}
+197        /* r1 - 8r0 */
+198        if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{
+199           goto ERR;
+200        \}
+201        if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{
+202           goto ERR;
+203        \}
+204        /* r3 - 8r4 */
+205        if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{
+206           goto ERR;
+207        \}
+208        if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{
+209           goto ERR;
+210        \}
+211        /* 3r2 - r1 - r3 */
+212        if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{
+213           goto ERR;
+214        \}
+215        if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{
+216           goto ERR;
+217        \}
+218        if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{
+219           goto ERR;
+220        \}
+221        /* r1 - r2 */
+222        if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
+223           goto ERR;
+224        \}
+225        /* r3 - r2 */
+226        if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
+227           goto ERR;
+228        \}
+229        /* r1/3 */
+230        if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{
+231           goto ERR;
+232        \}
+233        /* r3/3 */
+234        if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{
+235           goto ERR;
+236        \}
+237        
+238        /* at this point shift W[n] by B*n */
+239        if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{
+240           goto ERR;
+241        \}
+242        if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{
+243           goto ERR;
+244        \}
+245        if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{
+246           goto ERR;
+247        \}
+248        if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{
+249           goto ERR;
+250        \}     
+251        
+252        if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{
+253           goto ERR;
+254        \}
+255        if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{
+256           goto ERR;
+257        \}
+258        if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{
+259           goto ERR;
+260        \}
+261        if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{
+262           goto ERR;
+263        \}     
+264        
+265   ERR:
+266        mp_clear_multi(&w0, &w1, &w2, &w3, &w4, 
+267                       &a0, &a1, &a2, &b0, &b1, 
+268                       &b2, &tmp1, &tmp2, NULL);
+269        return res;
+270   \}     
+271        
 \end{alltt}
 \end{small}
 
@@ -4699,44 +4704,43 @@ s\_mp\_mul\_digs will clear it.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* high level multiplication (handles sign) */
-018   int mp_mul (mp_int * a, mp_int * b, mp_int * c)
-019   \{
-020     int     res, neg;
-021     neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
-022   
-023     /* use Toom-Cook? */
-024     if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{
-025       res = mp_toom_mul(a, b, c);
-026     /* use Karatsuba? */
-027     \} else if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{
-028       res = mp_karatsuba_mul (a, b, c);
-029     \} else \{
-030       /* can we use the fast multiplier?
-031        *
-032        * The fast multiplier can be used if the output will 
-033        * have less than MP_WARRAY digits and the number of 
-034        * digits won't affect carry propagation
-035        */
-036       int     digs = a->used + b->used + 1;
-037   
-038       if ((digs < MP_WARRAY) &&
-039           MIN(a->used, b->used) <= 
-040           (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
-041         res = fast_s_mp_mul_digs (a, b, c, digs);
-042       \} else \{
-043         res = s_mp_mul (a, b, c);
-044       \}
-045     \}
-046     c->sign = neg;
-047     return res;
-048   \}
+016   /* high level multiplication (handles sign) */
+017   int mp_mul (mp_int * a, mp_int * b, mp_int * c)
+018   \{
+019     int     res, neg;
+020     neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
+021   
+022     /* use Toom-Cook? */
+023     if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{
+024       res = mp_toom_mul(a, b, c);
+025     /* use Karatsuba? */
+026     \} else if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{
+027       res = mp_karatsuba_mul (a, b, c);
+028     \} else \{
+029       /* can we use the fast multiplier?
+030        *
+031        * The fast multiplier can be used if the output will 
+032        * have less than MP_WARRAY digits and the number of 
+033        * digits won't affect carry propagation
+034        */
+035       int     digs = a->used + b->used + 1;
+036   
+037       if ((digs < MP_WARRAY) &&
+038           MIN(a->used, b->used) <= 
+039           (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
+040         res = fast_s_mp_mul_digs (a, b, c, digs);
+041       \} else \{
+042         res = s_mp_mul (a, b, c);
+043       \}
+044     \}
+045     c->sign = (c->used > 0) ? neg : MP_ZPOS;
+046     return res;
+047   \}
 \end{alltt}
 \end{small}
 
-The implementation is rather simplistic and is not particularly noteworthy.  Line 23 computes the sign of the result using the ``?'' 
-operator from the C programming language.  Line 40 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.  
+The implementation is rather simplistic and is not particularly noteworthy.  Line 22 computes the sign of the result using the ``?'' 
+operator from the C programming language.  Line 39 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.  
 
 \section{Squaring}
 \label{sec:basesquare}
@@ -4837,76 +4841,75 @@ results calculated so far.  This involves expensive carry propagation which will
 \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
-018   int
-019   s_mp_sqr (mp_int * a, mp_int * b)
-020   \{
-021     mp_int  t;
-022     int     res, ix, iy, pa;
-023     mp_word r;
-024     mp_digit u, tmpx, *tmpt;
-025   
-026     pa = a->used;
-027     if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) \{
-028       return res;
-029     \}
-030   
-031     /* default used is maximum possible size */
-032     t.used = 2*pa + 1;
-033   
-034     for (ix = 0; ix < pa; ix++) \{
-035       /* first calculate the digit at 2*ix */
-036       /* calculate double precision result */
-037       r = ((mp_word) t.dp[2*ix]) +
-038           ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]);
-039   
-040       /* store lower part in result */
-041       t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK));
-042   
-043       /* get the carry */
-044       u           = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
-045   
-046       /* left hand side of A[ix] * A[iy] */
-047       tmpx        = a->dp[ix];
-048   
-049       /* alias for where to store the results */
-050       tmpt        = t.dp + (2*ix + 1);
-051       
-052       for (iy = ix + 1; iy < pa; iy++) \{
-053         /* first calculate the product */
-054         r       = ((mp_word)tmpx) * ((mp_word)a->dp[iy]);
-055   
-056         /* now calculate the double precision result, note we use
-057          * addition instead of *2 since it's easier to optimize
-058          */
-059         r       = ((mp_word) *tmpt) + r + r + ((mp_word) u);
-060   
-061         /* store lower part */
-062         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
-063   
-064         /* get carry */
-065         u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
-066       \}
-067       /* propagate upwards */
-068       while (u != ((mp_digit) 0)) \{
-069         r       = ((mp_word) *tmpt) + ((mp_word) u);
-070         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
-071         u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
-072       \}
-073     \}
-074   
-075     mp_clamp (&t);
-076     mp_exch (&t, b);
-077     mp_clear (&t);
-078     return MP_OKAY;
-079   \}
+016   /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
+017   int
+018   s_mp_sqr (mp_int * a, mp_int * b)
+019   \{
+020     mp_int  t;
+021     int     res, ix, iy, pa;
+022     mp_word r;
+023     mp_digit u, tmpx, *tmpt;
+024   
+025     pa = a->used;
+026     if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) \{
+027       return res;
+028     \}
+029   
+030     /* default used is maximum possible size */
+031     t.used = 2*pa + 1;
+032   
+033     for (ix = 0; ix < pa; ix++) \{
+034       /* first calculate the digit at 2*ix */
+035       /* calculate double precision result */
+036       r = ((mp_word) t.dp[2*ix]) +
+037           ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]);
+038   
+039       /* store lower part in result */
+040       t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK));
+041   
+042       /* get the carry */
+043       u           = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
+044   
+045       /* left hand side of A[ix] * A[iy] */
+046       tmpx        = a->dp[ix];
+047   
+048       /* alias for where to store the results */
+049       tmpt        = t.dp + (2*ix + 1);
+050       
+051       for (iy = ix + 1; iy < pa; iy++) \{
+052         /* first calculate the product */
+053         r       = ((mp_word)tmpx) * ((mp_word)a->dp[iy]);
+054   
+055         /* now calculate the double precision result, note we use
+056          * addition instead of *2 since it's easier to optimize
+057          */
+058         r       = ((mp_word) *tmpt) + r + r + ((mp_word) u);
+059   
+060         /* store lower part */
+061         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
+062   
+063         /* get carry */
+064         u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
+065       \}
+066       /* propagate upwards */
+067       while (u != ((mp_digit) 0)) \{
+068         r       = ((mp_word) *tmpt) + ((mp_word) u);
+069         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
+070         u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
+071       \}
+072     \}
+073   
+074     mp_clamp (&t);
+075     mp_exch (&t, b);
+076     mp_clear (&t);
+077     return MP_OKAY;
+078   \}
 \end{alltt}
 \end{small}
 
-Inside the outer loop (\textit{see line 34}) the square term is calculated on line 37.  Line 44 extracts the carry from the square
-term.  Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines 47 and 50 respectively.  The doubling is performed using two
-additions (\textit{see line 59}) since it is usually faster than shifting,if not at least as fast.  
+Inside the outer loop (\textit{see line 33}) the square term is calculated on line 36.  Line 43 extracts the carry from the square
+term.  Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines 46 and 49 respectively.  The doubling is performed using two
+additions (\textit{see line 58}) since it is usually faster than shifting,if not at least as fast.  
 
 \subsection{Faster Squaring by the ``Comba'' Method}
 A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop.  Squaring has an additional
@@ -4985,130 +4988,129 @@ squares in place.
 \hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* fast squaring
-018    *
-019    * This is the comba method where the columns of the product
-020    * are computed first then the carries are computed.  This
-021    * has the effect of making a very simple inner loop that
-022    * is executed the most
-023    *
-024    * W2 represents the outer products and W the inner.
-025    *
-026    * A further optimizations is made because the inner
-027    * products are of the form "A * B * 2".  The *2 part does
-028    * not need to be computed until the end which is good
-029    * because 64-bit shifts are slow!
-030    *
-031    * Based on Algorithm 14.16 on pp.597 of HAC.
-032    *
-033    */
-034   int fast_s_mp_sqr (mp_int * a, mp_int * b)
-035   \{
-036     int     olduse, newused, res, ix, pa;
-037     mp_word W2[MP_WARRAY], W[MP_WARRAY];
-038   
-039     /* calculate size of product and allocate as required */
-040     pa = a->used;
-041     newused = pa + pa + 1;
-042     if (b->alloc < newused) \{
-043       if ((res = mp_grow (b, newused)) != MP_OKAY) \{
-044         return res;
-045       \}
-046     \}
-047   
-048     /* zero temp buffer (columns)
-049      * Note that there are two buffers.  Since squaring requires
-050      * a outer and inner product and the inner product requires
-051      * computing a product and doubling it (a relatively expensive
-052      * op to perform n**2 times if you don't have to) the inner and
-053      * outer products are computed in different buffers.  This way
-054      * the inner product can be doubled using n doublings instead of
-055      * n**2
-056      */
-057     memset (W,  0, newused * sizeof (mp_word));
-058     memset (W2, 0, newused * sizeof (mp_word));
-059   
-060     /* This computes the inner product.  To simplify the inner N**2 loop
-061      * the multiplication by two is done afterwards in the N loop.
-062      */
-063     for (ix = 0; ix < pa; ix++) \{
-064       /* compute the outer product
-065        *
-066        * Note that every outer product is computed
-067        * for a particular column only once which means that
-068        * there is no need todo a double precision addition
-069        * into the W2[] array.
-070        */
-071       W2[ix + ix] = ((mp_word)a->dp[ix]) * ((mp_word)a->dp[ix]);
-072   
-073       \{
-074         register mp_digit tmpx, *tmpy;
-075         register mp_word *_W;
-076         register int iy;
-077   
-078         /* copy of left side */
-079         tmpx = a->dp[ix];
-080   
-081         /* alias for right side */
-082         tmpy = a->dp + (ix + 1);
-083   
-084         /* the column to store the result in */
-085         _W = W + (ix + ix + 1);
-086   
-087         /* inner products */
-088         for (iy = ix + 1; iy < pa; iy++) \{
-089             *_W++ += ((mp_word)tmpx) * ((mp_word)*tmpy++);
-090         \}
-091       \}
-092     \}
-093   
-094     /* setup dest */
-095     olduse  = b->used;
-096     b->used = newused;
-097   
-098     /* now compute digits
-099      *
-100      * We have to double the inner product sums, add in the
-101      * outer product sums, propagate carries and convert
-102      * to single precision.
-103      */
-104     \{
-105       register mp_digit *tmpb;
-106   
-107       /* double first value, since the inner products are
-108        * half of what they should be
-109        */
-110       W[0] += W[0] + W2[0];
-111   
-112       tmpb = b->dp;
-113       for (ix = 1; ix < newused; ix++) \{
-114         /* double/add next digit */
-115         W[ix] += W[ix] + W2[ix];
-116   
-117         /* propagate carry forwards [from the previous digit] */
-118         W[ix] = W[ix] + (W[ix - 1] >> ((mp_word) DIGIT_BIT));
-119   
-120         /* store the current digit now that the carry isn't
-121          * needed
-122          */
-123         *tmpb++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
-124       \}
-125       /* set the last value.  Note even if the carry is zero
-126        * this is required since the next step will not zero
-127        * it if b originally had a value at b->dp[2*a.used]
-128        */
-129       *tmpb++ = (mp_digit) (W[(newused) - 1] & ((mp_word) MP_MASK));
-130   
-131       /* clear high digits of b if there were any originally */
-132       for (; ix < olduse; ix++) \{
-133         *tmpb++ = 0;
-134       \}
-135     \}
-136   
-137     mp_clamp (b);
-138     return MP_OKAY;
-139   \}
+016   /* fast squaring
+017    *
+018    * This is the comba method where the columns of the product
+019    * are computed first then the carries are computed.  This
+020    * has the effect of making a very simple inner loop that
+021    * is executed the most
+022    *
+023    * W2 represents the outer products and W the inner.
+024    *
+025    * A further optimizations is made because the inner
+026    * products are of the form "A * B * 2".  The *2 part does
+027    * not need to be computed until the end which is good
+028    * because 64-bit shifts are slow!
+029    *
+030    * Based on Algorithm 14.16 on pp.597 of HAC.
+031    *
+032    */
+033   int fast_s_mp_sqr (mp_int * a, mp_int * b)
+034   \{
+035     int     olduse, newused, res, ix, pa;
+036     mp_word W2[MP_WARRAY], W[MP_WARRAY];
+037   
+038     /* calculate size of product and allocate as required */
+039     pa = a->used;
+040     newused = pa + pa + 1;
+041     if (b->alloc < newused) \{
+042       if ((res = mp_grow (b, newused)) != MP_OKAY) \{
+043         return res;
+044       \}
+045     \}
+046   
+047     /* zero temp buffer (columns)
+048      * Note that there are two buffers.  Since squaring requires
+049      * a outer and inner product and the inner product requires
+050      * computing a product and doubling it (a relatively expensive
+051      * op to perform n**2 times if you don't have to) the inner and
+052      * outer products are computed in different buffers.  This way
+053      * the inner product can be doubled using n doublings instead of
+054      * n**2
+055      */
+056     memset (W,  0, newused * sizeof (mp_word));
+057     memset (W2, 0, newused * sizeof (mp_word));
+058   
+059     /* This computes the inner product.  To simplify the inner N**2 loop
+060      * the multiplication by two is done afterwards in the N loop.
+061      */
+062     for (ix = 0; ix < pa; ix++) \{
+063       /* compute the outer product
+064        *
+065        * Note that every outer product is computed
+066        * for a particular column only once which means that
+067        * there is no need todo a double precision addition
+068        * into the W2[] array.
+069        */
+070       W2[ix + ix] = ((mp_word)a->dp[ix]) * ((mp_word)a->dp[ix]);
+071   
+072       \{
+073         register mp_digit tmpx, *tmpy;
+074         register mp_word *_W;
+075         register int iy;
+076   
+077         /* copy of left side */
+078         tmpx = a->dp[ix];
+079   
+080         /* alias for right side */
+081         tmpy = a->dp + (ix + 1);
+082   
+083         /* the column to store the result in */
+084         _W = W + (ix + ix + 1);
+085   
+086         /* inner products */
+087         for (iy = ix + 1; iy < pa; iy++) \{
+088             *_W++ += ((mp_word)tmpx) * ((mp_word)*tmpy++);
+089         \}
+090       \}
+091     \}
+092   
+093     /* setup dest */
+094     olduse  = b->used;
+095     b->used = newused;
+096   
+097     /* now compute digits
+098      *
+099      * We have to double the inner product sums, add in the
+100      * outer product sums, propagate carries and convert
+101      * to single precision.
+102      */
+103     \{
+104       register mp_digit *tmpb;
+105   
+106       /* double first value, since the inner products are
+107        * half of what they should be
+108        */
+109       W[0] += W[0] + W2[0];
+110   
+111       tmpb = b->dp;
+112       for (ix = 1; ix < newused; ix++) \{
+113         /* double/add next digit */
+114         W[ix] += W[ix] + W2[ix];
+115   
+116         /* propagate carry forwards [from the previous digit] */
+117         W[ix] = W[ix] + (W[ix - 1] >> ((mp_word) DIGIT_BIT));
+118   
+119         /* store the current digit now that the carry isn't
+120          * needed
+121          */
+122         *tmpb++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
+123       \}
+124       /* set the last value.  Note even if the carry is zero
+125        * this is required since the next step will not zero
+126        * it if b originally had a value at b->dp[2*a.used]
+127        */
+128       *tmpb++ = (mp_digit) (W[(newused) - 1] & ((mp_word) MP_MASK));
+129   
+130       /* clear high digits of b if there were any originally */
+131       for (; ix < olduse; ix++) \{
+132         *tmpb++ = 0;
+133       \}
+134     \}
+135   
+136     mp_clamp (b);
+137     return MP_OKAY;
+138   \}
 \end{alltt}
 \end{small}
 
