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added libtommath-0.19

This commit is contained in:
Tom St Denis 2003-06-06 19:35:48 +00:00 committed by Steffen Jaeckel
parent 0ef44cea9b
commit ef490f30f6
24 changed files with 15064 additions and 13807 deletions
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bn.pdf
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2
bn.tex
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@ -1,7 +1,7 @@
\documentclass[]{article}
\begin{document}
\title{LibTomMath v0.18 \\ A Free Multiple Precision Integer Library \\ http://math.libtomcrypt.org }
\title{LibTomMath v0.19 \\ A Free Multiple Precision Integer Library \\ http://math.libtomcrypt.org }
\author{Tom St Denis \\ tomstdenis@iahu.ca}
\maketitle
\newpage

6
bn_mp_exptmod.c
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@ -64,9 +64,9 @@ mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
if (dr == 0) {
dr = mp_reduce_is_2k(P) << 1;
}
/* if the modulus is odd use the fast method */
if ((mp_isodd (P) == 1 || dr != 0) && P->used > 4) {
/* if the modulus is odd or dr != 0 use the fast method */
if (mp_isodd (P) == 1 || dr != 0) {
return mp_exptmod_fast (G, X, P, Y, dr);
} else {
return s_mp_exptmod (G, X, P, Y);

1
bn_mp_exptmod_fast.c
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@ -80,6 +80,7 @@ mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode)
if (((P->used * 2 + 1) < MP_WARRAY) &&
P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
redux = fast_mp_montgomery_reduce;
} else {
/* use slower baselien method */
redux = mp_montgomery_reduce;

2
bn_mp_karatsuba_mul.c
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@ -49,6 +49,7 @@ mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
mp_int x0, x1, y0, y1, t1, x0y0, x1y1;
int B, err;
/* default the return code to an error */
err = MP_MEM;
/* min # of digits */
@ -149,6 +150,7 @@ mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
if (mp_add (&t1, &x1y1, c) != MP_OKAY)
goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */
/* Algorithm succeeded set the return code to MP_OKAY */
err = MP_OKAY;
X1Y1:mp_clear (&x1y1);

5
bn_mp_montgomery_reduce.c
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@ -68,14 +68,15 @@ mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
}
/* propagate carries */
while (u) {
*tmpx += u;
u = *tmpx >> DIGIT_BIT;
*tmpx += u;
u = *tmpx >> DIGIT_BIT;
*tmpx++ &= MP_MASK;
}
}
}
/* x = x/b**n.used */
mp_clamp(x);
mp_rshd (x, n->used);
/* if A >= m then A = A - m */

3
bn_mp_reduce_is_2k.c
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@ -27,7 +27,8 @@ mp_reduce_is_2k(mp_int *a)
} else if (a->used > 1) {
iy = mp_count_bits(a);
for (ix = DIGIT_BIT; ix < iy; ix++) {
if ((a->dp[ix/DIGIT_BIT] & ((mp_digit)1 << (mp_digit)(ix % DIGIT_BIT))) == 0) {
if ((a->dp[ix/DIGIT_BIT] &
((mp_digit)1 << (mp_digit)(ix % DIGIT_BIT))) == 0) {
return 0;
}
}

17
bn_mp_toom_mul.c
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@ -14,7 +14,7 @@
*/
#include <tommath.h>
/* multiplication using Toom-Cook 3-way algorithm */
/* multiplication using the Toom-Cook 3-way algorithm */
int
mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
{
@ -22,14 +22,16 @@ mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
int res, B;
/* init temps */
if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &b0, &b1, &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) {
if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4,
&a0, &a1, &a2, &b0, &b1,
&b2, &tmp1, &tmp2, NULL)) != MP_OKAY) {
return res;
}
/* B */
B = MIN(a->used, b->used) / 3;
/* a = a2 * B^2 + a1 * B + a0 */
/* a = a2 * B**2 + a1 * B + a0 */
if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) {
goto ERR;
}
@ -45,7 +47,7 @@ mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
}
mp_rshd(&a2, B*2);
/* b = b2 * B^2 + b1 * B + b0 */
/* b = b2 * B**2 + b1 * B + b0 */
if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) {
goto ERR;
}
@ -159,7 +161,8 @@ mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
16 8 4 2 1
1 0 0 0 0
using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication
using 12 subtractions, 4 shifts,
2 small divisions and 1 small multiplication
*/
/* r1 - r4 */
@ -262,7 +265,9 @@ mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
}
ERR:
mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &b0, &b1, &b2, &tmp1, &tmp2, NULL);
mp_clear_multi(&w0, &w1, &w2, &w3, &w4,
&a0, &a1, &a2, &b0, &b1,
&b2, &tmp1, &tmp2, NULL);
return res;
}

14
bn_radix.c
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@ -40,16 +40,16 @@ mp_read_radix (mp_int * a, char *str, int radix)
ch = (char) ((radix < 36) ? toupper (*str) : *str);
for (y = 0; y < 64; y++) {
if (ch == s_rmap[y]) {
break;
break;
}
}
if (y < radix) {
if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) {
return res;
return res;
}
if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) {
return res;
return res;
}
} else {
break;
@ -72,6 +72,14 @@ mp_toradix (mp_int * a, char *str, int radix)
if (radix < 2 || radix > 64) {
return MP_VAL;
}
/* quick out if its zero */
if (mp_iszero(a) == 1) {
*str++ = '0';
*str = '\0';
return MP_OKAY;
}
if ((res = mp_init_copy (&t, a)) != MP_OKAY) {
return res;

13
bn_s_mp_exptmod.c
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@ -65,21 +65,26 @@ s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
/* create M table
*
* The M table contains powers of the input base, e.g. M[x] = G**x mod P
* The M table contains powers of the base,
* e.g. M[x] = G**x mod P
*
* The first half of the table is not computed though accept for M[0] and M[1]
* The first half of the table is not
* computed though accept for M[0] and M[1]
*/
if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) {
goto __MU;
}
/* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */
/* compute the value at M[1<<(winsize-1)] by squaring
* M[1] (winsize-1) times
*/
if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) {
goto __MU;
}
for (x = 0; x < (winsize - 1); x++) {
if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != MP_OKAY) {
if ((err = mp_sqr (&M[1 << (winsize - 1)],
&M[1 << (winsize - 1)])) != MP_OKAY) {
goto __MU;
}
if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) {

12
bn_s_mp_sqr.c
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@ -24,19 +24,19 @@ s_mp_sqr (mp_int * a, mp_int * b)
mp_digit u, tmpx, *tmpt;
pa = a->used;
if ((res = mp_init_size (&t, pa + pa + 1)) != MP_OKAY) {
if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) {
return res;
}
t.used = pa + pa + 1;
t.used = 2*pa + 1;
for (ix = 0; ix < pa; ix++) {
/* first calculate the digit at 2*ix */
/* calculate double precision result */
r = ((mp_word) t.dp[ix + ix]) +
r = ((mp_word) t.dp[2*ix]) +
((mp_word) a->dp[ix]) * ((mp_word) a->dp[ix]);
/* store lower part in result */
t.dp[ix + ix] = (mp_digit) (r & ((mp_word) MP_MASK));
t.dp[2*ix] = (mp_digit) (r & ((mp_word) MP_MASK));
/* get the carry */
u = (r >> ((mp_word) DIGIT_BIT));
@ -45,14 +45,14 @@ s_mp_sqr (mp_int * a, mp_int * b)
tmpx = a->dp[ix];
/* alias for where to store the results */
tmpt = t.dp + (ix + ix + 1);
tmpt = t.dp + (2*ix + 1);
for (iy = ix + 1; iy < pa; iy++) {
/* first calculate the product */
r = ((mp_word) tmpx) * ((mp_word) a->dp[iy]);
/* now calculate the double precision result, note we use
* addition instead of *2 since its easier to optimize
* addition instead of *2 since it's easier to optimize
*/
r = ((mp_word) * tmpt) + r + r + ((mp_word) u);