@@ -5217,111 +5219,110 @@ ratio of 1:7.  } than simpler operations such as addition.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* Karatsuba squaring, computes b = a*a using three 
-018    * half size squarings
-019    *
-020    * See comments of mp_karatsuba_mul for details.  It 
-021    * is essentially the same algorithm but merely 
-022    * tuned to perform recursive squarings.
-023    */
-024   int mp_karatsuba_sqr (mp_int * a, mp_int * b)
-025   \{
-026     mp_int  x0, x1, t1, t2, x0x0, x1x1;
-027     int     B, err;
-028   
-029     err = MP_MEM;
-030   
-031     /* min # of digits */
-032     B = a->used;
-033   
-034     /* now divide in two */
-035     B = B >> 1;
-036   
-037     /* init copy all the temps */
-038     if (mp_init_size (&x0, B) != MP_OKAY)
-039       goto ERR;
-040     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
-041       goto X0;
-042   
-043     /* init temps */
-044     if (mp_init_size (&t1, a->used * 2) != MP_OKAY)
-045       goto X1;
-046     if (mp_init_size (&t2, a->used * 2) != MP_OKAY)
-047       goto T1;
-048     if (mp_init_size (&x0x0, B * 2) != MP_OKAY)
-049       goto T2;
-050     if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY)
-051       goto X0X0;
-052   
-053     \{
-054       register int x;
-055       register mp_digit *dst, *src;
-056   
-057       src = a->dp;
-058   
-059       /* now shift the digits */
-060       dst = x0.dp;
-061       for (x = 0; x < B; x++) \{
-062         *dst++ = *src++;
-063       \}
-064   
-065       dst = x1.dp;
-066       for (x = B; x < a->used; x++) \{
-067         *dst++ = *src++;
-068       \}
-069     \}
-070   
-071     x0.used = B;
-072     x1.used = a->used - B;
-073   
-074     mp_clamp (&x0);
-075   
-076     /* now calc the products x0*x0 and x1*x1 */
-077     if (mp_sqr (&x0, &x0x0) != MP_OKAY)
-078       goto X1X1;           /* x0x0 = x0*x0 */
-079     if (mp_sqr (&x1, &x1x1) != MP_OKAY)
-080       goto X1X1;           /* x1x1 = x1*x1 */
-081   
-082     /* now calc (x1-x0)**2 */
-083     if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
-084       goto X1X1;           /* t1 = x1 - x0 */
-085     if (mp_sqr (&t1, &t1) != MP_OKAY)
-086       goto X1X1;           /* t1 = (x1 - x0) * (x1 - x0) */
-087   
-088     /* add x0y0 */
-089     if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY)
-090       goto X1X1;           /* t2 = x0x0 + x1x1 */
-091     if (mp_sub (&t2, &t1, &t1) != MP_OKAY)
-092       goto X1X1;           /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */
-093   
-094     /* shift by B */
-095     if (mp_lshd (&t1, B) != MP_OKAY)
-096       goto X1X1;           /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */
-097     if (mp_lshd (&x1x1, B * 2) != MP_OKAY)
-098       goto X1X1;           /* x1x1 = x1x1 << 2*B */
-099   
-100     if (mp_add (&x0x0, &t1, &t1) != MP_OKAY)
-101       goto X1X1;           /* t1 = x0x0 + t1 */
-102     if (mp_add (&t1, &x1x1, b) != MP_OKAY)
-103       goto X1X1;           /* t1 = x0x0 + t1 + x1x1 */
-104   
-105     err = MP_OKAY;
-106   
-107   X1X1:mp_clear (&x1x1);
-108   X0X0:mp_clear (&x0x0);
-109   T2:mp_clear (&t2);
-110   T1:mp_clear (&t1);
-111   X1:mp_clear (&x1);
-112   X0:mp_clear (&x0);
-113   ERR:
-114     return err;
-115   \}
+016   /* Karatsuba squaring, computes b = a*a using three 
+017    * half size squarings
+018    *
+019    * See comments of mp_karatsuba_mul for details.  It 
+020    * is essentially the same algorithm but merely 
+021    * tuned to perform recursive squarings.
+022    */
+023   int mp_karatsuba_sqr (mp_int * a, mp_int * b)
+024   \{
+025     mp_int  x0, x1, t1, t2, x0x0, x1x1;
+026     int     B, err;
+027   
+028     err = MP_MEM;
+029   
+030     /* min # of digits */
+031     B = a->used;
+032   
+033     /* now divide in two */
+034     B = B >> 1;
+035   
+036     /* init copy all the temps */
+037     if (mp_init_size (&x0, B) != MP_OKAY)
+038       goto ERR;
+039     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
+040       goto X0;
+041   
+042     /* init temps */
+043     if (mp_init_size (&t1, a->used * 2) != MP_OKAY)
+044       goto X1;
+045     if (mp_init_size (&t2, a->used * 2) != MP_OKAY)
+046       goto T1;
+047     if (mp_init_size (&x0x0, B * 2) != MP_OKAY)
+048       goto T2;
+049     if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY)
+050       goto X0X0;
+051   
+052     \{
+053       register int x;
+054       register mp_digit *dst, *src;
+055   
+056       src = a->dp;
+057   
+058       /* now shift the digits */
+059       dst = x0.dp;
+060       for (x = 0; x < B; x++) \{
+061         *dst++ = *src++;
+062       \}
+063   
+064       dst = x1.dp;
+065       for (x = B; x < a->used; x++) \{
+066         *dst++ = *src++;
+067       \}
+068     \}
+069   
+070     x0.used = B;
+071     x1.used = a->used - B;
+072   
+073     mp_clamp (&x0);
+074   
+075     /* now calc the products x0*x0 and x1*x1 */
+076     if (mp_sqr (&x0, &x0x0) != MP_OKAY)
+077       goto X1X1;           /* x0x0 = x0*x0 */
+078     if (mp_sqr (&x1, &x1x1) != MP_OKAY)
+079       goto X1X1;           /* x1x1 = x1*x1 */
+080   
+081     /* now calc (x1-x0)**2 */
+082     if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
+083       goto X1X1;           /* t1 = x1 - x0 */
+084     if (mp_sqr (&t1, &t1) != MP_OKAY)
+085       goto X1X1;           /* t1 = (x1 - x0) * (x1 - x0) */
+086   
+087     /* add x0y0 */
+088     if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY)
+089       goto X1X1;           /* t2 = x0x0 + x1x1 */
+090     if (mp_sub (&t2, &t1, &t1) != MP_OKAY)
+091       goto X1X1;           /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */
+092   
+093     /* shift by B */
+094     if (mp_lshd (&t1, B) != MP_OKAY)
+095       goto X1X1;           /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */
+096     if (mp_lshd (&x1x1, B * 2) != MP_OKAY)
+097       goto X1X1;           /* x1x1 = x1x1 << 2*B */
+098   
+099     if (mp_add (&x0x0, &t1, &t1) != MP_OKAY)
+100       goto X1X1;           /* t1 = x0x0 + t1 */
+101     if (mp_add (&t1, &x1x1, b) != MP_OKAY)
+102       goto X1X1;           /* t1 = x0x0 + t1 + x1x1 */
+103   
+104     err = MP_OKAY;
+105   
+106   X1X1:mp_clear (&x1x1);
+107   X0X0:mp_clear (&x0x0);
+108   T2:mp_clear (&t2);
+109   T1:mp_clear (&t1);
+110   X1:mp_clear (&x1);
+111   X0:mp_clear (&x0);
+112   ERR:
+113     return err;
+114   \}
 \end{alltt}
 \end{small}
 
 This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul.  It uses the same inline style to copy and 
-shift the input into the two halves.  The loop from line 53 to line 69 has been modified since only one input exists.  The \textbf{used}
+shift the input into the two halves.  The loop from line 52 to line 68 has been modified since only one input exists.  The \textbf{used}
 count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin.  At this point $x1$ and $x0$ are valid equivalents
 to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.  
 
@@ -5376,32 +5377,31 @@ neither of the polynomial basis algorithms should be used then either the Comba
 \hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* computes b = a*a */
-018   int
-019   mp_sqr (mp_int * a, mp_int * b)
-020   \{
-021     int     res;
-022   
-023     /* use Toom-Cook? */
-024     if (a->used >= TOOM_SQR_CUTOFF) \{
-025       res = mp_toom_sqr(a, b);
-026     /* Karatsuba? */
-027     \} else if (a->used >= KARATSUBA_SQR_CUTOFF) \{
-028       res = mp_karatsuba_sqr (a, b);
-029     \} else \{
-030       /* can we use the fast comba multiplier? */
-031       if ((a->used * 2 + 1) < MP_WARRAY && 
-032            a->used < 
-033            (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) \{
-034         res = fast_s_mp_sqr (a, b);
-035       \} else \{
-036         res = s_mp_sqr (a, b);
-037       \}
-038     \}
-039     b->sign = MP_ZPOS;
-040     return res;
-041   \}
+016   /* computes b = a*a */
+017   int
+018   mp_sqr (mp_int * a, mp_int * b)
+019   \{
+020     int     res;
+021   
+022     /* use Toom-Cook? */
+023     if (a->used >= TOOM_SQR_CUTOFF) \{
+024       res = mp_toom_sqr(a, b);
+025     /* Karatsuba? */
+026     \} else if (a->used >= KARATSUBA_SQR_CUTOFF) \{
+027       res = mp_karatsuba_sqr (a, b);
+028     \} else \{
+029       /* can we use the fast comba multiplier? */
+030       if ((a->used * 2 + 1) < MP_WARRAY && 
+031            a->used < 
+032            (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) \{
+033         res = fast_s_mp_sqr (a, b);
+034       \} else \{
+035         res = s_mp_sqr (a, b);
+036       \}
+037     \}
+038     b->sign = MP_ZPOS;
+039     return res;
+040   \}
 \end{alltt}
 \end{small}
 
@@ -5650,81 +5650,80 @@ performed at most twice, and on average once. However, if $a \ge b^2$ than it wi
 \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* reduces x mod m, assumes 0 < x < m**2, mu is 
-018    * precomputed via mp_reduce_setup.
-019    * From HAC pp.604 Algorithm 14.42
-020    */
-021   int
-022   mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
-023   \{
-024     mp_int  q;
-025     int     res, um = m->used;
-026   
-027     /* q = x */
-028     if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{
-029       return res;
-030     \}
-031   
-032     /* q1 = x / b**(k-1)  */
-033     mp_rshd (&q, um - 1);         
-034   
-035     /* according to HAC this optimization is ok */
-036     if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{
-037       if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{
-038         goto CLEANUP;
-039       \}
-040     \} else \{
-041       if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{
-042         goto CLEANUP;
-043       \}
-044     \}
-045   
-046     /* q3 = q2 / b**(k+1) */
-047     mp_rshd (&q, um + 1);         
-048   
-049     /* x = x mod b**(k+1), quick (no division) */
-050     if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{
-051       goto CLEANUP;
-052     \}
-053   
-054     /* q = q * m mod b**(k+1), quick (no division) */
-055     if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{
-056       goto CLEANUP;
-057     \}
-058   
-059     /* x = x - q */
-060     if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{
-061       goto CLEANUP;
-062     \}
-063   
-064     /* If x < 0, add b**(k+1) to it */
-065     if (mp_cmp_d (x, 0) == MP_LT) \{
-066       mp_set (&q, 1);
-067       if ((res = mp_lshd (&q, um + 1)) != MP_OKAY)
-068         goto CLEANUP;
-069       if ((res = mp_add (x, &q, x)) != MP_OKAY)
-070         goto CLEANUP;
-071     \}
-072   
-073     /* Back off if it's too big */
-074     while (mp_cmp (x, m) != MP_LT) \{
-075       if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{
-076         goto CLEANUP;
-077       \}
-078     \}
-079     
-080   CLEANUP:
-081     mp_clear (&q);
-082   
-083     return res;
-084   \}
+016   /* reduces x mod m, assumes 0 < x < m**2, mu is 
+017    * precomputed via mp_reduce_setup.
+018    * From HAC pp.604 Algorithm 14.42
+019    */
+020   int
+021   mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
+022   \{
+023     mp_int  q;
+024     int     res, um = m->used;
+025   
+026     /* q = x */
+027     if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{
+028       return res;
+029     \}
+030   
+031     /* q1 = x / b**(k-1)  */
+032     mp_rshd (&q, um - 1);         
+033   
+034     /* according to HAC this optimization is ok */
+035     if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{
+036       if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{
+037         goto CLEANUP;
+038       \}
+039     \} else \{
+040       if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{
+041         goto CLEANUP;
+042       \}
+043     \}
+044   
+045     /* q3 = q2 / b**(k+1) */
+046     mp_rshd (&q, um + 1);         
+047   
+048     /* x = x mod b**(k+1), quick (no division) */
+049     if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{
+050       goto CLEANUP;
+051     \}
+052   
+053     /* q = q * m mod b**(k+1), quick (no division) */
+054     if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{
+055       goto CLEANUP;
+056     \}
+057   
+058     /* x = x - q */
+059     if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{
+060       goto CLEANUP;
+061     \}
+062   
+063     /* If x < 0, add b**(k+1) to it */
+064     if (mp_cmp_d (x, 0) == MP_LT) \{
+065       mp_set (&q, 1);
+066       if ((res = mp_lshd (&q, um + 1)) != MP_OKAY)
+067         goto CLEANUP;
+068       if ((res = mp_add (x, &q, x)) != MP_OKAY)
+069         goto CLEANUP;
+070     \}
+071   
+072     /* Back off if it's too big */
+073     while (mp_cmp (x, m) != MP_LT) \{
+074       if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{
+075         goto CLEANUP;
+076       \}
+077     \}
+078     
+079   CLEANUP:
+080     mp_clear (&q);
+081   
+082     return res;
+083   \}
 \end{alltt}
 \end{small}
 
 The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up.  This essentially halves
 the number of single precision multiplications required.  However, the optimization is only safe if $\beta$ is much larger than the number of digits
-in the modulus.  In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is
+in the modulus.  In the source code this is evaluated on lines 36 to 43 where algorithm s\_mp\_mul\_high\_digs is used when it is
 safe to do so.  
 
 \subsection{The Barrett Setup Algorithm}
@@ -5757,20 +5756,19 @@ is equivalent and much faster.  The final value is computed by taking the intege
 \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* pre-calculate the value required for Barrett reduction
-018    * For a given modulus "b" it calulates the value required in "a"
-019    */
-020   int
-021   mp_reduce_setup (mp_int * a, mp_int * b)
-022   \{
-023     int     res;
-024     
-025     if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{
-026       return res;
-027     \}
-028     return mp_div (a, b, a, NULL);
-029   \}
+016   /* pre-calculate the value required for Barrett reduction
+017    * For a given modulus "b" it calulates the value required in "a"
+018    */
+019   int
+020   mp_reduce_setup (mp_int * a, mp_int * b)
+021   \{
+022     int     res;
+023     
+024     if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{
+025       return res;
+026     \}
+027     return mp_div (a, b, a, NULL);
+028   \}
 \end{alltt}
 \end{small}
 
@@ -6028,108 +6026,107 @@ multiplications.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* computes xR**-1 == x (mod N) via Montgomery Reduction */
-018   int
-019   mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
-020   \{
-021     int     ix, res, digs;
-022     mp_digit mu;
-023   
-024     /* can the fast reduction [comba] method be used?
-025      *
-026      * Note that unlike in mp_mul you're safely allowed *less*
-027      * than the available columns [255 per default] since carries
-028      * are fixed up in the inner loop.
-029      */
-030     digs = n->used * 2 + 1;
-031     if ((digs < MP_WARRAY) &&
-032         n->used <
-033         (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
-034       return fast_mp_montgomery_reduce (x, n, rho);
-035     \}
-036   
-037     /* grow the input as required */
-038     if (x->alloc < digs) \{
-039       if ((res = mp_grow (x, digs)) != MP_OKAY) \{
-040         return res;
-041       \}
-042     \}
-043     x->used = digs;
-044   
-045     for (ix = 0; ix < n->used; ix++) \{
-046       /* mu = ai * rho mod b
-047        *
-048        * The value of rho must be precalculated via
-049        * bn_mp_montgomery_setup() such that
-050        * it equals -1/n0 mod b this allows the
-051        * following inner loop to reduce the
-052        * input one digit at a time
-053        */
-054       mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK);
-055   
-056       /* a = a + mu * m * b**i */
-057       \{
-058         register int iy;
-059         register mp_digit *tmpn, *tmpx, u;
-060         register mp_word r;
-061   
-062         /* alias for digits of the modulus */
-063         tmpn = n->dp;
-064   
-065         /* alias for the digits of x [the input] */
-066         tmpx = x->dp + ix;
-067   
-068         /* set the carry to zero */
-069         u = 0;
-070   
-071         /* Multiply and add in place */
-072         for (iy = 0; iy < n->used; iy++) \{
-073           /* compute product and sum */
-074           r       = ((mp_word)mu) * ((mp_word)*tmpn++) +
-075                     ((mp_word) u) + ((mp_word) * tmpx);
-076   
-077           /* get carry */
-078           u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
-079   
-080           /* fix digit */
-081           *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK));
-082         \}
-083         /* At this point the ix'th digit of x should be zero */
+016   /* computes xR**-1 == x (mod N) via Montgomery Reduction */
+017   int
+018   mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
+019   \{
+020     int     ix, res, digs;
+021     mp_digit mu;
+022   
+023     /* can the fast reduction [comba] method be used?
+024      *
+025      * Note that unlike in mp_mul you're safely allowed *less*
+026      * than the available columns [255 per default] since carries
+027      * are fixed up in the inner loop.
+028      */
+029     digs = n->used * 2 + 1;
+030     if ((digs < MP_WARRAY) &&
+031         n->used <
+032         (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
+033       return fast_mp_montgomery_reduce (x, n, rho);
+034     \}
+035   
+036     /* grow the input as required */
+037     if (x->alloc < digs) \{
+038       if ((res = mp_grow (x, digs)) != MP_OKAY) \{
+039         return res;
+040       \}
+041     \}
+042     x->used = digs;
+043   
+044     for (ix = 0; ix < n->used; ix++) \{
+045       /* mu = ai * rho mod b
+046        *
+047        * The value of rho must be precalculated via
+048        * bn_mp_montgomery_setup() such that
+049        * it equals -1/n0 mod b this allows the
+050        * following inner loop to reduce the
+051        * input one digit at a time
+052        */
+053       mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK);
+054   
+055       /* a = a + mu * m * b**i */
+056       \{
+057         register int iy;
+058         register mp_digit *tmpn, *tmpx, u;
+059         register mp_word r;
+060   
+061         /* alias for digits of the modulus */
+062         tmpn = n->dp;
+063   
+064         /* alias for the digits of x [the input] */
+065         tmpx = x->dp + ix;
+066   
+067         /* set the carry to zero */
+068         u = 0;
+069   
+070         /* Multiply and add in place */
+071         for (iy = 0; iy < n->used; iy++) \{
+072           /* compute product and sum */
+073           r       = ((mp_word)mu) * ((mp_word)*tmpn++) +
+074                     ((mp_word) u) + ((mp_word) * tmpx);
+075   
+076           /* get carry */
+077           u       = (mp_digit)(r >> ((mp_word) DIGIT_BIT));
+078   
+079           /* fix digit */
+080           *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK));
+081         \}
+082         /* At this point the ix'th digit of x should be zero */
+083   
 084   
-085   
-086         /* propagate carries upwards as required*/
-087         while (u) \{
-088           *tmpx   += u;
-089           u        = *tmpx >> DIGIT_BIT;
-090           *tmpx++ &= MP_MASK;
-091         \}
-092       \}
-093     \}
-094   
-095     /* at this point the n.used'th least
-096      * significant digits of x are all zero
-097      * which means we can shift x to the
-098      * right by n.used digits and the
-099      * residue is unchanged.
-100      */
-101   
-102     /* x = x/b**n.used */
-103     mp_clamp(x);
-104     mp_rshd (x, n->used);
-105   
-106     /* if x >= n then x = x - n */
-107     if (mp_cmp_mag (x, n) != MP_LT) \{
-108       return s_mp_sub (x, n, x);
-109     \}
-110   
-111     return MP_OKAY;
-112   \}
+085         /* propagate carries upwards as required*/
+086         while (u) \{
+087           *tmpx   += u;
+088           u        = *tmpx >> DIGIT_BIT;
+089           *tmpx++ &= MP_MASK;
+090         \}
+091       \}
+092     \}
+093   
+094     /* at this point the n.used'th least
+095      * significant digits of x are all zero
+096      * which means we can shift x to the
+097      * right by n.used digits and the
+098      * residue is unchanged.
+099      */
+100   
+101     /* x = x/b**n.used */
+102     mp_clamp(x);
+103     mp_rshd (x, n->used);
+104   
+105     /* if x >= n then x = x - n */
+106     if (mp_cmp_mag (x, n) != MP_LT) \{
+107       return s_mp_sub (x, n, x);
+108     \}
+109   
+110     return MP_OKAY;
+111   \}
 \end{alltt}
 \end{small}
 
-This is the baseline implementation of the Montgomery reduction algorithm.  Lines 30 to 35 determine if the Comba based
-routine can be used instead.  Line 48 computes the value of $\mu$ for that particular iteration of the outer loop.  
+This is the baseline implementation of the Montgomery reduction algorithm.  Lines 30 to 34 determine if the Comba based
+routine can be used instead.  Line 47 computes the value of $\mu$ for that particular iteration of the outer loop.  
 
 The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop.  The alias $tmpx$ refers to the $ix$'th digit of $x$ and
 the alias $tmpn$ refers to the modulus $n$.  
@@ -6217,169 +6214,168 @@ stored in the destination $x$.
 \hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* computes xR**-1 == x (mod N) via Montgomery Reduction
-018    *
-019    * This is an optimized implementation of mp_montgomery_reduce
-020    * which uses the comba method to quickly calculate the columns of the
-021    * reduction.
-022    *
-023    * Based on Algorithm 14.32 on pp.601 of HAC.
-024   */
-025   int
-026   fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
-027   \{
-028     int     ix, res, olduse;
-029     mp_word W[MP_WARRAY];
-030   
-031     /* get old used count */
-032     olduse = x->used;
-033   
-034     /* grow a as required */
-035     if (x->alloc < n->used + 1) \{
-036       if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{
-037         return res;
-038       \}
-039     \}
-040   
-041     /* first we have to get the digits of the input into
-042      * an array of double precision words W[...]
-043      */
-044     \{
-045       register mp_word *_W;
-046       register mp_digit *tmpx;
-047   
-048       /* alias for the W[] array */
-049       _W   = W;
-050   
-051       /* alias for the digits of  x*/
-052       tmpx = x->dp;
-053   
-054       /* copy the digits of a into W[0..a->used-1] */
-055       for (ix = 0; ix < x->used; ix++) \{
-056         *_W++ = *tmpx++;
-057       \}
-058   
-059       /* zero the high words of W[a->used..m->used*2] */
-060       for (; ix < n->used * 2 + 1; ix++) \{
-061         *_W++ = 0;
-062       \}
-063     \}
-064   
-065     /* now we proceed to zero successive digits
-066      * from the least significant upwards
-067      */
-068     for (ix = 0; ix < n->used; ix++) \{
-069       /* mu = ai * m' mod b
-070        *
-071        * We avoid a double precision multiplication (which isn't required)
-072        * by casting the value down to a mp_digit.  Note this requires
-073        * that W[ix-1] have  the carry cleared (see after the inner loop)
-074        */
-075       register mp_digit mu;
-076       mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK);
-077   
-078       /* a = a + mu * m * b**i
-079        *
-080        * This is computed in place and on the fly.  The multiplication
-081        * by b**i is handled by offseting which columns the results
-082        * are added to.
-083        *
-084        * Note the comba method normally doesn't handle carries in the
-085        * inner loop In this case we fix the carry from the previous
-086        * column since the Montgomery reduction requires digits of the
-087        * result (so far) [see above] to work.  This is
-088        * handled by fixing up one carry after the inner loop.  The
-089        * carry fixups are done in order so after these loops the
-090        * first m->used words of W[] have the carries fixed
-091        */
-092       \{
-093         register int iy;
-094         register mp_digit *tmpn;
-095         register mp_word *_W;
-096   
-097         /* alias for the digits of the modulus */
-098         tmpn = n->dp;
-099   
-100         /* Alias for the columns set by an offset of ix */
-101         _W = W + ix;
-102   
-103         /* inner loop */
-104         for (iy = 0; iy < n->used; iy++) \{
-105             *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++);
-106         \}
-107       \}
-108   
-109       /* now fix carry for next digit, W[ix+1] */
-110       W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT);
-111     \}
-112   
-113     /* now we have to propagate the carries and
-114      * shift the words downward [all those least
-115      * significant digits we zeroed].
-116      */
-117     \{
-118       register mp_digit *tmpx;
-119       register mp_word *_W, *_W1;
-120   
-121       /* nox fix rest of carries */
-122   
-123       /* alias for current word */
-124       _W1 = W + ix;
-125   
-126       /* alias for next word, where the carry goes */
-127       _W = W + ++ix;
-128   
-129       for (; ix <= n->used * 2 + 1; ix++) \{
-130         *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT);
-131       \}
-132   
-133       /* copy out, A = A/b**n
-134        *
-135        * The result is A/b**n but instead of converting from an
-136        * array of mp_word to mp_digit than calling mp_rshd
-137        * we just copy them in the right order
-138        */
-139   
-140       /* alias for destination word */
-141       tmpx = x->dp;
-142   
-143       /* alias for shifted double precision result */
-144       _W = W + n->used;
-145   
-146       for (ix = 0; ix < n->used + 1; ix++) \{
-147         *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK));
-148       \}
-149   
-150       /* zero oldused digits, if the input a was larger than
-151        * m->used+1 we'll have to clear the digits
-152        */
-153       for (; ix < olduse; ix++) \{
-154         *tmpx++ = 0;
-155       \}
-156     \}
-157   
-158     /* set the max used and clamp */
-159     x->used = n->used + 1;
-160     mp_clamp (x);
-161   
-162     /* if A >= m then A = A - m */
-163     if (mp_cmp_mag (x, n) != MP_LT) \{
-164       return s_mp_sub (x, n, x);
-165     \}
-166     return MP_OKAY;
-167   \}
+016   /* computes xR**-1 == x (mod N) via Montgomery Reduction
+017    *
+018    * This is an optimized implementation of mp_montgomery_reduce
+019    * which uses the comba method to quickly calculate the columns of the
+020    * reduction.
+021    *
+022    * Based on Algorithm 14.32 on pp.601 of HAC.
+023   */
+024   int
+025   fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
+026   \{
+027     int     ix, res, olduse;
+028     mp_word W[MP_WARRAY];
+029   
+030     /* get old used count */
+031     olduse = x->used;
+032   
+033     /* grow a as required */
+034     if (x->alloc < n->used + 1) \{
+035       if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{
+036         return res;
+037       \}
+038     \}
+039   
+040     /* first we have to get the digits of the input into
+041      * an array of double precision words W[...]
+042      */
+043     \{
+044       register mp_word *_W;
+045       register mp_digit *tmpx;
+046   
+047       /* alias for the W[] array */
+048       _W   = W;
+049   
+050       /* alias for the digits of  x*/
+051       tmpx = x->dp;
+052   
+053       /* copy the digits of a into W[0..a->used-1] */
+054       for (ix = 0; ix < x->used; ix++) \{
+055         *_W++ = *tmpx++;
+056       \}
+057   
+058       /* zero the high words of W[a->used..m->used*2] */
+059       for (; ix < n->used * 2 + 1; ix++) \{
+060         *_W++ = 0;
+061       \}
+062     \}
+063   
+064     /* now we proceed to zero successive digits
+065      * from the least significant upwards
+066      */
+067     for (ix = 0; ix < n->used; ix++) \{
+068       /* mu = ai * m' mod b
+069        *
+070        * We avoid a double precision multiplication (which isn't required)
+071        * by casting the value down to a mp_digit.  Note this requires
+072        * that W[ix-1] have  the carry cleared (see after the inner loop)
+073        */
+074       register mp_digit mu;
+075       mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK);
+076   
+077       /* a = a + mu * m * b**i
+078        *
+079        * This is computed in place and on the fly.  The multiplication
+080        * by b**i is handled by offseting which columns the results
+081        * are added to.
+082        *
+083        * Note the comba method normally doesn't handle carries in the
+084        * inner loop In this case we fix the carry from the previous
+085        * column since the Montgomery reduction requires digits of the
+086        * result (so far) [see above] to work.  This is
+087        * handled by fixing up one carry after the inner loop.  The
+088        * carry fixups are done in order so after these loops the
+089        * first m->used words of W[] have the carries fixed
+090        */
+091       \{
+092         register int iy;
+093         register mp_digit *tmpn;
+094         register mp_word *_W;
+095   
+096         /* alias for the digits of the modulus */
+097         tmpn = n->dp;
+098   
+099         /* Alias for the columns set by an offset of ix */
+100         _W = W + ix;
+101   
+102         /* inner loop */
+103         for (iy = 0; iy < n->used; iy++) \{
+104             *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++);
+105         \}
+106       \}
+107   
+108       /* now fix carry for next digit, W[ix+1] */
+109       W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT);
+110     \}
+111   
+112     /* now we have to propagate the carries and
+113      * shift the words downward [all those least
+114      * significant digits we zeroed].
+115      */
+116     \{
+117       register mp_digit *tmpx;
+118       register mp_word *_W, *_W1;
+119   
+120       /* nox fix rest of carries */
+121   
+122       /* alias for current word */
+123       _W1 = W + ix;
+124   
+125       /* alias for next word, where the carry goes */
+126       _W = W + ++ix;
+127   
+128       for (; ix <= n->used * 2 + 1; ix++) \{
+129         *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT);
+130       \}
+131   
+132       /* copy out, A = A/b**n
+133        *
+134        * The result is A/b**n but instead of converting from an
+135        * array of mp_word to mp_digit than calling mp_rshd
+136        * we just copy them in the right order
+137        */
+138   
+139       /* alias for destination word */
+140       tmpx = x->dp;
+141   
+142       /* alias for shifted double precision result */
+143       _W = W + n->used;
+144   
+145       for (ix = 0; ix < n->used + 1; ix++) \{
+146         *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK));
+147       \}
+148   
+149       /* zero oldused digits, if the input a was larger than
+150        * m->used+1 we'll have to clear the digits
+151        */
+152       for (; ix < olduse; ix++) \{
+153         *tmpx++ = 0;
+154       \}
+155     \}
+156   
+157     /* set the max used and clamp */
+158     x->used = n->used + 1;
+159     mp_clamp (x);
+160   
+161     /* if A >= m then A = A - m */
+162     if (mp_cmp_mag (x, n) != MP_LT) \{
+163       return s_mp_sub (x, n, x);
+164     \}
+165     return MP_OKAY;
+166   \}
 \end{alltt}
 \end{small}
 
-The $\hat W$ array is first filled with digits of $x$ on line 48 then the rest of the digits are zeroed on line 55.  Both loops share
+The $\hat W$ array is first filled with digits of $x$ on line 50 then the rest of the digits are zeroed on line 54.  Both loops share
 the same alias variables to make the code easier to read.  
 
 The value of $\mu$ is calculated in an interesting fashion.  First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit.  This
-forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision.   Line 110 fixes the carry 
+forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision.   Line 109 fixes the carry 
 for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.
 
-The for loop on line 109 propagates the rest of the carries upwards through the columns.  The for loop on line 126 reduces the columns
+The for loop on line 108 propagates the rest of the carries upwards through the columns.  The for loop on line 125 reduces the columns
 modulo $\beta$ and shifts them $k$ places at the same time.  The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
 digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.  
 
@@ -6416,44 +6412,43 @@ to calculate $1/n_0$ when $\beta$ is a power of two.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* setups the montgomery reduction stuff */
-018   int
-019   mp_montgomery_setup (mp_int * n, mp_digit * rho)
-020   \{
-021     mp_digit x, b;
-022   
-023   /* fast inversion mod 2**k
-024    *
-025    * Based on the fact that
-026    *
-027    * XA = 1 (mod 2**n)  =>  (X(2-XA)) A = 1 (mod 2**2n)
-028    *                    =>  2*X*A - X*X*A*A = 1
-029    *                    =>  2*(1) - (1)     = 1
-030    */
-031     b = n->dp[0];
-032   
-033     if ((b & 1) == 0) \{
-034       return MP_VAL;
-035     \}
-036   
-037     x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */
-038     x *= 2 - b * x;               /* here x*a==1 mod 2**8 */
-039   #if !defined(MP_8BIT)
-040     x *= 2 - b * x;               /* here x*a==1 mod 2**16 */
-041   #endif
-042   #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
-043     x *= 2 - b * x;               /* here x*a==1 mod 2**32 */
-044   #endif
-045   #ifdef MP_64BIT
-046     x *= 2 - b * x;               /* here x*a==1 mod 2**64 */
-047   #endif
-048   
-049     /* rho = -1/m mod b */
-050     *rho = (((mp_digit) 1 << ((mp_digit) DIGIT_BIT)) - x) & MP_MASK;
-051   
-052     return MP_OKAY;
-053   \}
+016   /* setups the montgomery reduction stuff */
+017   int
+018   mp_montgomery_setup (mp_int * n, mp_digit * rho)
+019   \{
+020     mp_digit x, b;
+021   
+022   /* fast inversion mod 2**k
+023    *
+024    * Based on the fact that
+025    *
+026    * XA = 1 (mod 2**n)  =>  (X(2-XA)) A = 1 (mod 2**2n)
+027    *                    =>  2*X*A - X*X*A*A = 1
+028    *                    =>  2*(1) - (1)     = 1
+029    */
+030     b = n->dp[0];
+031   
+032     if ((b & 1) == 0) \{
+033       return MP_VAL;
+034     \}
+035   
+036     x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */
+037     x *= 2 - b * x;               /* here x*a==1 mod 2**8 */
+038   #if !defined(MP_8BIT)
+039     x *= 2 - b * x;               /* here x*a==1 mod 2**16 */
+040   #endif
+041   #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
+042     x *= 2 - b * x;               /* here x*a==1 mod 2**32 */
+043   #endif
+044   #ifdef MP_64BIT
+045     x *= 2 - b * x;               /* here x*a==1 mod 2**64 */
+046   #endif
+047   
+048     /* rho = -1/m mod b */
+049     *rho = (((mp_digit) 1 << ((mp_digit) DIGIT_BIT)) - x) & MP_MASK;
+050   
+051     return MP_OKAY;
+052   \}
 \end{alltt}
 \end{small}
 
@@ -6646,95 +6641,94 @@ at step 3.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
-018    *
-019    * Based on algorithm from the paper
-020    *
-021    * "Generating Efficient Primes for Discrete Log Cryptosystems"
-022    *                 Chae Hoon Lim, Pil Loong Lee,
-023    *          POSTECH Information Research Laboratories
-024    *
-025    * The modulus must be of a special format [see manual]
-026    *
-027    * Has been modified to use algorithm 7.10 from the LTM book instead
-028    *
-029    * Input x must be in the range 0 <= x <= (n-1)**2
-030    */
-031   int
-032   mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k)
-033   \{
-034     int      err, i, m;
-035     mp_word  r;
-036     mp_digit mu, *tmpx1, *tmpx2;
-037   
-038     /* m = digits in modulus */
-039     m = n->used;
-040   
-041     /* ensure that "x" has at least 2m digits */
-042     if (x->alloc < m + m) \{
-043       if ((err = mp_grow (x, m + m)) != MP_OKAY) \{
-044         return err;
-045       \}
-046     \}
-047   
-048   /* top of loop, this is where the code resumes if
-049    * another reduction pass is required.
-050    */
-051   top:
-052     /* aliases for digits */
-053     /* alias for lower half of x */
-054     tmpx1 = x->dp;
-055   
-056     /* alias for upper half of x, or x/B**m */
-057     tmpx2 = x->dp + m;
-058   
-059     /* set carry to zero */
-060     mu = 0;
-061   
-062     /* compute (x mod B**m) + k * [x/B**m] inline and inplace */
-063     for (i = 0; i < m; i++) \{
-064         r         = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu;
-065         *tmpx1++  = (mp_digit)(r & MP_MASK);
-066         mu        = (mp_digit)(r >> ((mp_word)DIGIT_BIT));
-067     \}
-068   
-069     /* set final carry */
-070     *tmpx1++ = mu;
-071   
-072     /* zero words above m */
-073     for (i = m + 1; i < x->used; i++) \{
-074         *tmpx1++ = 0;
-075     \}
-076   
-077     /* clamp, sub and return */
-078     mp_clamp (x);
-079   
-080     /* if x >= n then subtract and reduce again
-081      * Each successive "recursion" makes the input smaller and smaller.
-082      */
-083     if (mp_cmp_mag (x, n) != MP_LT) \{
-084       s_mp_sub(x, n, x);
-085       goto top;
-086     \}
-087     return MP_OKAY;
-088   \}
+016   /* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
+017    *
+018    * Based on algorithm from the paper
+019    *
+020    * "Generating Efficient Primes for Discrete Log Cryptosystems"
+021    *                 Chae Hoon Lim, Pil Loong Lee,
+022    *          POSTECH Information Research Laboratories
+023    *
+024    * The modulus must be of a special format [see manual]
+025    *
+026    * Has been modified to use algorithm 7.10 from the LTM book instead
+027    *
+028    * Input x must be in the range 0 <= x <= (n-1)**2
+029    */
+030   int
+031   mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k)
+032   \{
+033     int      err, i, m;
+034     mp_word  r;
+035     mp_digit mu, *tmpx1, *tmpx2;
+036   
+037     /* m = digits in modulus */
+038     m = n->used;
+039   
+040     /* ensure that "x" has at least 2m digits */
+041     if (x->alloc < m + m) \{
+042       if ((err = mp_grow (x, m + m)) != MP_OKAY) \{
+043         return err;
+044       \}
+045     \}
+046   
+047   /* top of loop, this is where the code resumes if
+048    * another reduction pass is required.
+049    */
+050   top:
+051     /* aliases for digits */
+052     /* alias for lower half of x */
+053     tmpx1 = x->dp;
+054   
+055     /* alias for upper half of x, or x/B**m */
+056     tmpx2 = x->dp + m;
+057   
+058     /* set carry to zero */
+059     mu = 0;
+060   
+061     /* compute (x mod B**m) + k * [x/B**m] inline and inplace */
+062     for (i = 0; i < m; i++) \{
+063         r         = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu;
+064         *tmpx1++  = (mp_digit)(r & MP_MASK);
+065         mu        = (mp_digit)(r >> ((mp_word)DIGIT_BIT));
+066     \}
+067   
+068     /* set final carry */
+069     *tmpx1++ = mu;
+070   
+071     /* zero words above m */
+072     for (i = m + 1; i < x->used; i++) \{
+073         *tmpx1++ = 0;
+074     \}
+075   
+076     /* clamp, sub and return */
+077     mp_clamp (x);
+078   
+079     /* if x >= n then subtract and reduce again
+080      * Each successive "recursion" makes the input smaller and smaller.
+081      */
+082     if (mp_cmp_mag (x, n) != MP_LT) \{
+083       s_mp_sub(x, n, x);
+084       goto top;
+085     \}
+086     return MP_OKAY;
+087   \}
 \end{alltt}
 \end{small}
 
-The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$.  The label on line 51 is where
+The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$.  The label on line 50 is where
 the algorithm will resume if further reduction passes are required.  In theory it could be placed at the top of the function however, the size of
 the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.  
 
 The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits.  By reading digits from $x$ offset by $m$ digits
-a division by $\beta^m$ can be simulated virtually for free.  The loop on line 63 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
+a division by $\beta^m$ can be simulated virtually for free.  The loop on line 62 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
 in this algorithm.
 
-By line 70 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed.  Similarly by line 73 the 
+By line 69 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed.  Similarly by line 72 the 
 same pointer will point to the $m+1$'th digit where the zeroes will be placed.  
 
 Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.  
-With the same logic at line 84 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
+With the same logic at line 83 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
 as well.  Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
 does not need to be checked.
 
@@ -6762,17 +6756,16 @@ completeness.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* determines the setup value */
-018   void mp_dr_setup(mp_int *a, mp_digit *d)
-019   \{
-020      /* the casts are required if DIGIT_BIT is one less than
-021       * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
-022       */
-023      *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - 
-024           ((mp_word)a->dp[0]));
-025   \}
-026   
+016   /* determines the setup value */
+017   void mp_dr_setup(mp_int *a, mp_digit *d)
+018   \{
+019      /* the casts are required if DIGIT_BIT is one less than
+020       * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
+021       */
+022      *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - 
+023           ((mp_word)a->dp[0]));
+024   \}
+025   
 \end{alltt}
 \end{small}
 
@@ -6808,28 +6801,27 @@ step 3 then $n$ must be of Diminished Radix form.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* determines if a number is a valid DR modulus */
-018   int mp_dr_is_modulus(mp_int *a)
-019   \{
-020      int ix;
-021   
-022      /* must be at least two digits */
-023      if (a->used < 2) \{
-024         return 0;
-025      \}
-026   
-027      /* must be of the form b**k - a [a <= b] so all
-028       * but the first digit must be equal to -1 (mod b).
-029       */
-030      for (ix = 1; ix < a->used; ix++) \{
-031          if (a->dp[ix] != MP_MASK) \{
-032             return 0;
-033          \}
-034      \}
-035      return 1;
-036   \}
-037   
+016   /* determines if a number is a valid DR modulus */
+017   int mp_dr_is_modulus(mp_int *a)
+018   \{
+019      int ix;
+020   
+021      /* must be at least two digits */
+022      if (a->used < 2) \{
+023         return 0;
+024      \}
+025   
+026      /* must be of the form b**k - a [a <= b] so all
+027       * but the first digit must be equal to -1 (mod b).
+028       */
+029      for (ix = 1; ix < a->used; ix++) \{
+030          if (a->dp[ix] != MP_MASK) \{
+031             return 0;
+032          \}
+033      \}
+034      return 1;
+035   \}
+036   
 \end{alltt}
 \end{small}
 
@@ -6873,52 +6865,51 @@ shift which makes the algorithm fairly inexpensive to use.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* reduces a modulo n where n is of the form 2**p - d */
-018   int
-019   mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d)
-020   \{
-021      mp_int q;
-022      int    p, res;
-023      
-024      if ((res = mp_init(&q)) != MP_OKAY) \{
-025         return res;
-026      \}
-027      
-028      p = mp_count_bits(n);    
-029   top:
-030      /* q = a/2**p, a = a mod 2**p */
-031      if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{
-032         goto ERR;
-033      \}
-034      
-035      if (d != 1) \{
-036         /* q = q * d */
-037         if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) \{ 
-038            goto ERR;
-039         \}
-040      \}
-041      
-042      /* a = a + q */
-043      if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{
-044         goto ERR;
-045      \}
-046      
-047      if (mp_cmp_mag(a, n) != MP_LT) \{
-048         s_mp_sub(a, n, a);
-049         goto top;
-050      \}
-051      
-052   ERR:
-053      mp_clear(&q);
-054      return res;
-055   \}
-056   
+016   /* reduces a modulo n where n is of the form 2**p - d */
+017   int
+018   mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d)
+019   \{
+020      mp_int q;
+021      int    p, res;
+022      
+023      if ((res = mp_init(&q)) != MP_OKAY) \{
+024         return res;
+025      \}
+026      
+027      p = mp_count_bits(n);    
+028   top:
+029      /* q = a/2**p, a = a mod 2**p */
+030      if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{
+031         goto ERR;
+032      \}
+033      
+034      if (d != 1) \{
+035         /* q = q * d */
+036         if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) \{ 
+037            goto ERR;
+038         \}
+039      \}
+040      
+041      /* a = a + q */
+042      if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{
+043         goto ERR;
+044      \}
+045      
+046      if (mp_cmp_mag(a, n) != MP_LT) \{
+047         s_mp_sub(a, n, a);
+048         goto top;
+049      \}
+050      
+051   ERR:
+052      mp_clear(&q);
+053      return res;
+054   \}
+055   
 \end{alltt}
 \end{small}
 
 The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$.  The call to mp\_div\_2d
-on line 31 calculates both the quotient $q$ and the remainder $a$ required.  By doing both in a single function call the code size
+on line 30 calculates both the quotient $q$ and the remainder $a$ required.  By doing both in a single function call the code size
 is kept fairly small.  The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
 any multiplications.  
 