7
changes.txt
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@ -1,3 +1,10 @@
June 6th, 2003
v0.19 -- Fixed a bug in mp_montgomery_reduce() which was introduced when I tweaked mp_rshd() in the previous release.
Essentially the digits were not trimmed before the compare which cause a subtraction to occur all the time.
-- Fixed up etc/tune.c a bit to stop testing new cutoffs after 16 failures [to find more optimal points].
Brute force ho!
May 29th, 2003
v0.18 -- Fixed a bug in s_mp_sqr which would handle carries properly just not very elegantly.
(e.g. correct result, just bad looking code)

6
etc/drprimes.txt Normal file
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@ -0,0 +1,6 @@
224-bit prime:
p == 26959946667150639794667015087019630673637144422540572481103341844143
532-bit prime:
p == 14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368691747

2
etc/mersenne.c
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@ -57,7 +57,7 @@ is_mersenne (long s, int *pp)
/* if u == 0 then its prime */
if (mp_iszero (&u) == 1) {
mp_prime_is_prime(&n, 3, pp);
mp_prime_is_prime(&n, 8, pp);
if (*pp != 1) printf("FAILURE\n");
}

55
etc/tune.c
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@ -5,6 +5,12 @@
#include <tommath.h>
#include <time.h>
/* how many times todo each size mult. Depends on your computer. For slow computers
* this can be low like 5 or 10. For fast [re: Athlon] should be 25 - 50 or so
*/
#define TIMES 50
#ifndef X86_TIMER
/* generic ISO C timer */
@ -31,7 +37,7 @@ time_mult (int max)
for (x = 32; x <= max; x += 4) {
mp_rand (&a, x);
mp_rand (&b, x);
for (y = 0; y < 100; y++) {
for (y = 0; y < TIMES; y++) {
mp_mul (&a, &b, &c);
}
}
@ -53,7 +59,7 @@ time_sqr (int max)
t_start();
for (x = 32; x <= max; x += 4) {
mp_rand (&a, x);
for (y = 0; y < 100; y++) {
for (y = 0; y < TIMES; y++) {
mp_sqr (&a, &b);
}
}
@ -65,7 +71,7 @@ time_sqr (int max)
int
main (void)
{
int best_kmult, best_tmult, best_ksquare, best_tsquare;
int best_kmult, best_tmult, best_ksquare, best_tsquare, counter;
ulong64 best, ti;
FILE *log;
@ -77,73 +83,94 @@ main (void)
log = fopen ("mult.log", "w");
best = -1;
counter = 16;
for (KARATSUBA_MUL_CUTOFF = 8; KARATSUBA_MUL_CUTOFF <= 200; KARATSUBA_MUL_CUTOFF++) {
ti = time_mult (300);
printf ("%4d : %9llu\r", KARATSUBA_MUL_CUTOFF, ti);
printf ("%4d : %9llu \r", KARATSUBA_MUL_CUTOFF, ti);
fprintf (log, "%d, %llu\n", KARATSUBA_MUL_CUTOFF, ti);
fflush (stdout);
if (ti < best) {
printf ("New best: %llu, %d \n", ti, KARATSUBA_MUL_CUTOFF);
printf ("New best: %llu, %d \r", ti, KARATSUBA_MUL_CUTOFF);
best = ti;
best_kmult = KARATSUBA_MUL_CUTOFF;
counter = 16;
} else if (--counter == 0) {
printf("No better found in 16 trials.\n");
break;
}
}
fclose (log);
printf("Karatsuba Multiplier Cutoff (KARATSUBA_MUL_CUTOFF) == %d\n", best_kmult);