@@ -6956,33 +6947,32 @@ is sufficient to solve for $k$.  Alternatively if $n$ has more than one digit th
 \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* determines the setup value */
-018   int 
-019   mp_reduce_2k_setup(mp_int *a, mp_digit *d)
-020   \{
-021      int res, p;
-022      mp_int tmp;
-023      
-024      if ((res = mp_init(&tmp)) != MP_OKAY) \{
-025         return res;
-026      \}
-027      
-028      p = mp_count_bits(a);
-029      if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{
-030         mp_clear(&tmp);
-031         return res;
-032      \}
-033      
-034      if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{
-035         mp_clear(&tmp);
-036         return res;
-037      \}
-038      
-039      *d = tmp.dp[0];
-040      mp_clear(&tmp);
-041      return MP_OKAY;
-042   \}
+016   /* determines the setup value */
+017   int 
+018   mp_reduce_2k_setup(mp_int *a, mp_digit *d)
+019   \{
+020      int res, p;
+021      mp_int tmp;
+022      
+023      if ((res = mp_init(&tmp)) != MP_OKAY) \{
+024         return res;
+025      \}
+026      
+027      p = mp_count_bits(a);
+028      if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{
+029         mp_clear(&tmp);
+030         return res;
+031      \}
+032      
+033      if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{
+034         mp_clear(&tmp);
+035         return res;
+036      \}
+037      
+038      *d = tmp.dp[0];
+039      mp_clear(&tmp);
+040      return MP_OKAY;
+041   \}
 \end{alltt}
 \end{small}
 
@@ -7027,11 +7017,11 @@ This algorithm quickly determines if a modulus is of the form required for algor
 \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* determines if mp_reduce_2k can be used */
-018   int mp_reduce_is_2k(mp_int *a)
-019   \{
-020      int ix, iy, iz, iw;
+016   /* determines if mp_reduce_2k can be used */
+017   int mp_reduce_is_2k(mp_int *a)
+018   \{
+019      int ix, iy, iw;
+020      mp_digit iz;
 021      
 022      if (a->used == 0) \{
 023         return 0;
@@ -7048,7 +7038,7 @@ This algorithm quickly determines if a modulus is of the form required for algor
 034                return 0;
 035             \}
 036             iz <<= 1;
-037             if (iz > (int)MP_MASK) \{
+037             if (iz > (mp_digit)MP_MASK) \{
 038                ++iw;
 039                iz = 1;
 040             \}
@@ -7229,49 +7219,48 @@ iteration of the loop moves the bits of the exponent $b$ upwards to the most sig
 \hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* calculate c = a**b  using a square-multiply algorithm */
-018   int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
-019   \{
-020     int     res, x;
-021     mp_int  g;
-022   
-023     if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{
-024       return res;
-025     \}
-026   
-027     /* set initial result */
-028     mp_set (c, 1);
-029   
-030     for (x = 0; x < (int) DIGIT_BIT; x++) \{
-031       /* square */
-032       if ((res = mp_sqr (c, c)) != MP_OKAY) \{
-033         mp_clear (&g);
-034         return res;
-035       \}
-036   
-037       /* if the bit is set multiply */
-038       if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{
-039         if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{
-040            mp_clear (&g);
-041            return res;
-042         \}
-043       \}
-044   
-045       /* shift to next bit */
-046       b <<= 1;
-047     \}
-048   
-049     mp_clear (&g);
-050     return MP_OKAY;
-051   \}
+016   /* calculate c = a**b  using a square-multiply algorithm */
+017   int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
+018   \{
+019     int     res, x;
+020     mp_int  g;
+021   
+022     if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{
+023       return res;
+024     \}
+025   
+026     /* set initial result */
+027     mp_set (c, 1);
+028   
+029     for (x = 0; x < (int) DIGIT_BIT; x++) \{
+030       /* square */
+031       if ((res = mp_sqr (c, c)) != MP_OKAY) \{
+032         mp_clear (&g);
+033         return res;
+034       \}
+035   
+036       /* if the bit is set multiply */
+037       if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{
+038         if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{
+039            mp_clear (&g);
+040            return res;
+041         \}
+042       \}
+043   
+044       /* shift to next bit */
+045       b <<= 1;
+046     \}
+047   
+048     mp_clear (&g);
+049     return MP_OKAY;
+050   \}
 \end{alltt}
 \end{small}
 
-Line 28 sets the initial value of the result to $1$.  Next the loop on line 30 steps through each bit of the exponent starting from
-the most significant down towards the least significant. The invariant squaring operation placed on line 32 is performed first.  After 
+Line 27 sets the initial value of the result to $1$.  Next the loop on line 29 steps through each bit of the exponent starting from
+the most significant down towards the least significant. The invariant squaring operation placed on line 31 is performed first.  After 
 the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set.  The shift on line
-46 moves all of the bits of the exponent upwards towards the most significant location.  
+45 moves all of the bits of the exponent upwards towards the most significant location.  
 
 \section{$k$-ary Exponentiation}
 When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
@@ -7453,78 +7442,77 @@ algorithm since their arguments are essentially the same (\textit{two mp\_ints a
 \vspace{-3mm}
 \begin{alltt}
 016   
-017   
-018   /* this is a shell function that calls either the normal or Montgomery
-019    * exptmod functions.  Originally the call to the montgomery code was
-020    * embedded in the normal function but that wasted alot of stack space
-021    * for nothing (since 99% of the time the Montgomery code would be called)
-022    */
-023   int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
-024   \{
-025     int dr;
-026   
-027     /* modulus P must be positive */
-028     if (P->sign == MP_NEG) \{
-029        return MP_VAL;
-030     \}
-031   
-032     /* if exponent X is negative we have to recurse */
-033     if (X->sign == MP_NEG) \{
-034        mp_int tmpG, tmpX;
-035        int err;
-036   
-037        /* first compute 1/G mod P */
-038        if ((err = mp_init(&tmpG)) != MP_OKAY) \{
-039           return err;
-040        \}
-041        if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{
-042           mp_clear(&tmpG);
-043           return err;
-044        \}
-045   
-046        /* now get |X| */
-047        if ((err = mp_init(&tmpX)) != MP_OKAY) \{
-048           mp_clear(&tmpG);
-049           return err;
-050        \}
-051        if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{
-052           mp_clear_multi(&tmpG, &tmpX, NULL);
-053           return err;
-054        \}
-055   
-056        /* and now compute (1/G)**|X| instead of G**X [X < 0] */
-057        err = mp_exptmod(&tmpG, &tmpX, P, Y);
-058        mp_clear_multi(&tmpG, &tmpX, NULL);
-059        return err;
-060     \}
-061   
-062     /* is it a DR modulus? */
-063     dr = mp_dr_is_modulus(P);
-064   
-065     /* if not, is it a uDR modulus? */
-066     if (dr == 0) \{
-067        dr = mp_reduce_is_2k(P) << 1;
-068     \}
-069       
-070     /* if the modulus is odd or dr != 0 use the fast method */
-071     if (mp_isodd (P) == 1 || dr !=  0) \{
-072       return mp_exptmod_fast (G, X, P, Y, dr);
-073     \} else \{
-074       /* otherwise use the generic Barrett reduction technique */
-075       return s_mp_exptmod (G, X, P, Y);
-076     \}
-077   \}
-078   
+017   /* this is a shell function that calls either the normal or Montgomery
+018    * exptmod functions.  Originally the call to the montgomery code was
+019    * embedded in the normal function but that wasted alot of stack space
+020    * for nothing (since 99% of the time the Montgomery code would be called)
+021    */
+022   int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
+023   \{
+024     int dr;
+025   
+026     /* modulus P must be positive */
+027     if (P->sign == MP_NEG) \{
+028        return MP_VAL;
+029     \}
+030   
+031     /* if exponent X is negative we have to recurse */
+032     if (X->sign == MP_NEG) \{
+033        mp_int tmpG, tmpX;
+034        int err;
+035   
+036        /* first compute 1/G mod P */
+037        if ((err = mp_init(&tmpG)) != MP_OKAY) \{
+038           return err;
+039        \}
+040        if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{
+041           mp_clear(&tmpG);
+042           return err;
+043        \}
+044   
+045        /* now get |X| */
+046        if ((err = mp_init(&tmpX)) != MP_OKAY) \{
+047           mp_clear(&tmpG);
+048           return err;
+049        \}
+050        if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{
+051           mp_clear_multi(&tmpG, &tmpX, NULL);
+052           return err;
+053        \}
+054   
+055        /* and now compute (1/G)**|X| instead of G**X [X < 0] */
+056        err = mp_exptmod(&tmpG, &tmpX, P, Y);
+057        mp_clear_multi(&tmpG, &tmpX, NULL);
+058        return err;
+059     \}
+060   
+061     /* is it a DR modulus? */
+062     dr = mp_dr_is_modulus(P);
+063   
+064     /* if not, is it a uDR modulus? */
+065     if (dr == 0) \{
+066        dr = mp_reduce_is_2k(P) << 1;
+067     \}
+068       
+069     /* if the modulus is odd or dr != 0 use the fast method */
+070     if (mp_isodd (P) == 1 || dr !=  0) \{
+071       return mp_exptmod_fast (G, X, P, Y, dr);
+072     \} else \{
+073       /* otherwise use the generic Barrett reduction technique */
+074       return s_mp_exptmod (G, X, P, Y);
+075     \}
+076   \}
+077   
 \end{alltt}
 \end{small}
 
-In order to keep the algorithms in a known state the first step on line 28 is to reject any negative modulus as input.  If the exponent is
+In order to keep the algorithms in a known state the first step on line 31 is to reject any negative modulus as input.  If the exponent is
 negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$.  The temporary variable $tmpG$ is assigned
 the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$.  The algorithm will recuse with these new values with a positive
 exponent.
 
-If the exponent is positive the algorithm resumes the exponentiation.  Line 63 determines if the modulus is of the restricted Diminished Radix 
-form.  If it is not line 67 attempts to determine if it is of a unrestricted Diminished Radix form.  The integer $dr$ will take on one
+If the exponent is positive the algorithm resumes the exponentiation.  Line 62 determines if the modulus is of the restricted Diminished Radix 
+form.  If it is not line 66 attempts to determine if it is of a unrestricted Diminished Radix form.  The integer $dr$ will take on one
 of three values.
 
 \begin{enumerate}
@@ -7533,7 +7521,7 @@ of three values.
 \item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
 \end{enumerate}
 
-Line 70 determines if the fast modular exponentiation algorithm can be used.  It is allowed if $dr \ne 0$ or if the modulus is odd.  Otherwise,
+Line 69 determines if the fast modular exponentiation algorithm can be used.  It is allowed if $dr \ne 0$ or if the modulus is odd.  Otherwise,
 the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.  
 
 \subsection{Barrett Modular Exponentiation}
@@ -7694,236 +7682,235 @@ a Left-to-Right algorithm is used to process the remaining few bits.
 \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   #ifdef MP_LOW_MEM
-018      #define TAB_SIZE 32
-019   #else
-020      #define TAB_SIZE 256
-021   #endif
-022   
-023   int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
-024   \{
-025     mp_int  M[TAB_SIZE], res, mu;
-026     mp_digit buf;
-027     int     err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;
-028   
-029     /* find window size */
-030     x = mp_count_bits (X);
-031     if (x <= 7) \{
-032       winsize = 2;
-033     \} else if (x <= 36) \{
-034       winsize = 3;
-035     \} else if (x <= 140) \{
-036       winsize = 4;
-037     \} else if (x <= 450) \{
-038       winsize = 5;
-039     \} else if (x <= 1303) \{
-040       winsize = 6;
-041     \} else if (x <= 3529) \{
-042       winsize = 7;
-043     \} else \{
-044       winsize = 8;
-045     \}
-046   
-047   #ifdef MP_LOW_MEM
-048       if (winsize > 5) \{
-049          winsize = 5;
-050       \}
-051   #endif
-052   
-053     /* init M array */
-054     /* init first cell */
-055     if ((err = mp_init(&M[1])) != MP_OKAY) \{
-056        return err; 
-057     \}
-058   
-059     /* now init the second half of the array */
-060     for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{
-061       if ((err = mp_init(&M[x])) != MP_OKAY) \{
-062         for (y = 1<<(winsize-1); y < x; y++) \{
-063           mp_clear (&M[y]);
-064         \}
-065         mp_clear(&M[1]);
-066         return err;
-067       \}
-068     \}
-069   
-070     /* create mu, used for Barrett reduction */
-071     if ((err = mp_init (&mu)) != MP_OKAY) \{
-072       goto __M;
-073     \}
-074     if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{
-075       goto __MU;
-076     \}
-077   
-078     /* create M table
-079      *
-080      * The M table contains powers of the base, 
-081      * e.g. M[x] = G**x mod P
-082      *
-083      * The first half of the table is not 
-084      * computed though accept for M[0] and M[1]
-085      */
-086     if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{
-087       goto __MU;
-088     \}
-089   
-090     /* compute the value at M[1<<(winsize-1)] by squaring 
-091      * M[1] (winsize-1) times 
-092      */
-093     if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{
-094       goto __MU;
-095     \}
-096   
-097     for (x = 0; x < (winsize - 1); x++) \{
-098       if ((err = mp_sqr (&M[1 << (winsize - 1)], 
-099                          &M[1 << (winsize - 1)])) != MP_OKAY) \{
-100         goto __MU;
-101       \}
-102       if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{
-103         goto __MU;
-104       \}
-105     \}
-106   
-107     /* create upper table, that is M[x] = M[x-1] * M[1] (mod P)
-108      * for x = (2**(winsize - 1) + 1) to (2**winsize - 1)
-109      */
-110     for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{
-111       if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{
-112         goto __MU;
-113       \}
-114       if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) \{
-115         goto __MU;
-116       \}
-117     \}
-118   
-119     /* setup result */
-120     if ((err = mp_init (&res)) != MP_OKAY) \{
-121       goto __MU;
-122     \}
-123     mp_set (&res, 1);
-124   
-125     /* set initial mode and bit cnt */
-126     mode   = 0;
-127     bitcnt = 1;
-128     buf    = 0;
-129     digidx = X->used - 1;
-130     bitcpy = 0;
-131     bitbuf = 0;
-132   
-133     for (;;) \{
-134       /* grab next digit as required */
-135       if (--bitcnt == 0) \{
-136         /* if digidx == -1 we are out of digits */
-137         if (digidx == -1) \{
-138           break;
-139         \}
-140         /* read next digit and reset the bitcnt */
-141         buf    = X->dp[digidx--];
-142         bitcnt = (int) DIGIT_BIT;
-143       \}
-144   
-145       /* grab the next msb from the exponent */
-146       y     = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1;
-147       buf <<= (mp_digit)1;
-148   
-149       /* if the bit is zero and mode == 0 then we ignore it
-150        * These represent the leading zero bits before the first 1 bit
-151        * in the exponent.  Technically this opt is not required but it
-152        * does lower the # of trivial squaring/reductions used
-153        */
-154       if (mode == 0 && y == 0) \{
-155         continue;
-156       \}
-157   
-158       /* if the bit is zero and mode == 1 then we square */
-159       if (mode == 1 && y == 0) \{
-160         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
-161           goto __RES;
-162         \}
-163         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-164           goto __RES;
-165         \}
-166         continue;
-167       \}
-168   
-169       /* else we add it to the window */
-170       bitbuf |= (y << (winsize - ++bitcpy));
-171       mode    = 2;
-172   
-173       if (bitcpy == winsize) \{
-174         /* ok window is filled so square as required and multiply  */
-175         /* square first */
-176         for (x = 0; x < winsize; x++) \{
-177           if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
-178             goto __RES;
-179           \}
-180           if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-181             goto __RES;
-182           \}
-183         \}
-184   
-185         /* then multiply */
-186         if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{
-187           goto __RES;
-188         \}
-189         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-190           goto __RES;
-191         \}
-192   
-193         /* empty window and reset */
-194         bitcpy = 0;
-195         bitbuf = 0;
-196         mode   = 1;
-197       \}
-198     \}
-199   
-200     /* if bits remain then square/multiply */
-201     if (mode == 2 && bitcpy > 0) \{
-202       /* square then multiply if the bit is set */
-203       for (x = 0; x < bitcpy; x++) \{
-204         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
-205           goto __RES;
-206         \}
-207         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-208           goto __RES;
-209         \}
-210   
-211         bitbuf <<= 1;
-212         if ((bitbuf & (1 << winsize)) != 0) \{
-213           /* then multiply */
-214           if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{
-215             goto __RES;
-216           \}
-217           if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-218             goto __RES;
-219           \}
-220         \}
-221       \}
-222     \}
-223   
-224     mp_exch (&res, Y);
-225     err = MP_OKAY;
-226   __RES:mp_clear (&res);
-227   __MU:mp_clear (&mu);
-228   __M:
-229     mp_clear(&M[1]);
-230     for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{
-231       mp_clear (&M[x]);
-232     \}
-233     return err;
-234   \}
+016   #ifdef MP_LOW_MEM
+017      #define TAB_SIZE 32
+018   #else
+019      #define TAB_SIZE 256
+020   #endif
+021   
+022   int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
+023   \{
+024     mp_int  M[TAB_SIZE], res, mu;
+025     mp_digit buf;
+026     int     err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;
+027   
+028     /* find window size */
+029     x = mp_count_bits (X);
+030     if (x <= 7) \{
+031       winsize = 2;
+032     \} else if (x <= 36) \{
+033       winsize = 3;
+034     \} else if (x <= 140) \{
+035       winsize = 4;
+036     \} else if (x <= 450) \{
+037       winsize = 5;
+038     \} else if (x <= 1303) \{
+039       winsize = 6;
+040     \} else if (x <= 3529) \{
+041       winsize = 7;
+042     \} else \{
+043       winsize = 8;
+044     \}
+045   
+046   #ifdef MP_LOW_MEM
+047       if (winsize > 5) \{
+048          winsize = 5;
+049       \}
+050   #endif
+051   
+052     /* init M array */
+053     /* init first cell */
+054     if ((err = mp_init(&M[1])) != MP_OKAY) \{
+055        return err; 
+056     \}
+057   
+058     /* now init the second half of the array */
+059     for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{
+060       if ((err = mp_init(&M[x])) != MP_OKAY) \{
+061         for (y = 1<<(winsize-1); y < x; y++) \{
+062           mp_clear (&M[y]);
+063         \}
+064         mp_clear(&M[1]);
+065         return err;
+066       \}
+067     \}
+068   
+069     /* create mu, used for Barrett reduction */
+070     if ((err = mp_init (&mu)) != MP_OKAY) \{
+071       goto __M;
+072     \}
+073     if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{
+074       goto __MU;
+075     \}
+076   
+077     /* create M table
+078      *
+079      * The M table contains powers of the base, 
+080      * e.g. M[x] = G**x mod P
+081      *
+082      * The first half of the table is not 
+083      * computed though accept for M[0] and M[1]
+084      */
+085     if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{
+086       goto __MU;
+087     \}
+088   
+089     /* compute the value at M[1<<(winsize-1)] by squaring 
+090      * M[1] (winsize-1) times 
+091      */
+092     if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{
+093       goto __MU;
+094     \}
+095   
+096     for (x = 0; x < (winsize - 1); x++) \{
+097       if ((err = mp_sqr (&M[1 << (winsize - 1)], 
+098                          &M[1 << (winsize - 1)])) != MP_OKAY) \{
+099         goto __MU;
+100       \}
+101       if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{
+102         goto __MU;
+103       \}
+104     \}
+105   
+106     /* create upper table, that is M[x] = M[x-1] * M[1] (mod P)
+107      * for x = (2**(winsize - 1) + 1) to (2**winsize - 1)
+108      */
+109     for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{
+110       if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{
+111         goto __MU;
+112       \}
+113       if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) \{
+114         goto __MU;
+115       \}
+116     \}
+117   
+118     /* setup result */
+119     if ((err = mp_init (&res)) != MP_OKAY) \{
+120       goto __MU;
+121     \}
+122     mp_set (&res, 1);
+123   
+124     /* set initial mode and bit cnt */
+125     mode   = 0;
+126     bitcnt = 1;
+127     buf    = 0;
+128     digidx = X->used - 1;
+129     bitcpy = 0;
+130     bitbuf = 0;
+131   
+132     for (;;) \{
+133       /* grab next digit as required */
+134       if (--bitcnt == 0) \{
+135         /* if digidx == -1 we are out of digits */
+136         if (digidx == -1) \{
+137           break;
+138         \}
+139         /* read next digit and reset the bitcnt */
+140         buf    = X->dp[digidx--];
+141         bitcnt = (int) DIGIT_BIT;
+142       \}
+143   
+144       /* grab the next msb from the exponent */
+145       y     = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1;
+146       buf <<= (mp_digit)1;
+147   
+148       /* if the bit is zero and mode == 0 then we ignore it
+149        * These represent the leading zero bits before the first 1 bit
+150        * in the exponent.  Technically this opt is not required but it
+151        * does lower the # of trivial squaring/reductions used
+152        */
+153       if (mode == 0 && y == 0) \{
+154         continue;
+155       \}
+156   
+157       /* if the bit is zero and mode == 1 then we square */
+158       if (mode == 1 && y == 0) \{
+159         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
+160           goto __RES;
+161         \}
+162         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
+163           goto __RES;
+164         \}
+165         continue;
+166       \}
+167   
+168       /* else we add it to the window */
+169       bitbuf |= (y << (winsize - ++bitcpy));
+170       mode    = 2;
+171   
+172       if (bitcpy == winsize) \{
+173         /* ok window is filled so square as required and multiply  */
+174         /* square first */
+175         for (x = 0; x < winsize; x++) \{
+176           if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
+177             goto __RES;
+178           \}
+179           if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
+180             goto __RES;
+181           \}
+182         \}
+183   
+184         /* then multiply */
+185         if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{
+186           goto __RES;
+187         \}
+188         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
+189           goto __RES;
+190         \}
+191   
+192         /* empty window and reset */
+193         bitcpy = 0;
+194         bitbuf = 0;
+195         mode   = 1;
+196       \}
+197     \}
+198   
+199     /* if bits remain then square/multiply */
+200     if (mode == 2 && bitcpy > 0) \{
+201       /* square then multiply if the bit is set */
+202       for (x = 0; x < bitcpy; x++) \{
+203         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
+204           goto __RES;
+205         \}
+206         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
+207           goto __RES;
+208         \}
+209   
+210         bitbuf <<= 1;
+211         if ((bitbuf & (1 << winsize)) != 0) \{
+212           /* then multiply */
+213           if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{
+214             goto __RES;
+215           \}
+216           if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
+217             goto __RES;
+218           \}
+219         \}
+220       \}
+221     \}
+222   
+223     mp_exch (&res, Y);
+224     err = MP_OKAY;
+225   __RES:mp_clear (&res);
+226   __MU:mp_clear (&mu);
+227   __M:
+228     mp_clear(&M[1]);
+229     for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{
+230       mp_clear (&M[x]);
+231     \}
+232     return err;
+233   \}
 \end{alltt}
 \end{small}
 
-Lines 31 through 41 determine the optimal window size based on the length of the exponent in bits.  The window divisions are sorted
+Lines 30 through 40 determine the optimal window size based on the length of the exponent in bits.  The window divisions are sorted
 from smallest to greatest so that in each \textbf{if} statement only one condition must be tested.  For example, by the \textbf{if} statement 
-on line 33 the value of $x$ is already known to be greater than $140$.  
+on line 32 the value of $x$ is already known to be greater than $140$.  
 