/* tune squaring */
log = fopen ("sqr.log", "w");
best = -1;
counter = 16;
for (KARATSUBA_SQR_CUTOFF = 8; KARATSUBA_SQR_CUTOFF <= 200; KARATSUBA_SQR_CUTOFF++) {
ti = time_sqr (300);
printf ("%4d : %9llu\r", KARATSUBA_SQR_CUTOFF, ti);
printf ("%4d : %9llu \r", KARATSUBA_SQR_CUTOFF, ti);
fprintf (log, "%d, %llu\n", KARATSUBA_SQR_CUTOFF, ti);
fflush (stdout);
if (ti < best) {
printf ("New best: %llu, %d \n", ti, KARATSUBA_SQR_CUTOFF);
printf ("New best: %llu, %d \r", ti, KARATSUBA_SQR_CUTOFF);
best = ti;
best_ksquare = KARATSUBA_SQR_CUTOFF;
counter = 16;
} else if (--counter == 0) {
printf("No better found in 16 trials.\n");
break;
}
}
fclose (log);
printf("Karatsuba Squaring Cutoff (KARATSUBA_SQR_CUTOFF) == %d\n", best_ksquare);
KARATSUBA_MUL_CUTOFF = best_kmult;
KARATSUBA_SQR_CUTOFF = best_ksquare;
/* tune TOOM mult */
counter = 16;
log = fopen ("tmult.log", "w");
best = -1;
for (TOOM_MUL_CUTOFF = best_kmult*5; TOOM_MUL_CUTOFF <= 800; TOOM_MUL_CUTOFF++) {
ti = time_mult (1200);
printf ("%4d : %9llu\r", TOOM_MUL_CUTOFF, ti);
printf ("%4d : %9llu \r", TOOM_MUL_CUTOFF, ti);
fprintf (log, "%d, %llu\n", TOOM_MUL_CUTOFF, ti);
fflush (stdout);
if (ti < best) {
printf ("New best: %llu, %d \n", ti, TOOM_MUL_CUTOFF);
printf ("New best: %llu, %d \r", ti, TOOM_MUL_CUTOFF);
best = ti;
best_tmult = TOOM_MUL_CUTOFF;
counter = 16;
} else if (--counter == 0) {
printf("No better found in 16 trials.\n");
break;
}
}
fclose (log);
printf("Toom-Cook Multiplier Cutoff (TOOM_MUL_CUTOFF) == %d\n", best_tmult);
/* tune TOOM sqr */
log = fopen ("tsqr.log", "w");
best = -1;
counter = 16;
for (TOOM_SQR_CUTOFF = best_ksquare*3; TOOM_SQR_CUTOFF <= 800; TOOM_SQR_CUTOFF++) {
ti = time_sqr (1200);
printf ("%4d : %9llu\r", TOOM_SQR_CUTOFF, ti);
printf ("%4d : %9llu \r", TOOM_SQR_CUTOFF, ti);
fprintf (log, "%d, %llu\n", TOOM_SQR_CUTOFF, ti);
fflush (stdout);
if (ti < best) {
printf ("New best: %llu, %d \n", ti, TOOM_SQR_CUTOFF);
printf ("New best: %llu, %d \r", ti, TOOM_SQR_CUTOFF);
best = ti;
best_tsquare = TOOM_SQR_CUTOFF;
counter = 16;
} else if (--counter == 0) {
printf("No better found in 16 trials.\n");
break;
}
}
fclose (log);
printf("Toom-Cook Squaring Cutoff (TOOM_SQR_CUTOFF) == %d\n", best_tsquare);
printf
("\n\n\nKaratsuba Multiplier Cutoff: %d\nKaratsuba Squaring Cutoff: %d\nToom Multiplier Cutoff: %d\nToom Squaring Cutoff: %d\n",
best_kmult, best_ksquare, best_tmult, best_tsquare);
return 0;
}

2
makefile
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@ -1,6 +1,6 @@
CFLAGS += -I./ -Wall -W -Wshadow -O3 -fomit-frame-pointer -funroll-loops
VERSION=0.18
VERSION=0.19
default: libtommath.a

2
makefile.bcc
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@ -32,6 +32,6 @@ TARGET = libtommath.lib
$(TARGET): $(OBJECTS)
.c.objbj:
.c.obj:
$(CC) $(CFLAGS) $<
$(LIB) $(TARGET) -+$@

0
demo/test.c → poster.out
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12
poster.tex
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@ -1,14 +1,15 @@
\documentclass[landscape,11pt]{article}
\usepackage{amsmath, amssymb}
\usepackage{hyperref}
\begin{document}
\hspace*{-3in}
\begin{tabular}{llllll}
$c = a + b$ & {\tt mp\_add(\&a, \&b, \&c)} & $b = 2a$ & {\tt mp\_mul\_2(\&a, \&b)} & Greater Than & MP\_GT \\
$c = a - b$ & {\tt mp\_sub(\&a, \&b, \&c)} & $b = a/2$ & {\tt mp\_div\_2(\&a, \&b)} & Equal To & MP\_EQ \\
$c = ab $ & {\tt mp\_mul(\&a, \&b, \&c)} & $c = 2^ba$ & {\tt mp\_mul\_2d(\&a, b, \&c)} & Less Than & MP\_LT \\
$c = a + b$ & {\tt mp\_add(\&a, \&b, \&c)} & $b = 2a$ & {\tt mp\_mul\_2(\&a, \&b)} & \\
$c = a - b$ & {\tt mp\_sub(\&a, \&b, \&c)} & $b = a/2$ & {\tt mp\_div\_2(\&a, \&b)} & \\
$c = ab $ & {\tt mp\_mul(\&a, \&b, \&c)} & $c = 2^ba$ & {\tt mp\_mul\_2d(\&a, b, \&c)} \\
$b = a^2 $ & {\tt mp\_sqr(\&a, \&b)} & $c = a/2^b, d = a \mod 2^b$ & {\tt mp\_div\_2d(\&a, b, \&c, \&d)} \\
$c = \lfloor a/b \rfloor, d = a \mod b$ & {\tt mp\_div(\&a, \&b, \&c, \&d)} & $c = a \mod 2^b $ & {\tt mp\_mod\_2d(\&a, b, \&c)} & Bits per digit & DIGIT\_BIT \\
$c = \lfloor a/b \rfloor, d = a \mod b$ & {\tt mp\_div(\&a, \&b, \&c, \&d)} & $c = a \mod 2^b $ & {\tt mp\_mod\_2d(\&a, b, \&c)} \\
&& \\
$a = b $ & {\tt mp\_set\_int(\&a, b)} & $c = a \vee b$ & {\tt mp\_or(\&a, \&b, \&c)} \\
$b = a $ & {\tt mp\_copy(\&a, \&b)} & $c = a \wedge b$ & {\tt mp\_and(\&a, \&b, \&c)} \\
@ -28,5 +29,8 @@ $a \leftarrow buf[0..len-1]$ & {\tt mp\_read\_unsigned\_bin(\&a, buf, l
&\\
$b = \sqrt{a}$ & {\tt mp\_sqrt(\&a, \&b)} & $c = \mbox{gcd}(a, b)$ & {\tt mp\_gcd(\&a, \&b, \&c)} \\
$c = a^{1/b}$ & {\tt mp\_n\_root(\&a, b, \&c)} & $c = \mbox{lcm}(a, b)$ & {\tt mp\_lcm(\&a, \&b, \&c)} \\
&\\
Greater Than & MP\_GT & Equal To & MP\_EQ \\
Less Than & MP\_LT & Bits per digit & DIGIT\_BIT \\
\end{tabular}
\end{document}