-The conditional piece of code beginning on line 47 allows the window size to be restricted to five bits.  This logic is used to ensure
+The conditional piece of code beginning on line 46 allows the window size to be restricted to five bits.  This logic is used to ensure
 the table of precomputed powers of $G$ remains relatively small.  
 
-The for loop on line 60 initializes the $M$ array while lines 61 and 74 compute the value of $\mu$ required for
+The for loop on line 59 initializes the $M$ array while lines 60 and 73 compute the value of $\mu$ required for
 Barrett reduction.  
 
 -- More later.
@@ -7958,33 +7945,32 @@ equivalent to $m \cdot 2^k$.  By this logic when $m = 1$ a quick power of two ca
 \hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* computes a = 2**b 
-018    *
-019    * Simple algorithm which zeroes the int, grows it then just sets one bit
-020    * as required.
-021    */
-022   int
-023   mp_2expt (mp_int * a, int b)
-024   \{
-025     int     res;
-026   
-027     /* zero a as per default */
-028     mp_zero (a);
-029   
-030     /* grow a to accomodate the single bit */
-031     if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) \{
-032       return res;
-033     \}
-034   
-035     /* set the used count of where the bit will go */
-036     a->used = b / DIGIT_BIT + 1;
-037   
-038     /* put the single bit in its place */
-039     a->dp[b / DIGIT_BIT] = 1 << (b % DIGIT_BIT);
-040   
-041     return MP_OKAY;
-042   \}
+016   /* computes a = 2**b 
+017    *
+018    * Simple algorithm which zeroes the int, grows it then just sets one bit
+019    * as required.
+020    */
+021   int
+022   mp_2expt (mp_int * a, int b)
+023   \{
+024     int     res;
+025   
+026     /* zero a as per default */
+027     mp_zero (a);
+028   
+029     /* grow a to accomodate the single bit */
+030     if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) \{
+031       return res;
+032     \}
+033   
+034     /* set the used count of where the bit will go */
+035     a->used = b / DIGIT_BIT + 1;
+036   
+037     /* put the single bit in its place */
+038     a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT);
+039   
+040     return MP_OKAY;
+041   \}
 \end{alltt}
 \end{small}
 
@@ -8233,202 +8219,201 @@ respectively be replaced with a zero.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_div.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* integer signed division. 
-018    * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
-019    * HAC pp.598 Algorithm 14.20
-020    *
-021    * Note that the description in HAC is horribly 
-022    * incomplete.  For example, it doesn't consider 
-023    * the case where digits are removed from 'x' in 
-024    * the inner loop.  It also doesn't consider the 
-025    * case that y has fewer than three digits, etc..
-026    *
-027    * The overall algorithm is as described as 
-028    * 14.20 from HAC but fixed to treat these cases.
-029   */
-030   int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
-031   \{
-032     mp_int  q, x, y, t1, t2;
-033     int     res, n, t, i, norm, neg;
-034   
-035     /* is divisor zero ? */
-036     if (mp_iszero (b) == 1) \{
-037       return MP_VAL;
-038     \}
-039   
-040     /* if a < b then q=0, r = a */
-041     if (mp_cmp_mag (a, b) == MP_LT) \{
-042       if (d != NULL) \{
-043         res = mp_copy (a, d);
-044       \} else \{
-045         res = MP_OKAY;
-046       \}
-047       if (c != NULL) \{
-048         mp_zero (c);
-049       \}
-050       return res;
-051     \}
-052   
-053     if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) \{
-054       return res;
-055     \}
-056     q.used = a->used + 2;
-057   
-058     if ((res = mp_init (&t1)) != MP_OKAY) \{
-059       goto __Q;
-060     \}
-061   
-062     if ((res = mp_init (&t2)) != MP_OKAY) \{
-063       goto __T1;
-064     \}
-065   
-066     if ((res = mp_init_copy (&x, a)) != MP_OKAY) \{
-067       goto __T2;
-068     \}
-069   
-070     if ((res = mp_init_copy (&y, b)) != MP_OKAY) \{
-071       goto __X;
-072     \}
-073   
-074     /* fix the sign */
-075     neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
-076     x.sign = y.sign = MP_ZPOS;
-077   
-078     /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */
-079     norm = mp_count_bits(&y) % DIGIT_BIT;
-080     if (norm < (int)(DIGIT_BIT-1)) \{
-081        norm = (DIGIT_BIT-1) - norm;
-082        if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) \{
-083          goto __Y;
-084        \}
-085        if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) \{
-086          goto __Y;
-087        \}
-088     \} else \{
-089        norm = 0;
-090     \}
-091   
-092     /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */
-093     n = x.used - 1;
-094     t = y.used - 1;
-095   
-096     /* while (x >= y*b**n-t) do \{ q[n-t] += 1; x -= y*b**\{n-t\} \} */
-097     if ((res = mp_lshd (&y, n - t)) != MP_OKAY) \{ /* y = y*b**\{n-t\} */
-098       goto __Y;
-099     \}
-100   
-101     while (mp_cmp (&x, &y) != MP_LT) \{
-102       ++(q.dp[n - t]);
-103       if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) \{
-104         goto __Y;
-105       \}
-106     \}
-107   
-108     /* reset y by shifting it back down */
-109     mp_rshd (&y, n - t);
-110   
-111     /* step 3. for i from n down to (t + 1) */
-112     for (i = n; i >= (t + 1); i--) \{
-113       if (i > x.used) \{
-114         continue;
-115       \}
-116   
-117       /* step 3.1 if xi == yt then set q\{i-t-1\} to b-1, 
-118        * otherwise set q\{i-t-1\} to (xi*b + x\{i-1\})/yt */
-119       if (x.dp[i] == y.dp[t]) \{
-120         q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1);
-121       \} else \{
-122         mp_word tmp;
-123         tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT);
-124         tmp |= ((mp_word) x.dp[i - 1]);
-125         tmp /= ((mp_word) y.dp[t]);
-126         if (tmp > (mp_word) MP_MASK)
-127           tmp = MP_MASK;
-128         q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK));
-129       \}
-130   
-131       /* while (q\{i-t-1\} * (yt * b + y\{t-1\})) > 
-132                xi * b**2 + xi-1 * b + xi-2 
-133        
-134          do q\{i-t-1\} -= 1; 
-135       */
-136       q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK;
-137       do \{
-138         q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK;
-139   
-140         /* find left hand */
-141         mp_zero (&t1);
-142         t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1];
-143         t1.dp[1] = y.dp[t];
-144         t1.used = 2;
-145         if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) \{
-146           goto __Y;
-147         \}
-148   
-149         /* find right hand */
-150         t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2];
-151         t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1];
-152         t2.dp[2] = x.dp[i];
-153         t2.used = 3;
-154       \} while (mp_cmp_mag(&t1, &t2) == MP_GT);
-155   
-156       /* step 3.3 x = x - q\{i-t-1\} * y * b**\{i-t-1\} */
-157       if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) \{
-158         goto __Y;
-159       \}
-160   
-161       if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{
-162         goto __Y;
-163       \}
-164   
-165       if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) \{
-166         goto __Y;
-167       \}
-168   
-169       /* if x < 0 then \{ x = x + y*b**\{i-t-1\}; q\{i-t-1\} -= 1; \} */
-170       if (x.sign == MP_NEG) \{
-171         if ((res = mp_copy (&y, &t1)) != MP_OKAY) \{
-172           goto __Y;
-173         \}
-174         if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{
-175           goto __Y;
-176         \}
-177         if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) \{
-178           goto __Y;
-179         \}
-180   
-181         q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK;
-182       \}
-183     \}
-184   
-185     /* now q is the quotient and x is the remainder 
-186      * [which we have to normalize] 
-187      */
-188     
-189     /* get sign before writing to c */
-190     x.sign = a->sign;
-191   
-192     if (c != NULL) \{
-193       mp_clamp (&q);
-194       mp_exch (&q, c);
-195       c->sign = neg;
-196     \}
-197   
-198     if (d != NULL) \{
-199       mp_div_2d (&x, norm, &x, NULL);
-200       mp_exch (&x, d);
-201     \}
-202   
-203     res = MP_OKAY;
-204   
-205   __Y:mp_clear (&y);
-206   __X:mp_clear (&x);
-207   __T2:mp_clear (&t2);
-208   __T1:mp_clear (&t1);
-209   __Q:mp_clear (&q);
-210     return res;
-211   \}
+016   /* integer signed division. 
+017    * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
+018    * HAC pp.598 Algorithm 14.20
+019    *
+020    * Note that the description in HAC is horribly 
+021    * incomplete.  For example, it doesn't consider 
+022    * the case where digits are removed from 'x' in 
+023    * the inner loop.  It also doesn't consider the 
+024    * case that y has fewer than three digits, etc..
+025    *
+026    * The overall algorithm is as described as 
+027    * 14.20 from HAC but fixed to treat these cases.
+028   */
+029   int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d)
+030   \{
+031     mp_int  q, x, y, t1, t2;
+032     int     res, n, t, i, norm, neg;
+033   
+034     /* is divisor zero ? */
+035     if (mp_iszero (b) == 1) \{
+036       return MP_VAL;
+037     \}
+038   
+039     /* if a < b then q=0, r = a */
+040     if (mp_cmp_mag (a, b) == MP_LT) \{
+041       if (d != NULL) \{
+042         res = mp_copy (a, d);
+043       \} else \{
+044         res = MP_OKAY;
+045       \}
+046       if (c != NULL) \{
+047         mp_zero (c);
+048       \}
+049       return res;
+050     \}
+051   
+052     if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) \{
+053       return res;
+054     \}
+055     q.used = a->used + 2;
+056   
+057     if ((res = mp_init (&t1)) != MP_OKAY) \{
+058       goto __Q;
+059     \}
+060   
+061     if ((res = mp_init (&t2)) != MP_OKAY) \{
+062       goto __T1;
+063     \}
+064   
+065     if ((res = mp_init_copy (&x, a)) != MP_OKAY) \{
+066       goto __T2;
+067     \}
+068   
+069     if ((res = mp_init_copy (&y, b)) != MP_OKAY) \{
+070       goto __X;
+071     \}
+072   
+073     /* fix the sign */
+074     neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
+075     x.sign = y.sign = MP_ZPOS;
+076   
+077     /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */
+078     norm = mp_count_bits(&y) % DIGIT_BIT;
+079     if (norm < (int)(DIGIT_BIT-1)) \{
+080        norm = (DIGIT_BIT-1) - norm;
+081        if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) \{
+082          goto __Y;
+083        \}
+084        if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) \{
+085          goto __Y;
+086        \}
+087     \} else \{
+088        norm = 0;
+089     \}
+090   
+091     /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */
+092     n = x.used - 1;
+093     t = y.used - 1;
+094   
+095     /* while (x >= y*b**n-t) do \{ q[n-t] += 1; x -= y*b**\{n-t\} \} */
+096     if ((res = mp_lshd (&y, n - t)) != MP_OKAY) \{ /* y = y*b**\{n-t\} */
+097       goto __Y;
+098     \}
+099   
+100     while (mp_cmp (&x, &y) != MP_LT) \{
+101       ++(q.dp[n - t]);
+102       if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) \{
+103         goto __Y;
+104       \}
+105     \}
+106   
+107     /* reset y by shifting it back down */
+108     mp_rshd (&y, n - t);
+109   
+110     /* step 3. for i from n down to (t + 1) */
+111     for (i = n; i >= (t + 1); i--) \{
+112       if (i > x.used) \{
+113         continue;
+114       \}
+115   
+116       /* step 3.1 if xi == yt then set q\{i-t-1\} to b-1, 
+117        * otherwise set q\{i-t-1\} to (xi*b + x\{i-1\})/yt */
+118       if (x.dp[i] == y.dp[t]) \{
+119         q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1);
+120       \} else \{
+121         mp_word tmp;
+122         tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT);
+123         tmp |= ((mp_word) x.dp[i - 1]);
+124         tmp /= ((mp_word) y.dp[t]);
+125         if (tmp > (mp_word) MP_MASK)
+126           tmp = MP_MASK;
+127         q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK));
+128       \}
+129   
+130       /* while (q\{i-t-1\} * (yt * b + y\{t-1\})) > 
+131                xi * b**2 + xi-1 * b + xi-2 
+132        
+133          do q\{i-t-1\} -= 1; 
+134       */
+135       q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK;
+136       do \{
+137         q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK;
+138   
+139         /* find left hand */
+140         mp_zero (&t1);
+141         t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1];
+142         t1.dp[1] = y.dp[t];
+143         t1.used = 2;
+144         if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) \{
+145           goto __Y;
+146         \}
+147   
+148         /* find right hand */
+149         t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2];
+150         t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1];
+151         t2.dp[2] = x.dp[i];
+152         t2.used = 3;
+153       \} while (mp_cmp_mag(&t1, &t2) == MP_GT);
+154   
+155       /* step 3.3 x = x - q\{i-t-1\} * y * b**\{i-t-1\} */
+156       if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) \{
+157         goto __Y;
+158       \}
+159   
+160       if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{
+161         goto __Y;
+162       \}
+163   
+164       if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) \{
+165         goto __Y;
+166       \}
+167   
+168       /* if x < 0 then \{ x = x + y*b**\{i-t-1\}; q\{i-t-1\} -= 1; \} */
+169       if (x.sign == MP_NEG) \{
+170         if ((res = mp_copy (&y, &t1)) != MP_OKAY) \{
+171           goto __Y;
+172         \}
+173         if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{
+174           goto __Y;
+175         \}
+176         if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) \{
+177           goto __Y;
+178         \}
+179   
+180         q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK;
+181       \}
+182     \}
+183   
+184     /* now q is the quotient and x is the remainder 
+185      * [which we have to normalize] 
+186      */
+187     
+188     /* get sign before writing to c */
+189     x.sign = x.used == 0 ? MP_ZPOS : a->sign;
+190   
+191     if (c != NULL) \{
+192       mp_clamp (&q);
+193       mp_exch (&q, c);
+194       c->sign = neg;
+195     \}
+196   
+197     if (d != NULL) \{
+198       mp_div_2d (&x, norm, &x, NULL);
+199       mp_exch (&x, d);
+200     \}
+201   
+202     res = MP_OKAY;
+203   
+204   __Y:mp_clear (&y);
+205   __X:mp_clear (&x);
+206   __T2:mp_clear (&t2);
+207   __T1:mp_clear (&t1);
+208   __Q:mp_clear (&q);
+209     return res;
+210   \}
 \end{alltt}
 \end{small}
 
@@ -8440,9 +8425,9 @@ algorithm with only the quotient is
 mp_div(&a, &b, &c, NULL);  /* c = [a/b] */
 \end{verbatim}
 
-Lines 36 and 42 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor 
-respectively.  After the two trivial cases all of the temporary variables are initialized.  Line 75 determines the sign of 
-the quotient and line 76 ensures that both $x$ and $y$ are positive.  
+Lines 39 and 41 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor 
+respectively.  After the two trivial cases all of the temporary variables are initialized.  Line 74 determines the sign of 
+the quotient and line 75 ensures that both $x$ and $y$ are positive.  
 
 The number of bits in the leading digit is calculated on line 80.  Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits
 of precision which when reduced modulo $lg(\beta)$ produces the value of $k$.  In this case $k$ is the number of bits in the leading digit which is
@@ -8450,9 +8435,9 @@ exactly what is required.  For the algorithm to operate $k$ must equal $lg(\beta
 them to the left by $lg(\beta) - 1 - k$ bits.
 
 Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively.  These are first used to produce the 
-leading digit of the quotient.  The loop beginning on line 112 will produce the remainder of the quotient digits.
+leading digit of the quotient.  The loop beginning on line 111 will produce the remainder of the quotient digits.
 
-The conditional ``continue'' on line 113 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the
+The conditional ``continue'' on line 116 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the
 algorithm eliminates multiple non-zero digits in a single iteration.  This ensures that $x_i$ is always non-zero since by definition the digits
 above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}.  
 
@@ -8494,94 +8479,93 @@ This algorithm initiates a temporary mp\_int with the value of the single digit
 \hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* single digit addition */
-018   int
-019   mp_add_d (mp_int * a, mp_digit b, mp_int * c)
-020   \{
-021     int     res, ix, oldused;
-022     mp_digit *tmpa, *tmpc, mu;
-023   
-024     /* grow c as required */
-025     if (c->alloc < a->used + 1) \{
-026        if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) \{
-027           return res;
-028        \}
-029     \}
-030   
-031     /* if a is negative and |a| >= b, call c = |a| - b */
-032     if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) \{
-033        /* temporarily fix sign of a */
-034        a->sign = MP_ZPOS;
-035   
-036        /* c = |a| - b */
-037        res = mp_sub_d(a, b, c);
-038   
-039        /* fix sign  */
-040        a->sign = c->sign = MP_NEG;
-041   
-042        return res;
-043     \}
-044   
-045     /* old number of used digits in c */
-046     oldused = c->used;
-047   
-048     /* sign always positive */
-049     c->sign = MP_ZPOS;
-050   
-051     /* source alias */
-052     tmpa    = a->dp;
-053   
-054     /* destination alias */
-055     tmpc    = c->dp;
-056   
-057     /* if a is positive */
-058     if (a->sign == MP_ZPOS) \{
-059        /* add digit, after this we're propagating
-060         * the carry.
-061         */
-062        *tmpc   = *tmpa++ + b;
-063        mu      = *tmpc >> DIGIT_BIT;
-064        *tmpc++ &= MP_MASK;
-065   
-066        /* now handle rest of the digits */
-067        for (ix = 1; ix < a->used; ix++) \{
-068           *tmpc   = *tmpa++ + mu;
-069           mu      = *tmpc >> DIGIT_BIT;
-070           *tmpc++ &= MP_MASK;
-071        \}
-072        /* set final carry */
-073        ix++;
-074        *tmpc++  = mu;
-075   
-076        /* setup size */
-077        c->used = a->used + 1;
-078     \} else \{
-079        /* a was negative and |a| < b */
-080        c->used  = 1;
-081   
-082        /* the result is a single digit */
-083        if (a->used == 1) \{
-084           *tmpc++  =  b - a->dp[0];
-085        \} else \{
-086           *tmpc++  =  b;
-087        \}
-088   
-089        /* setup count so the clearing of oldused
-090         * can fall through correctly
-091         */
-092        ix       = 1;
-093     \}
-094   
-095     /* now zero to oldused */
-096     while (ix++ < oldused) \{
-097        *tmpc++ = 0;
-098     \}
-099     mp_clamp(c);
-100   
-101     return MP_OKAY;
-102   \}
-103   
+016   /* single digit addition */
+017   int
+018   mp_add_d (mp_int * a, mp_digit b, mp_int * c)
+019   \{
+020     int     res, ix, oldused;
+021     mp_digit *tmpa, *tmpc, mu;
+022   
+023     /* grow c as required */
+024     if (c->alloc < a->used + 1) \{
+025        if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) \{
+026           return res;
+027        \}
+028     \}
+029   
+030     /* if a is negative and |a| >= b, call c = |a| - b */
+031     if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) \{
+032        /* temporarily fix sign of a */
+033        a->sign = MP_ZPOS;
+034   
+035        /* c = |a| - b */
+036        res = mp_sub_d(a, b, c);
+037   
+038        /* fix sign  */
+039        a->sign = c->sign = MP_NEG;
+040   
+041        return res;
+042     \}
+043   
+044     /* old number of used digits in c */
+045     oldused = c->used;
+046   
+047     /* sign always positive */
+048     c->sign = MP_ZPOS;
+049   
+050     /* source alias */
+051     tmpa    = a->dp;
+052   
+053     /* destination alias */
+054     tmpc    = c->dp;
+055   
+056     /* if a is positive */
+057     if (a->sign == MP_ZPOS) \{
+058        /* add digit, after this we're propagating
+059         * the carry.
+060         */
+061        *tmpc   = *tmpa++ + b;
+062        mu      = *tmpc >> DIGIT_BIT;
+063        *tmpc++ &= MP_MASK;
+064   
+065        /* now handle rest of the digits */
+066        for (ix = 1; ix < a->used; ix++) \{
+067           *tmpc   = *tmpa++ + mu;
+068           mu      = *tmpc >> DIGIT_BIT;
+069           *tmpc++ &= MP_MASK;
+070        \}
+071        /* set final carry */
+072        ix++;
+073        *tmpc++  = mu;
+074   
+075        /* setup size */
+076        c->used = a->used + 1;
+077     \} else \{
+078        /* a was negative and |a| < b */
+079        c->used  = 1;
+080   
+081        /* the result is a single digit */
+082        if (a->used == 1) \{
+083           *tmpc++  =  b - a->dp[0];
+084        \} else \{
+085           *tmpc++  =  b;
+086        \}
+087   
+088        /* setup count so the clearing of oldused
+089         * can fall through correctly
+090         */
+091        ix       = 1;
+092     \}
+093   
+094     /* now zero to oldused */
+095     while (ix++ < oldused) \{
+096        *tmpc++ = 0;
+097     \}
+098     mp_clamp(c);
+099   
+100     return MP_OKAY;
+101   \}
+102   
 \end{alltt}
 \end{small}
 
@@ -8632,63 +8616,62 @@ Unlike the full multiplication algorithms this algorithm does not require any si
 \hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* multiply by a digit */
-018   int
-019   mp_mul_d (mp_int * a, mp_digit b, mp_int * c)
-020   \{
-021     mp_digit u, *tmpa, *tmpc;
-022     mp_word  r;
-023     int      ix, res, olduse;
-024   
-025     /* make sure c is big enough to hold a*b */
-026     if (c->alloc < a->used + 1) \{
-027       if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) \{
-028         return res;
-029       \}
-030     \}
-031   
-032     /* get the original destinations used count */
-033     olduse = c->used;
-034   
-035     /* set the sign */
-036     c->sign = a->sign;
-037   
-038     /* alias for a->dp [source] */
-039     tmpa = a->dp;
-040   
-041     /* alias for c->dp [dest] */
-042     tmpc = c->dp;
-043   
-044     /* zero carry */
-045     u = 0;
-046   
-047     /* compute columns */
-048     for (ix = 0; ix < a->used; ix++) \{
-049       /* compute product and carry sum for this term */
-050       r       = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b);
-051   
-052       /* mask off higher bits to get a single digit */
-053       *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK));
-054   
-055       /* send carry into next iteration */
-056       u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
-057     \}
-058   
-059     /* store final carry [if any] */
-060     *tmpc++ = u;
-061   
-062     /* now zero digits above the top */
-063     while (ix++ < olduse) \{
-064        *tmpc++ = 0;
-065     \}
-066   
-067     /* set used count */
-068     c->used = a->used + 1;
-069     mp_clamp(c);
-070   
-071     return MP_OKAY;
-072   \}
+016   /* multiply by a digit */
+017   int
+018   mp_mul_d (mp_int * a, mp_digit b, mp_int * c)
+019   \{
+020     mp_digit u, *tmpa, *tmpc;
+021     mp_word  r;
+022     int      ix, res, olduse;
+023   
+024     /* make sure c is big enough to hold a*b */
+025     if (c->alloc < a->used + 1) \{
+026       if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) \{
+027         return res;
+028       \}
+029     \}
+030   
+031     /* get the original destinations used count */
+032     olduse = c->used;
+033   
+034     /* set the sign */
+035     c->sign = a->sign;
+036   
+037     /* alias for a->dp [source] */
+038     tmpa = a->dp;
+039   
+040     /* alias for c->dp [dest] */
+041     tmpc = c->dp;
+042   
+043     /* zero carry */
+044     u = 0;
+045   
+046     /* compute columns */
+047     for (ix = 0; ix < a->used; ix++) \{
+048       /* compute product and carry sum for this term */
+049       r       = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b);
+050   
+051       /* mask off higher bits to get a single digit */
+052       *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK));
+053   
+054       /* send carry into next iteration */
+055       u       = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
+056     \}
+057   
+058     /* store final carry [if any] */
+059     *tmpc++ = u;
+060   
+061     /* now zero digits above the top */
+062     while (ix++ < olduse) \{
+063        *tmpc++ = 0;
+064     \}
+065   
+066     /* set used count */
+067     c->used = a->used + 1;
+068     mp_clamp(c);
+069   
+070     return MP_OKAY;
+071   \}
 \end{alltt}
 \end{small}
 
@@ -8744,100 +8727,99 @@ from chapter seven.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   static int s_is_power_of_two(mp_digit b, int *p)
-018   \{
-019      int x;
-020   
-021      for (x = 1; x < DIGIT_BIT; x++) \{
-022         if (b == (((mp_digit)1)<<x)) \{
-023            *p = x;
-024            return 1;
-025         \}
-026      \}
-027      return 0;
-028   \}
-029   
-030   /* single digit division (based on routine from MPI) */
-031   int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d)
-032   \{
-033     mp_int  q;
-034     mp_word w;
-035     mp_digit t;
-036     int     res, ix;
-037   
-038     /* cannot divide by zero */
-039     if (b == 0) \{
-040        return MP_VAL;
-041     \}
-042   
-043     /* quick outs */
-044     if (b == 1 || mp_iszero(a) == 1) \{
-045        if (d != NULL) \{
-046           *d = 0;
-047        \}
-048        if (c != NULL) \{
-049           return mp_copy(a, c);
-050        \}
-051        return MP_OKAY;
-052     \}
-053   
-054     /* power of two ? */
-055     if (s_is_power_of_two(b, &ix) == 1) \{
-056        if (d != NULL) \{
-057           *d = a->dp[0] & ((1<<ix) - 1);
-058        \}
-059        if (c != NULL) \{
-060           return mp_div_2d(a, ix, c, NULL);
-061        \}
-062        return MP_OKAY;
-063     \}
-064   
-065     /* three? */
-066     if (b == 3) \{
-067        return mp_div_3(a, c, d);
-068     \}
-069   
-070     /* no easy answer [c'est la vie].  Just division */
-071     if ((res = mp_init_size(&q, a->used)) != MP_OKAY) \{
-072        return res;
-073     \}
-074     
-075     q.used = a->used;
-076     q.sign = a->sign;
-077     w = 0;
-078     for (ix = a->used - 1; ix >= 0; ix--) \{
-079        w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]);
-080        
-081        if (w >= b) \{
-082           t = (mp_digit)(w / b);
-083           w -= ((mp_word)t) * ((mp_word)b);
-084         \} else \{
-085           t = 0;
-086         \}
-087         q.dp[ix] = (mp_digit)t;
-088     \}
-089     
-090     if (d != NULL) \{
-091        *d = (mp_digit)w;
-092     \}
-093     
-094     if (c != NULL) \{
-095        mp_clamp(&q);
-096        mp_exch(&q, c);
-097     \}
-098     mp_clear(&q);
-099     
-100     return res;
-101   \}
-102   
+016   static int s_is_power_of_two(mp_digit b, int *p)
+017   \{
+018      int x;
+019   
+020      for (x = 1; x < DIGIT_BIT; x++) \{
+021         if (b == (((mp_digit)1)<<x)) \{
+022            *p = x;
+023            return 1;
+024         \}
+025      \}
+026      return 0;
+027   \}
+028   
+029   /* single digit division (based on routine from MPI) */
+030   int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d)
+031   \{
+032     mp_int  q;
+033     mp_word w;
+034     mp_digit t;
+035     int     res, ix;
+036   
+037     /* cannot divide by zero */
+038     if (b == 0) \{
+039        return MP_VAL;
+040     \}
+041   
+042     /* quick outs */
+043     if (b == 1 || mp_iszero(a) == 1) \{
+044        if (d != NULL) \{
+045           *d = 0;
+046        \}
+047        if (c != NULL) \{
+048           return mp_copy(a, c);
+049        \}
+050        return MP_OKAY;
+051     \}
+052   
+053     /* power of two ? */
+054     if (s_is_power_of_two(b, &ix) == 1) \{
+055        if (d != NULL) \{
+056           *d = a->dp[0] & ((1<<ix) - 1);
+057        \}
+058        if (c != NULL) \{
+059           return mp_div_2d(a, ix, c, NULL);
+060        \}
+061        return MP_OKAY;
+062     \}
+063   
+064     /* three? */
+065     if (b == 3) \{
+066        return mp_div_3(a, c, d);
+067     \}
+068   
+069     /* no easy answer [c'est la vie].  Just division */
+070     if ((res = mp_init_size(&q, a->used)) != MP_OKAY) \{
+071        return res;
+072     \}
+073     
+074     q.used = a->used;
+075     q.sign = a->sign;
+076     w = 0;
+077     for (ix = a->used - 1; ix >= 0; ix--) \{
+078        w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]);
+079        
+080        if (w >= b) \{
+081           t = (mp_digit)(w / b);
+082           w -= ((mp_word)t) * ((mp_word)b);
+083         \} else \{
+084           t = 0;
+085         \}
+086         q.dp[ix] = (mp_digit)t;
+087     \}
+088     
+089     if (d != NULL) \{
+090        *d = (mp_digit)w;
+091     \}
+092     
+093     if (c != NULL) \{
+094        mp_clamp(&q);
+095        mp_exch(&q, c);
+096     \}
+097     mp_clear(&q);
+098     
+099     return res;
+100   \}
+101   
 \end{alltt}
 \end{small}
 
 Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to
 indicate the respective value is not required.  This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created.
 
-The division and remainder on lines 43 and @45,%@ can be replaced often by a single division on most processors.  For example, the 32-bit x86 based 
+The division and remainder on lines 42 and @45,%@ can be replaced often by a single division on most processors.  For example, the 32-bit x86 based 
 processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously.  Unfortunately the GCC 
 compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively.  
 
@@ -8905,117 +8887,116 @@ root.  Ideally this algorithm is meant to find the $n$'th root of an input where
 \hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* find the n'th root of an integer 
-018    *
-019    * Result found such that (c)**b <= a and (c+1)**b > a 
-020    *
-021    * This algorithm uses Newton's approximation 
-022    * x[i+1] = x[i] - f(x[i])/f'(x[i]) 
-023    * which will find the root in log(N) time where 
-024    * each step involves a fair bit.  This is not meant to 
-025    * find huge roots [square and cube, etc].
-026    */
-027   int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
-028   \{
-029     mp_int  t1, t2, t3;
-030     int     res, neg;
-031   
-032     /* input must be positive if b is even */
-033     if ((b & 1) == 0 && a->sign == MP_NEG) \{
-034       return MP_VAL;
-035     \}
-036   
-037     if ((res = mp_init (&t1)) != MP_OKAY) \{
-038       return res;
-039     \}
-040   
-041     if ((res = mp_init (&t2)) != MP_OKAY) \{
-042       goto __T1;
-043     \}
-044   
-045     if ((res = mp_init (&t3)) != MP_OKAY) \{
-046       goto __T2;
-047     \}
-048   
-049     /* if a is negative fudge the sign but keep track */
-050     neg     = a->sign;
-051     a->sign = MP_ZPOS;
-052   
-053     /* t2 = 2 */
-054     mp_set (&t2, 2);
-055   
-056     do \{
-057       /* t1 = t2 */
-058       if ((res = mp_copy (&t2, &t1)) != MP_OKAY) \{
-059         goto __T3;
-060       \}
-061   
-062       /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */
-063       
-064       /* t3 = t1**(b-1) */
-065       if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) \{   
-066         goto __T3;
-067       \}
-068   
-069       /* numerator */
-070       /* t2 = t1**b */
-071       if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) \{    
-072         goto __T3;
-073       \}
-074   
-075       /* t2 = t1**b - a */
-076       if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) \{  
-077         goto __T3;
-078       \}
-079   
-080       /* denominator */
-081       /* t3 = t1**(b-1) * b  */
-082       if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) \{    
-083         goto __T3;
-084       \}
-085   
-086       /* t3 = (t1**b - a)/(b * t1**(b-1)) */
-087       if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) \{  
-088         goto __T3;
-089       \}
-090   
-091       if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) \{
-092         goto __T3;
-093       \}
-094     \}  while (mp_cmp (&t1, &t2) != MP_EQ);
-095   
-096     /* result can be off by a few so check */
-097     for (;;) \{
-098       if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) \{
-099         goto __T3;
-100       \}
-101   
-102       if (mp_cmp (&t2, a) == MP_GT) \{
-103         if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) \{
-104            goto __T3;
-105         \}
-106       \} else \{
-107         break;
-108       \}
-109     \}
-110   
-111     /* reset the sign of a first */
-112     a->sign = neg;
-113   
-114     /* set the result */
-115     mp_exch (&t1, c);
-116   
-117     /* set the sign of the result */
-118     c->sign = neg;
-119   
-120     res = MP_OKAY;
-121   
-122   __T3:mp_clear (&t3);
-123   __T2:mp_clear (&t2);
-124   __T1:mp_clear (&t1);
-125     return res;
-126   \}
+016   /* find the n'th root of an integer 
+017    *
+018    * Result found such that (c)**b <= a and (c+1)**b > a 
+019    *
+020    * This algorithm uses Newton's approximation 
+021    * x[i+1] = x[i] - f(x[i])/f'(x[i]) 
+022    * which will find the root in log(N) time where 
+023    * each step involves a fair bit.  This is not meant to 
+024    * find huge roots [square and cube, etc].
+025    */
+026   int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
+027   \{
+028     mp_int  t1, t2, t3;
+029     int     res, neg;
+030   
+031     /* input must be positive if b is even */
+032     if ((b & 1) == 0 && a->sign == MP_NEG) \{
+033       return MP_VAL;
+034     \}
+035   
+036     if ((res = mp_init (&t1)) != MP_OKAY) \{
+037       return res;
+038     \}
+039   
+040     if ((res = mp_init (&t2)) != MP_OKAY) \{
+041       goto __T1;
+042     \}
+043   
+044     if ((res = mp_init (&t3)) != MP_OKAY) \{
+045       goto __T2;
+046     \}
+047   
+048     /* if a is negative fudge the sign but keep track */
+049     neg     = a->sign;
+050     a->sign = MP_ZPOS;
+051   
+052     /* t2 = 2 */
+053     mp_set (&t2, 2);
+054   
+055     do \{
+056       /* t1 = t2 */
+057       if ((res = mp_copy (&t2, &t1)) != MP_OKAY) \{
+058         goto __T3;
+059       \}
+060   
+061       /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */
+062       
+063       /* t3 = t1**(b-1) */
+064       if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) \{   
+065         goto __T3;
+066       \}
+067   
+068       /* numerator */
+069       /* t2 = t1**b */
+070       if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) \{    
+071         goto __T3;
+072       \}
+073   
+074       /* t2 = t1**b - a */
+075       if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) \{  
+076         goto __T3;
+077       \}
+078   
+079       /* denominator */
+080       /* t3 = t1**(b-1) * b  */
+081       if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) \{    
+082         goto __T3;
+083       \}
+084   
+085       /* t3 = (t1**b - a)/(b * t1**(b-1)) */
+086       if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) \{  
+087         goto __T3;
+088       \}
+089   
+090       if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) \{
+091         goto __T3;
+092       \}
+093     \}  while (mp_cmp (&t1, &t2) != MP_EQ);
+094   
+095     /* result can be off by a few so check */
+096     for (;;) \{
+097       if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) \{
+098         goto __T3;
+099       \}
+100   
+101       if (mp_cmp (&t2, a) == MP_GT) \{
+102         if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) \{
+103            goto __T3;
+104         \}
+105       \} else \{
+106         break;
+107       \}
+108     \}
+109   
+110     /* reset the sign of a first */
+111     a->sign = neg;
+112   
+113     /* set the result */
+114     mp_exch (&t1, c);
+115   
+116     /* set the sign of the result */
+117     c->sign = neg;
+118   
+119     res = MP_OKAY;
+120   
+121   __T3:mp_clear (&t3);
+122   __T2:mp_clear (&t2);
+123   __T1:mp_clear (&t1);
+124     return res;
+125   \}
 \end{alltt}
 \end{small}
 
@@ -9057,40 +9038,39 @@ the integers from $0$ to $\beta - 1$.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* makes a pseudo-random int of a given size */
-018   int
-019   mp_rand (mp_int * a, int digits)
-020   \{
-021     int     res;
-022     mp_digit d;
-023   
-024     mp_zero (a);
-025     if (digits <= 0) \{
-026       return MP_OKAY;
-027     \}
-028   
-029     /* first place a random non-zero digit */
-030     do \{
-031       d = ((mp_digit) abs (rand ()));
-032     \} while (d == 0);
-033   
-034     if ((res = mp_add_d (a, d, a)) != MP_OKAY) \{
-035       return res;
-036     \}
-037   
-038     while (digits-- > 0) \{
-039       if ((res = mp_lshd (a, 1)) != MP_OKAY) \{
-040         return res;
-041       \}
-042   
-043       if ((res = mp_add_d (a, ((mp_digit) abs (rand ())), a)) != MP_OKAY) \{
-044         return res;
-045       \}
-046     \}
-047   
-048     return MP_OKAY;
-049   \}
+016   /* makes a pseudo-random int of a given size */
+017   int
+018   mp_rand (mp_int * a, int digits)
+019   \{
+020     int     res;
+021     mp_digit d;
+022   
+023     mp_zero (a);
+024     if (digits <= 0) \{
+025       return MP_OKAY;
+026     \}
+027   
+028     /* first place a random non-zero digit */
+029     do \{
+030       d = ((mp_digit) abs (rand ()));
+031     \} while (d == 0);
+032   
+033     if ((res = mp_add_d (a, d, a)) != MP_OKAY) \{
+034       return res;
+035     \}
+036   
+037     while (digits-- > 0) \{
+038       if ((res = mp_lshd (a, 1)) != MP_OKAY) \{
+039         return res;
+040       \}
+041   
+042       if ((res = mp_add_d (a, ((mp_digit) abs (rand ())), a)) != MP_OKAY) \{
+043         return res;
+044       \}
+045     \}
+046   
+047     return MP_OKAY;
+048   \}
 \end{alltt}
 \end{small}
 
@@ -9173,67 +9153,66 @@ as part of larger input without any significant problem.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* read a string [ASCII] in a given radix */
-018   int mp_read_radix (mp_int * a, char *str, int radix)
-019   \{
-020     int     y, res, neg;
-021     char    ch;
-022   
-023     /* make sure the radix is ok */
-024     if (radix < 2 || radix > 64) \{
-025       return MP_VAL;
-026     \}
-027   
-028     /* if the leading digit is a 
-029      * minus set the sign to negative. 
-030      */
-031     if (*str == '-') \{
-032       ++str;
-033       neg = MP_NEG;
-034     \} else \{
-035       neg = MP_ZPOS;
-036     \}
-037   
-038     /* set the integer to the default of zero */
-039     mp_zero (a);
-040     
-041     /* process each digit of the string */
-042     while (*str) \{
-043       /* if the radix < 36 the conversion is case insensitive
-044        * this allows numbers like 1AB and 1ab to represent the same  value
-045        * [e.g. in hex]
-046        */
-047       ch = (char) ((radix < 36) ? toupper (*str) : *str);
-048       for (y = 0; y < 64; y++) \{
-049         if (ch == mp_s_rmap[y]) \{
-050            break;
-051         \}
-052       \}
-053   
-054       /* if the char was found in the map 
-055        * and is less than the given radix add it
-056        * to the number, otherwise exit the loop. 
-057        */
-058       if (y < radix) \{
-059         if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) \{
-060            return res;
-061         \}
-062         if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) \{
-063            return res;
-064         \}
-065       \} else \{
-066         break;
-067       \}
-068       ++str;
-069     \}
-070     
-071     /* set the sign only if a != 0 */
-072     if (mp_iszero(a) != 1) \{
-073        a->sign = neg;
-074     \}
-075     return MP_OKAY;
-076   \}
+016   /* read a string [ASCII] in a given radix */
+017   int mp_read_radix (mp_int * a, char *str, int radix)
+018   \{
+019     int     y, res, neg;
+020     char    ch;
+021   
+022     /* make sure the radix is ok */
+023     if (radix < 2 || radix > 64) \{
+024       return MP_VAL;
+025     \}
+026   
+027     /* if the leading digit is a 
+028      * minus set the sign to negative. 
+029      */
+030     if (*str == '-') \{
+031       ++str;
+032       neg = MP_NEG;
+033     \} else \{
+034       neg = MP_ZPOS;
+035     \}
+036   
+037     /* set the integer to the default of zero */
+038     mp_zero (a);
+039     
+040     /* process each digit of the string */
+041     while (*str) \{
+042       /* if the radix < 36 the conversion is case insensitive
+043        * this allows numbers like 1AB and 1ab to represent the same  value
+044        * [e.g. in hex]
+045        */
+046       ch = (char) ((radix < 36) ? toupper (*str) : *str);
+047       for (y = 0; y < 64; y++) \{
+048         if (ch == mp_s_rmap[y]) \{
+049            break;
+050         \}
+051       \}
+052   
+053       /* if the char was found in the map 
+054        * and is less than the given radix add it
+055        * to the number, otherwise exit the loop. 
+056        */
+057       if (y < radix) \{
+058         if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) \{
+059            return res;
+060         \}
+061         if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) \{
+062            return res;
+063         \}
+064       \} else \{
+065         break;
+066       \}
+067       ++str;
+068     \}
+069     
+070     /* set the sign only if a != 0 */
+071     if (mp_iszero(a) != 1) \{
+072        a->sign = neg;
+073     \}
+074     return MP_OKAY;
+075   \}
 \end{alltt}
 \end{small}
 
@@ -9298,60 +9277,59 @@ are required instead of a series of $n \times k$ divisions.  One design flaw of
 \hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* stores a bignum as a ASCII string in a given radix (2..64) */
-018   int mp_toradix (mp_int * a, char *str, int radix)
-019   \{
-020     int     res, digs;
-021     mp_int  t;
-022     mp_digit d;
-023     char   *_s = str;
-024   
-025     /* check range of the radix */
-026     if (radix < 2 || radix > 64) \{
-027       return MP_VAL;
-028     \}
-029   
-030     /* quick out if its zero */
-031     if (mp_iszero(a) == 1) \{
-032        *str++ = '0';
-033        *str = '\symbol{92}0';
-034        return MP_OKAY;
-035     \}
-036   
-037     if ((res = mp_init_copy (&t, a)) != MP_OKAY) \{
-038       return res;
-039     \}
-040   
-041     /* if it is negative output a - */
-042     if (t.sign == MP_NEG) \{
-043       ++_s;
-044       *str++ = '-';
-045       t.sign = MP_ZPOS;
-046     \}
-047   
-048     digs = 0;
-049     while (mp_iszero (&t) == 0) \{
-050       if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) \{
-051         mp_clear (&t);
-052         return res;
-053       \}
-054       *str++ = mp_s_rmap[d];
-055       ++digs;
-056     \}
-057   
-058     /* reverse the digits of the string.  In this case _s points
-059      * to the first digit [exluding the sign] of the number]
-060      */
-061     bn_reverse ((unsigned char *)_s, digs);
-062   
-063     /* append a NULL so the string is properly terminated */
-064     *str = '\symbol{92}0';
-065   
-066     mp_clear (&t);
-067     return MP_OKAY;
-068   \}
-069   
+016   /* stores a bignum as a ASCII string in a given radix (2..64) */
+017   int mp_toradix (mp_int * a, char *str, int radix)
+018   \{
+019     int     res, digs;
+020     mp_int  t;
+021     mp_digit d;
+022     char   *_s = str;
+023   
+024     /* check range of the radix */
+025     if (radix < 2 || radix > 64) \{
+026       return MP_VAL;
+027     \}
+028   
+029     /* quick out if its zero */
+030     if (mp_iszero(a) == 1) \{
+031        *str++ = '0';
+032        *str = '\symbol{92}0';
+033        return MP_OKAY;
+034     \}
+035   
+036     if ((res = mp_init_copy (&t, a)) != MP_OKAY) \{
+037       return res;
+038     \}
+039   
+040     /* if it is negative output a - */
+041     if (t.sign == MP_NEG) \{
+042       ++_s;
+043       *str++ = '-';
+044       t.sign = MP_ZPOS;
+045     \}
+046   
+047     digs = 0;
+048     while (mp_iszero (&t) == 0) \{
+049       if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) \{
+050         mp_clear (&t);
+051         return res;
+052       \}
+053       *str++ = mp_s_rmap[d];
+054       ++digs;
+055     \}
+056   
+057     /* reverse the digits of the string.  In this case _s points
+058      * to the first digit [exluding the sign] of the number]
+059      */
+060     bn_reverse ((unsigned char *)_s, digs);
+061   
+062     /* append a NULL so the string is properly terminated */
+063     *str = '\symbol{92}0';
+064   
+065     mp_clear (&t);
+066     return MP_OKAY;
+067   \}
+068   
 \end{alltt}
 \end{small}
 
@@ -9539,115 +9517,114 @@ must be adjusted by multiplying by the common factors of two ($2^k$) removed ear
 \hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* Greatest Common Divisor using the binary method */
-018   int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
-019   \{
-020     mp_int  u, v;
-021     int     k, u_lsb, v_lsb, res;
-022   
-023     /* either zero than gcd is the largest */
-024     if (mp_iszero (a) == 1 && mp_iszero (b) == 0) \{
-025       return mp_abs (b, c);
-026     \}
-027     if (mp_iszero (a) == 0 && mp_iszero (b) == 1) \{
-028       return mp_abs (a, c);
-029     \}
-030   
-031     /* optimized.  At this point if a == 0 then
-032      * b must equal zero too
-033      */
-034     if (mp_iszero (a) == 1) \{
-035       mp_zero(c);
-036       return MP_OKAY;
-037     \}
-038   
-039     /* get copies of a and b we can modify */
-040     if ((res = mp_init_copy (&u, a)) != MP_OKAY) \{
-041       return res;
-042     \}
-043   
-044     if ((res = mp_init_copy (&v, b)) != MP_OKAY) \{
-045       goto __U;
-046     \}
-047   
-048     /* must be positive for the remainder of the algorithm */
-049     u.sign = v.sign = MP_ZPOS;
-050   
-051     /* B1.  Find the common power of two for u and v */
-052     u_lsb = mp_cnt_lsb(&u);
-053     v_lsb = mp_cnt_lsb(&v);
-054     k     = MIN(u_lsb, v_lsb);
-055   
-056     if (k > 0) \{
-057        /* divide the power of two out */
-058        if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) \{
-059           goto __V;
-060        \}
-061   
-062        if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) \{
-063           goto __V;
-064        \}
-065     \}
-066   
-067     /* divide any remaining factors of two out */
-068     if (u_lsb != k) \{
-069        if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) \{
-070           goto __V;
-071        \}
-072     \}
-073   
-074     if (v_lsb != k) \{
-075        if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) \{
-076           goto __V;
-077        \}
-078     \}
-079   
-080     while (mp_iszero(&v) == 0) \{
-081        /* make sure v is the largest */
-082        if (mp_cmp_mag(&u, &v) == MP_GT) \{
-083           /* swap u and v to make sure v is >= u */
-084           mp_exch(&u, &v);
-085        \}
-086        
-087        /* subtract smallest from largest */
-088        if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) \{
-089           goto __V;
-090        \}
-091        
-092        /* Divide out all factors of two */
-093        if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) \{
-094           goto __V;
-095        \} 
-096     \} 
-097   
-098     /* multiply by 2**k which we divided out at the beginning */
-099     if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) \{
-100        goto __V;
-101     \}
-102     c->sign = MP_ZPOS;
-103     res = MP_OKAY;
-104   __V:mp_clear (&u);
-105   __U:mp_clear (&v);
-106     return res;
-107   \}
+016   /* Greatest Common Divisor using the binary method */
+017   int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
+018   \{
+019     mp_int  u, v;
+020     int     k, u_lsb, v_lsb, res;
+021   
+022     /* either zero than gcd is the largest */
+023     if (mp_iszero (a) == 1 && mp_iszero (b) == 0) \{
+024       return mp_abs (b, c);
+025     \}
+026     if (mp_iszero (a) == 0 && mp_iszero (b) == 1) \{
+027       return mp_abs (a, c);
+028     \}
+029   
+030     /* optimized.  At this point if a == 0 then
+031      * b must equal zero too
+032      */
+033     if (mp_iszero (a) == 1) \{
+034       mp_zero(c);
+035       return MP_OKAY;
+036     \}
+037   
+038     /* get copies of a and b we can modify */
+039     if ((res = mp_init_copy (&u, a)) != MP_OKAY) \{
+040       return res;
+041     \}
+042   
+043     if ((res = mp_init_copy (&v, b)) != MP_OKAY) \{
+044       goto __U;
+045     \}
+046   
+047     /* must be positive for the remainder of the algorithm */
+048     u.sign = v.sign = MP_ZPOS;
+049   
+050     /* B1.  Find the common power of two for u and v */
+051     u_lsb = mp_cnt_lsb(&u);
+052     v_lsb = mp_cnt_lsb(&v);
+053     k     = MIN(u_lsb, v_lsb);
+054   
+055     if (k > 0) \{
+056        /* divide the power of two out */
+057        if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) \{
+058           goto __V;
+059        \}
+060   
+061        if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) \{
+062           goto __V;
+063        \}
+064     \}
+065   
+066     /* divide any remaining factors of two out */
+067     if (u_lsb != k) \{
+068        if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) \{
+069           goto __V;
+070        \}
+071     \}
+072   
+073     if (v_lsb != k) \{
+074        if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) \{
+075           goto __V;
+076        \}
+077     \}
+078   
+079     while (mp_iszero(&v) == 0) \{
+080        /* make sure v is the largest */
+081        if (mp_cmp_mag(&u, &v) == MP_GT) \{
+082           /* swap u and v to make sure v is >= u */
+083           mp_exch(&u, &v);
+084        \}
+085        
+086        /* subtract smallest from largest */
+087        if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) \{
+088           goto __V;
+089        \}
+090        
+091        /* Divide out all factors of two */
+092        if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) \{
+093           goto __V;
+094        \} 
+095     \} 
+096   
+097     /* multiply by 2**k which we divided out at the beginning */
+098     if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) \{
+099        goto __V;
+100     \}
+101     c->sign = MP_ZPOS;
+102     res = MP_OKAY;
+103   __V:mp_clear (&u);
+104   __U:mp_clear (&v);
+105     return res;
+106   \}
 \end{alltt}
 \end{small}
 
 This function makes use of the macros mp\_iszero and mp\_iseven.  The former evaluates to $1$ if the input mp\_int is equivalent to the 
 integer zero otherwise it evaluates to $0$.  The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise
 it evaluates to $0$.  Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero.  The three 
-trivial cases of inputs are handled on lines 24 through 37.  After those lines the inputs are assumed to be non-zero.
+trivial cases of inputs are handled on lines 26 through 36.  After those lines the inputs are assumed to be non-zero.
 
-Lines 34 and 40 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively.  At this point the common factors of two 
-must be divided out of the two inputs.  The while loop on line 80 iterates so long as both are even.  The local integer $k$ is used to
+Lines 38 and 39 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively.  At this point the common factors of two 
+must be divided out of the two inputs.  The while loop on line 79 iterates so long as both are even.  The local integer $k$ is used to
 keep track of how many factors of $2$ are pulled out of both values.  It is assumed that the number of factors will not exceed the maximum 
 value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than entries than are accessible by an ``int'' so this is not 
 a limitation.}.  
 
-At this point there are no more common factors of two in the two values.  The while loops on lines 80 and 80 remove any independent
+At this point there are no more common factors of two in the two values.  The while loops on lines 79 and 79 remove any independent
 factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm.  The while loop
-on line 80 performs the reduction of the pair until $v$ is equal to zero.  The unsigned comparison and subtraction algorithms are used in
+on line 79 performs the reduction of the pair until $v$ is equal to zero.  The unsigned comparison and subtraction algorithms are used in
 place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative.
 
 \section{Least Common Multiple}
@@ -9686,45 +9663,44 @@ dividing the product of the two inputs by their greatest common divisor.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* computes least common multiple as |a*b|/(a, b) */
-018   int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
-019   \{
-020     int     res;
-021     mp_int  t1, t2;
+016   /* computes least common multiple as |a*b|/(a, b) */
+017   int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
+018   \{
+019     int     res;
+020     mp_int  t1, t2;
+021   
 022   
-023   
-024     if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) \{
-025       return res;
-026     \}
-027   
-028     /* t1 = get the GCD of the two inputs */
-029     if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) \{
-030       goto __T;
-031     \}
-032   
-033     /* divide the smallest by the GCD */
-034     if (mp_cmp_mag(a, b) == MP_LT) \{
-035        /* store quotient in t2 such that t2 * b is the LCM */
-036        if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) \{
-037           goto __T;
-038        \}
-039        res = mp_mul(b, &t2, c);
-040     \} else \{
-041        /* store quotient in t2 such that t2 * a is the LCM */
-042        if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) \{
-043           goto __T;
-044        \}
-045        res = mp_mul(a, &t2, c);
-046     \}
-047   
-048     /* fix the sign to positive */
-049     c->sign = MP_ZPOS;
-050   
-051   __T:
-052     mp_clear_multi (&t1, &t2, NULL);
-053     return res;
-054   \}
+023     if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) \{
+024       return res;
+025     \}
+026   
+027     /* t1 = get the GCD of the two inputs */
+028     if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) \{
+029       goto __T;
+030     \}
+031   
+032     /* divide the smallest by the GCD */
+033     if (mp_cmp_mag(a, b) == MP_LT) \{
+034        /* store quotient in t2 such that t2 * b is the LCM */
+035        if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) \{
+036           goto __T;
+037        \}
+038        res = mp_mul(b, &t2, c);
+039     \} else \{
+040        /* store quotient in t2 such that t2 * a is the LCM */
+041        if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) \{
+042           goto __T;
+043        \}
+044        res = mp_mul(a, &t2, c);
+045     \}
+046   
+047     /* fix the sign to positive */
+048     c->sign = MP_ZPOS;
+049   
+050   __T:
+051     mp_clear_multi (&t1, &t2, NULL);
+052     return res;
+053   \}
 \end{alltt}
 \end{small}
 
@@ -9882,90 +9858,89 @@ $\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi
 \hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* computes the jacobi c = (a | n) (or Legendre if n is prime)
-018    * HAC pp. 73 Algorithm 2.149
-019    */
-020   int mp_jacobi (mp_int * a, mp_int * p, int *c)
-021   \{
-022     mp_int  a1, p1;
-023     int     k, s, r, res;
-024     mp_digit residue;
-025   
-026     /* if p <= 0 return MP_VAL */
-027     if (mp_cmp_d(p, 0) != MP_GT) \{
-028        return MP_VAL;
-029     \}
-030   
-031     /* step 1.  if a == 0, return 0 */
-032     if (mp_iszero (a) == 1) \{
-033       *c = 0;
-034       return MP_OKAY;
-035     \}
-036   
-037     /* step 2.  if a == 1, return 1 */
-038     if (mp_cmp_d (a, 1) == MP_EQ) \{
-039       *c = 1;
-040       return MP_OKAY;
-041     \}
-042   
-043     /* default */
-044     s = 0;
-045   
-046     /* step 3.  write a = a1 * 2**k  */
-047     if ((res = mp_init_copy (&a1, a)) != MP_OKAY) \{
-048       return res;
-049     \}
-050   
-051     if ((res = mp_init (&p1)) != MP_OKAY) \{
-052       goto __A1;
-053     \}
-054   
-055     /* divide out larger power of two */
-056     k = mp_cnt_lsb(&a1);
-057     if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) \{
-058        goto __P1;
-059     \}
-060   
-061     /* step 4.  if e is even set s=1 */
-062     if ((k & 1) == 0) \{
-063       s = 1;
-064     \} else \{
-065       /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */
-066       residue = p->dp[0] & 7;
-067   
-068       if (residue == 1 || residue == 7) \{
-069         s = 1;
-070       \} else if (residue == 3 || residue == 5) \{
-071         s = -1;
-072       \}
-073     \}
-074   
-075     /* step 5.  if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */
-076     if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) \{
-077       s = -s;
-078     \}
-079   
-080     /* if a1 == 1 we're done */
-081     if (mp_cmp_d (&a1, 1) == MP_EQ) \{
-082       *c = s;
-083     \} else \{
-084       /* n1 = n mod a1 */
-085       if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) \{
-086         goto __P1;
-087       \}
-088       if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) \{
-089         goto __P1;
-090       \}
-091       *c = s * r;
-092     \}
-093   
-094     /* done */
-095     res = MP_OKAY;
-096   __P1:mp_clear (&p1);
-097   __A1:mp_clear (&a1);
-098     return res;
-099   \}
+016   /* computes the jacobi c = (a | n) (or Legendre if n is prime)
+017    * HAC pp. 73 Algorithm 2.149
+018    */
+019   int mp_jacobi (mp_int * a, mp_int * p, int *c)
+020   \{
+021     mp_int  a1, p1;
+022     int     k, s, r, res;
+023     mp_digit residue;
+024   
+025     /* if p <= 0 return MP_VAL */
+026     if (mp_cmp_d(p, 0) != MP_GT) \{
+027        return MP_VAL;
+028     \}
+029   
+030     /* step 1.  if a == 0, return 0 */
+031     if (mp_iszero (a) == 1) \{
+032       *c = 0;
+033       return MP_OKAY;
+034     \}
+035   
+036     /* step 2.  if a == 1, return 1 */
+037     if (mp_cmp_d (a, 1) == MP_EQ) \{
+038       *c = 1;
+039       return MP_OKAY;
+040     \}
+041   
+042     /* default */
+043     s = 0;
+044   
+045     /* step 3.  write a = a1 * 2**k  */
+046     if ((res = mp_init_copy (&a1, a)) != MP_OKAY) \{
+047       return res;
+048     \}
+049   
+050     if ((res = mp_init (&p1)) != MP_OKAY) \{
+051       goto __A1;
+052     \}
+053   
+054     /* divide out larger power of two */
+055     k = mp_cnt_lsb(&a1);
+056     if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) \{
+057        goto __P1;
+058     \}
+059   
+060     /* step 4.  if e is even set s=1 */
+061     if ((k & 1) == 0) \{
+062       s = 1;
+063     \} else \{
+064       /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */
+065       residue = p->dp[0] & 7;
+066   
+067       if (residue == 1 || residue == 7) \{
+068         s = 1;
+069       \} else if (residue == 3 || residue == 5) \{
+070         s = -1;
+071       \}
+072     \}
+073   
+074     /* step 5.  if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */
+075     if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) \{
+076       s = -s;
+077     \}
+078   
+079     /* if a1 == 1 we're done */
+080     if (mp_cmp_d (&a1, 1) == MP_EQ) \{
+081       *c = s;
+082     \} else \{
+083       /* n1 = n mod a1 */
+084       if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) \{
+085         goto __P1;
+086       \}
+087       if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) \{
+088         goto __P1;
+089       \}
+090       *c = s * r;
+091     \}
+092   
+093     /* done */
+094     res = MP_OKAY;
+095   __P1:mp_clear (&p1);
+096   __A1:mp_clear (&a1);
+097     return res;
+098   \}
 \end{alltt}
 \end{small}
 
@@ -9980,9 +9955,9 @@ After a local copy of $a$ is made all of the factors of two are divided out and
 bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same 
 processor requirements and neither is faster than the other.
 
-Line 61 through 70 determines the value of $\left ( { 2 \over p } \right )^k$.  If the least significant bit of $k$ is zero than
+Line 60 through 71 determines the value of $\left ( { 2 \over p } \right )^k$.  If the least significant bit of $k$ is zero than
 $k$ is even and the value is one.  Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight.  The value of
-$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 75 through 73.  
+$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 74 through 77.  
 
 Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$.  
 
@@ -10091,165 +10066,164 @@ then only a couple of additions or subtractions will be required to adjust the i
 \hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* hac 14.61, pp608 */
-018   int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
-019   \{
-020     mp_int  x, y, u, v, A, B, C, D;
-021     int     res;
-022   
-023     /* b cannot be negative */
-024     if (b->sign == MP_NEG || mp_iszero(b) == 1) \{
-025       return MP_VAL;
-026     \}
-027   
-028     /* if the modulus is odd we can use a faster routine instead */
-029     if (mp_isodd (b) == 1) \{
-030       return fast_mp_invmod (a, b, c);
-031     \}
-032     
-033     /* init temps */
-034     if ((res = mp_init_multi(&x, &y, &u, &v, 
-035                              &A, &B, &C, &D, NULL)) != MP_OKAY) \{
-036        return res;
-037     \}
-038   
-039     /* x = a, y = b */
-040     if ((res = mp_copy (a, &x)) != MP_OKAY) \{
-041       goto __ERR;
-042     \}
-043     if ((res = mp_copy (b, &y)) != MP_OKAY) \{
-044       goto __ERR;
-045     \}
-046   
-047     /* 2. [modified] if x,y are both even then return an error! */
-048     if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) \{
-049       res = MP_VAL;
-050       goto __ERR;
-051     \}
-052   
-053     /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */
-054     if ((res = mp_copy (&x, &u)) != MP_OKAY) \{
-055       goto __ERR;
-056     \}
-057     if ((res = mp_copy (&y, &v)) != MP_OKAY) \{
-058       goto __ERR;
-059     \}
-060     mp_set (&A, 1);
-061     mp_set (&D, 1);
-062   
-063   top:
-064     /* 4.  while u is even do */
-065     while (mp_iseven (&u) == 1) \{
-066       /* 4.1 u = u/2 */
-067       if ((res = mp_div_2 (&u, &u)) != MP_OKAY) \{
-068         goto __ERR;
-069       \}
-070       /* 4.2 if A or B is odd then */
-071       if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) \{
-072         /* A = (A+y)/2, B = (B-x)/2 */
-073         if ((res = mp_add (&A, &y, &A)) != MP_OKAY) \{
-074            goto __ERR;
-075         \}
-076         if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) \{
-077            goto __ERR;
-078         \}
-079       \}
-080       /* A = A/2, B = B/2 */
-081       if ((res = mp_div_2 (&A, &A)) != MP_OKAY) \{
-082         goto __ERR;
-083       \}
-084       if ((res = mp_div_2 (&B, &B)) != MP_OKAY) \{
-085         goto __ERR;
-086       \}
-087     \}
-088   
-089     /* 5.  while v is even do */
-090     while (mp_iseven (&v) == 1) \{
-091       /* 5.1 v = v/2 */
-092       if ((res = mp_div_2 (&v, &v)) != MP_OKAY) \{
-093         goto __ERR;
-094       \}
-095       /* 5.2 if C or D is odd then */
-096       if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) \{
-097         /* C = (C+y)/2, D = (D-x)/2 */
-098         if ((res = mp_add (&C, &y, &C)) != MP_OKAY) \{
-099            goto __ERR;
-100         \}
-101         if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) \{
-102            goto __ERR;
-103         \}
-104       \}
-105       /* C = C/2, D = D/2 */
-106       if ((res = mp_div_2 (&C, &C)) != MP_OKAY) \{
-107         goto __ERR;
-108       \}
-109       if ((res = mp_div_2 (&D, &D)) != MP_OKAY) \{
-110         goto __ERR;
-111       \}
-112     \}
-113   
-114     /* 6.  if u >= v then */
-115     if (mp_cmp (&u, &v) != MP_LT) \{
-116       /* u = u - v, A = A - C, B = B - D */
-117       if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) \{
-118         goto __ERR;
-119       \}
-120   
-121       if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) \{
-122         goto __ERR;
-123       \}
-124   
-125       if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) \{
-126         goto __ERR;
-127       \}
-128     \} else \{
-129       /* v - v - u, C = C - A, D = D - B */
-130       if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) \{
-131         goto __ERR;
-132       \}
-133   
-134       if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) \{
-135         goto __ERR;
-136       \}
-137   
-138       if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) \{
-139         goto __ERR;
-140       \}
-141     \}
-142   
-143     /* if not zero goto step 4 */
-144     if (mp_iszero (&u) == 0)
-145       goto top;
-146   
-147     /* now a = C, b = D, gcd == g*v */
-148   
-149     /* if v != 1 then there is no inverse */
-150     if (mp_cmp_d (&v, 1) != MP_EQ) \{
-151       res = MP_VAL;
-152       goto __ERR;
-153     \}
-154   
-155     /* if its too low */
-156     while (mp_cmp_d(&C, 0) == MP_LT) \{
-157         if ((res = mp_add(&C, b, &C)) != MP_OKAY) \{
-158            goto __ERR;
-159         \}
-160     \}
-161     
-162     /* too big */
-163     while (mp_cmp_mag(&C, b) != MP_LT) \{
-164         if ((res = mp_sub(&C, b, &C)) != MP_OKAY) \{
-165            goto __ERR;
-166         \}
-167     \}
-168     
-169     /* C is now the inverse */
-170     mp_exch (&C, c);
-171     res = MP_OKAY;
-172   __ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL);
-173     return res;
-174   \}
+016   /* hac 14.61, pp608 */
+017   int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
+018   \{
+019     mp_int  x, y, u, v, A, B, C, D;
+020     int     res;
+021   
+022     /* b cannot be negative */
+023     if (b->sign == MP_NEG || mp_iszero(b) == 1) \{
+024       return MP_VAL;
+025     \}
+026   
+027     /* if the modulus is odd we can use a faster routine instead */
+028     if (mp_isodd (b) == 1) \{
+029       return fast_mp_invmod (a, b, c);
+030     \}
+031     
+032     /* init temps */
+033     if ((res = mp_init_multi(&x, &y, &u, &v, 
+034                              &A, &B, &C, &D, NULL)) != MP_OKAY) \{
+035        return res;
+036     \}
+037   
+038     /* x = a, y = b */
+039     if ((res = mp_copy (a, &x)) != MP_OKAY) \{
+040       goto __ERR;
+041     \}
+042     if ((res = mp_copy (b, &y)) != MP_OKAY) \{
+043       goto __ERR;
+044     \}
+045   
+046     /* 2. [modified] if x,y are both even then return an error! */
+047     if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) \{
+048       res = MP_VAL;
+049       goto __ERR;
+050     \}
+051   
+052     /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */
+053     if ((res = mp_copy (&x, &u)) != MP_OKAY) \{
+054       goto __ERR;
+055     \}
+056     if ((res = mp_copy (&y, &v)) != MP_OKAY) \{
+057       goto __ERR;
+058     \}
+059     mp_set (&A, 1);
+060     mp_set (&D, 1);
+061   
+062   top:
+063     /* 4.  while u is even do */
+064     while (mp_iseven (&u) == 1) \{
+065       /* 4.1 u = u/2 */
+066       if ((res = mp_div_2 (&u, &u)) != MP_OKAY) \{
+067         goto __ERR;
+068       \}
+069       /* 4.2 if A or B is odd then */
+070       if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) \{
+071         /* A = (A+y)/2, B = (B-x)/2 */
+072         if ((res = mp_add (&A, &y, &A)) != MP_OKAY) \{
+073            goto __ERR;
+074         \}
+075         if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) \{
+076            goto __ERR;
+077         \}
+078       \}
+079       /* A = A/2, B = B/2 */
+080       if ((res = mp_div_2 (&A, &A)) != MP_OKAY) \{
+081         goto __ERR;
+082       \}
+083       if ((res = mp_div_2 (&B, &B)) != MP_OKAY) \{
+084         goto __ERR;
+085       \}
+086     \}
+087   
+088     /* 5.  while v is even do */
+089     while (mp_iseven (&v) == 1) \{
+090       /* 5.1 v = v/2 */
+091       if ((res = mp_div_2 (&v, &v)) != MP_OKAY) \{
+092         goto __ERR;
+093       \}
+094       /* 5.2 if C or D is odd then */
+095       if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) \{
+096         /* C = (C+y)/2, D = (D-x)/2 */
+097         if ((res = mp_add (&C, &y, &C)) != MP_OKAY) \{
+098            goto __ERR;
+099         \}
+100         if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) \{
+101            goto __ERR;
+102         \}
+103       \}
+104       /* C = C/2, D = D/2 */
+105       if ((res = mp_div_2 (&C, &C)) != MP_OKAY) \{
+106         goto __ERR;
+107       \}
+108       if ((res = mp_div_2 (&D, &D)) != MP_OKAY) \{
+109         goto __ERR;
+110       \}
+111     \}
+112   
+113     /* 6.  if u >= v then */
+114     if (mp_cmp (&u, &v) != MP_LT) \{
+115       /* u = u - v, A = A - C, B = B - D */
+116       if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) \{
+117         goto __ERR;
+118       \}
+119   
+120       if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) \{
+121         goto __ERR;
+122       \}
+123   
+124       if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) \{
+125         goto __ERR;
+126       \}
+127     \} else \{
+128       /* v - v - u, C = C - A, D = D - B */
+129       if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) \{
+130         goto __ERR;
+131       \}
+132   
+133       if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) \{
+134         goto __ERR;
+135       \}
+136   
+137       if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) \{
+138         goto __ERR;
+139       \}
+140     \}
+141   
+142     /* if not zero goto step 4 */
+143     if (mp_iszero (&u) == 0)
+144       goto top;
+145   
+146     /* now a = C, b = D, gcd == g*v */
+147   
+148     /* if v != 1 then there is no inverse */
+149     if (mp_cmp_d (&v, 1) != MP_EQ) \{
+150       res = MP_VAL;
+151       goto __ERR;
+152     \}
+153   
+154     /* if its too low */
+155     while (mp_cmp_d(&C, 0) == MP_LT) \{
+156         if ((res = mp_add(&C, b, &C)) != MP_OKAY) \{
+157            goto __ERR;
+158         \}
+159     \}
+160     
+161     /* too big */
+162     while (mp_cmp_mag(&C, b) != MP_LT) \{
+163         if ((res = mp_sub(&C, b, &C)) != MP_OKAY) \{
+164            goto __ERR;
+165         \}
+166     \}
+167     
+168     /* C is now the inverse */
+169     mp_exch (&C, c);
+170     res = MP_OKAY;
+171   __ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL);
+172     return res;
+173   \}
 \end{alltt}
 \end{small}
 
@@ -10321,35 +10295,34 @@ This algorithm attempts to determine if a candidate integer $n$ is composite by
 \hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* determines if an integers is divisible by one 
-018    * of the first PRIME_SIZE primes or not
-019    *
-020    * sets result to 0 if not, 1 if yes
-021    */
-022   int mp_prime_is_divisible (mp_int * a, int *result)
-023   \{
-024     int     err, ix;
-025     mp_digit res;
-026   
-027     /* default to not */
-028     *result = MP_NO;
-029   
-030     for (ix = 0; ix < PRIME_SIZE; ix++) \{
-031       /* what is a mod __prime_tab[ix] */
-032       if ((err = mp_mod_d (a, __prime_tab[ix], &res)) != MP_OKAY) \{
-033         return err;
-034       \}
-035   
-036       /* is the residue zero? */
-037       if (res == 0) \{
-038         *result = MP_YES;
-039         return MP_OKAY;
-040       \}
-041     \}
-042   
-043     return MP_OKAY;
-044   \}
+016   /* determines if an integers is divisible by one 
+017    * of the first PRIME_SIZE primes or not
+018    *
+019    * sets result to 0 if not, 1 if yes
+020    */
+021   int mp_prime_is_divisible (mp_int * a, int *result)
+022   \{
+023     int     err, ix;
+024     mp_digit res;
+025   
+026     /* default to not */
+027     *result = MP_NO;
+028   
+029     for (ix = 0; ix < PRIME_SIZE; ix++) \{
+030       /* what is a mod __prime_tab[ix] */
+031       if ((err = mp_mod_d (a, __prime_tab[ix], &res)) != MP_OKAY) \{
+032         return err;
+033       \}
+034   
+035       /* is the residue zero? */
+036       if (res == 0) \{
+037         *result = MP_YES;
+038         return MP_OKAY;
+039       \}
+040     \}
+041   
+042     return MP_OKAY;
+043   \}
 \end{alltt}
 \end{small}
 
@@ -10360,46 +10333,45 @@ mp\_digit.  The table \_\_prime\_tab is defined in the following file.
 \hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c
 \vspace{-3mm}
 \begin{alltt}
-016   const mp_digit __prime_tab[] = \{
-017     0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013,
-018     0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035,
-019     0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059,
-020     0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F,
-021   #ifndef MP_8BIT
-022     0x0083,
-023     0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD,
-024     0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF,
-025     0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107,
-026     0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137,
-027   
-028     0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167,
-029     0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199,
-030     0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9,
-031     0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7,
-032     0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239,
-033     0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265,
-034     0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293,
-035     0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF,
-036   
-037     0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301,
-038     0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B,
-039     0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371,
-040     0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD,
-041     0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5,
-042     0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419,
-043     0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449,
-044     0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B,
-045   
-046     0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7,
-047     0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503,
-048     0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529,
-049     0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F,
-050     0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
-051     0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
-052     0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
-053     0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
-054   #endif
-055   \};
+016     0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013,
+017     0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035,
+018     0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059,
+019     0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F,
+020   #ifndef MP_8BIT
+021     0x0083,
+022     0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD,
+023     0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF,
+024     0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107,
+025     0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137,
+026   
+027     0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167,
+028     0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199,
+029     0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9,
+030     0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7,
+031     0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239,
+032     0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265,
+033     0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293,
+034     0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF,
+035   
+036     0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301,
+037     0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B,
+038     0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371,
+039     0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD,
+040     0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5,
+041     0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419,
+042     0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449,
+043     0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B,
+044   
+045     0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7,
+046     0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503,
+047     0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529,
+048     0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F,
+049     0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
+050     0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
+051     0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
+052     0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
+053   #endif
+054   \};
 \end{alltt}
 \end{small}
 
@@ -10446,47 +10418,46 @@ determine the result.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* performs one Fermat test.
-018    * 
-019    * If "a" were prime then b**a == b (mod a) since the order of
-020    * the multiplicative sub-group would be phi(a) = a-1.  That means
-021    * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
-022    *
-023    * Sets result to 1 if the congruence holds, or zero otherwise.
-024    */
-025   int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
-026   \{
-027     mp_int  t;
-028     int     err;
-029   
-030     /* default to composite  */
-031     *result = MP_NO;
-032   
-033     /* ensure b > 1 */
-034     if (mp_cmp_d(b, 1) != MP_GT) \{
-035        return MP_VAL;
-036     \}
-037   
-038     /* init t */
-039     if ((err = mp_init (&t)) != MP_OKAY) \{
-040       return err;
-041     \}
-042   
-043     /* compute t = b**a mod a */
-044     if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) \{
-045       goto __T;
-046     \}
-047   
-048     /* is it equal to b? */
-049     if (mp_cmp (&t, b) == MP_EQ) \{
-050       *result = MP_YES;
-051     \}
-052   
-053     err = MP_OKAY;
-054   __T:mp_clear (&t);
-055     return err;
-056   \}
+016   /* performs one Fermat test.
+017    * 
+018    * If "a" were prime then b**a == b (mod a) since the order of
+019    * the multiplicative sub-group would be phi(a) = a-1.  That means
+020    * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
+021    *
+022    * Sets result to 1 if the congruence holds, or zero otherwise.
+023    */
+024   int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
+025   \{
+026     mp_int  t;
+027     int     err;
+028   
+029     /* default to composite  */
+030     *result = MP_NO;
+031   
+032     /* ensure b > 1 */
+033     if (mp_cmp_d(b, 1) != MP_GT) \{
+034        return MP_VAL;
+035     \}
+036   
+037     /* init t */
+038     if ((err = mp_init (&t)) != MP_OKAY) \{
+039       return err;
+040     \}
+041   
+042     /* compute t = b**a mod a */
+043     if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) \{
+044       goto __T;
+045     \}
+046   
+047     /* is it equal to b? */
+048     if (mp_cmp (&t, b) == MP_EQ) \{
+049       *result = MP_YES;
+050     \}
+051   
+052     err = MP_OKAY;
+053   __T:mp_clear (&t);
+054     return err;
+055   \}
 \end{alltt}
 \end{small}
 
@@ -10539,88 +10510,87 @@ composite then it is \textit{probably} prime.
 \hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c
 \vspace{-3mm}
 \begin{alltt}
-016   
-017   /* Miller-Rabin test of "a" to the base of "b" as described in 
-018    * HAC pp. 139 Algorithm 4.24
-019    *
-020    * Sets result to 0 if definitely composite or 1 if probably prime.
-021    * Randomly the chance of error is no more than 1/4 and often 
-022    * very much lower.
-023    */
-024   int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
-025   \{
-026     mp_int  n1, y, r;
-027     int     s, j, err;
-028   
-029     /* default */
-030     *result = MP_NO;
-031   
-032     /* ensure b > 1 */
-033     if (mp_cmp_d(b, 1) != MP_GT) \{
-034        return MP_VAL;
-035     \}     
-036   
-037     /* get n1 = a - 1 */
-038     if ((err = mp_init_copy (&n1, a)) != MP_OKAY) \{
-039       return err;
-040     \}
-041     if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) \{
-042       goto __N1;
-043     \}
-044   
-045     /* set 2**s * r = n1 */
-046     if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) \{
-047       goto __N1;
-048     \}
-049   
-050     /* count the number of least significant bits
-051      * which are zero
-052      */
-053     s = mp_cnt_lsb(&r);
-054   
-055     /* now divide n - 1 by 2**s */
-056     if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) \{
-057       goto __R;
-058     \}
-059   
-060     /* compute y = b**r mod a */
-061     if ((err = mp_init (&y)) != MP_OKAY) \{
-062       goto __R;
-063     \}
-064     if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) \{
-065       goto __Y;
-066     \}
-067   
-068     /* if y != 1 and y != n1 do */
-069     if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) \{
-070       j = 1;
-071       /* while j <= s-1 and y != n1 */
-072       while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) \{
-073         if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) \{
-074            goto __Y;
-075         \}
-076   
-077         /* if y == 1 then composite */
-078         if (mp_cmp_d (&y, 1) == MP_EQ) \{
-079            goto __Y;
-080         \}
-081   
-082         ++j;
-083       \}
-084   
-085       /* if y != n1 then composite */
-086       if (mp_cmp (&y, &n1) != MP_EQ) \{
-087         goto __Y;
-088       \}
-089     \}
-090   
-091     /* probably prime now */
-092     *result = MP_YES;
-093   __Y:mp_clear (&y);
-094   __R:mp_clear (&r);
-095   __N1:mp_clear (&n1);
-096     return err;
-097   \}
+016   /* Miller-Rabin test of "a" to the base of "b" as described in 
+017    * HAC pp. 139 Algorithm 4.24
+018    *
+019    * Sets result to 0 if definitely composite or 1 if probably prime.
+020    * Randomly the chance of error is no more than 1/4 and often 
+021    * very much lower.
+022    */
+023   int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
+024   \{
+025     mp_int  n1, y, r;
+026     int     s, j, err;
+027   
+028     /* default */
+029     *result = MP_NO;
+030   
+031     /* ensure b > 1 */
+032     if (mp_cmp_d(b, 1) != MP_GT) \{
+033        return MP_VAL;
+034     \}     
+035   
+036     /* get n1 = a - 1 */
+037     if ((err = mp_init_copy (&n1, a)) != MP_OKAY) \{
+038       return err;
+039     \}
+040     if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) \{
+041       goto __N1;
+042     \}
+043   
+044     /* set 2**s * r = n1 */
+045     if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) \{
+046       goto __N1;
+047     \}
+048   
+049     /* count the number of least significant bits
+050      * which are zero
+051      */
+052     s = mp_cnt_lsb(&r);
+053   
+054     /* now divide n - 1 by 2**s */
+055     if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) \{
+056       goto __R;
+057     \}
+058   
+059     /* compute y = b**r mod a */
+060     if ((err = mp_init (&y)) != MP_OKAY) \{
+061       goto __R;
+062     \}
+063     if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) \{
+064       goto __Y;
+065     \}
+066   
+067     /* if y != 1 and y != n1 do */
+068     if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) \{
+069       j = 1;
+070       /* while j <= s-1 and y != n1 */
+071       while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) \{
+072         if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) \{
+073            goto __Y;
+074         \}
+075   
+076         /* if y == 1 then composite */
+077         if (mp_cmp_d (&y, 1) == MP_EQ) \{
+078            goto __Y;
+079         \}
+080   
+081         ++j;
+082       \}
+083   
+084       /* if y != n1 then composite */
+085       if (mp_cmp (&y, &n1) != MP_EQ) \{
+086         goto __Y;
+087       \}
+088     \}
+089   
+090     /* probably prime now */
+091     *result = MP_YES;
+092   __Y:mp_clear (&y);
+093   __R:mp_clear (&r);
+094   __N1:mp_clear (&n1);
+095     return err;
+096   \}
 \end{alltt}
 \end{small}