12367
pre_gen/mpi.c
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143
tommath.out Normal file
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@ -0,0 +1,143 @@
\BOOKMARK [0][-]{chapter.1}{Introduction}{}
\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1}
\BOOKMARK [2][-]{subsection.1.1.1}{The Need for Multiple Precision Arithmetic}{section.1.1}
\BOOKMARK [2][-]{subsection.1.1.2}{Multiple Precision Arithmetic}{section.1.1}
\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1}
\BOOKMARK [2][-]{subsection.1.1.4}{Basis of Operations}{section.1.1}
\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1}
\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1}
\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.2}{Work Effort}{section.1.3}
\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1}
\BOOKMARK [0][-]{chapter.2}{Introduction to LibTomMath}{}
\BOOKMARK [1][-]{section.2.1}{What is LibTomMath?}{chapter.2}
\BOOKMARK [1][-]{section.2.2}{Goals of LibTomMath}{chapter.2}
\BOOKMARK [1][-]{section.2.3}{Choice of LibTomMath}{chapter.2}
\BOOKMARK [2][-]{subsection.2.3.1}{Code Base}{section.2.3}
\BOOKMARK [2][-]{subsection.2.3.2}{API Simplicity}{section.2.3}
\BOOKMARK [2][-]{subsection.2.3.3}{Optimizations}{section.2.3}
\BOOKMARK [2][-]{subsection.2.3.4}{Portability and Stability}{section.2.3}
\BOOKMARK [2][-]{subsection.2.3.5}{Choice}{section.2.3}
\BOOKMARK [0][-]{chapter.3}{Getting Started}{}
\BOOKMARK [1][-]{section.3.1}{Library Basics}{chapter.3}
\BOOKMARK [1][-]{section.3.2}{What is a Multiple Precision Integer?}{chapter.3}
\BOOKMARK [2][-]{subsection.3.2.1}{The mp\137int structure}{section.3.2}
\BOOKMARK [1][-]{section.3.3}{Argument Passing}{chapter.3}
\BOOKMARK [1][-]{section.3.4}{Return Values}{chapter.3}
\BOOKMARK [1][-]{section.3.5}{Initialization and Clearing}{chapter.3}
\BOOKMARK [2][-]{subsection.3.5.1}{Initializing an mp\137int}{section.3.5}
\BOOKMARK [2][-]{subsection.3.5.2}{Clearing an mp\137int}{section.3.5}
\BOOKMARK [1][-]{section.3.6}{Other Initialization Routines}{chapter.3}
\BOOKMARK [2][-]{subsection.3.6.1}{Initializing Variable Sized mp\137int Structures}{section.3.6}
\BOOKMARK [2][-]{subsection.3.6.2}{Creating a Clone}{section.3.6}
\BOOKMARK [2][-]{subsection.3.6.3}{Multiple Integer Initializations And Clearings}{section.3.6}
\BOOKMARK [1][-]{section.3.7}{Maintenance}{chapter.3}
\BOOKMARK [2][-]{subsection.3.7.1}{Augmenting Integer Precision}{section.3.7}
\BOOKMARK [2][-]{subsection.3.7.2}{Clamping Excess Digits}{section.3.7}
\BOOKMARK [0][-]{chapter.4}{Basic Operations}{}
\BOOKMARK [1][-]{section.4.1}{Copying an Integer}{chapter.4}
\BOOKMARK [1][-]{section.4.2}{Zeroing an Integer}{chapter.4}
\BOOKMARK [1][-]{section.4.3}{Sign Manipulation}{chapter.4}
\BOOKMARK [2][-]{subsection.4.3.1}{Absolute Value}{section.4.3}
\BOOKMARK [2][-]{subsection.4.3.2}{Integer Negation}{section.4.3}
\BOOKMARK [1][-]{section.4.4}{Small Constants}{chapter.4}
\BOOKMARK [2][-]{subsection.4.4.1}{Setting Small Constants}{section.4.4}
\BOOKMARK [2][-]{subsection.4.4.2}{Setting Large Constants}{section.4.4}
\BOOKMARK [1][-]{section.4.5}{Comparisons}{chapter.4}
\BOOKMARK [2][-]{subsection.4.5.1}{Unsigned Comparisions}{section.4.5}
\BOOKMARK [2][-]{subsection.4.5.2}{Signed Comparisons}{section.4.5}
\BOOKMARK [0][-]{chapter.5}{Basic Arithmetic}{}
\BOOKMARK [1][-]{section.5.1}{Building Blocks}{chapter.5}
\BOOKMARK [1][-]{section.5.2}{Addition and Subtraction}{chapter.5}
\BOOKMARK [2][-]{subsection.5.2.1}{Low Level Addition}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.2}{Low Level Subtraction}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.3}{High Level Addition}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.4}{High Level Subtraction}{section.5.2}
\BOOKMARK [1][-]{section.5.3}{Bit and Digit Shifting}{chapter.5}
\BOOKMARK [2][-]{subsection.5.3.1}{Multiplication by Two}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.2}{Division by Two}{section.5.3}
\BOOKMARK [1][-]{section.5.4}{Polynomial Basis Operations}{chapter.5}
\BOOKMARK [2][-]{subsection.5.4.1}{Multiplication by x}{section.5.4}
\BOOKMARK [2][-]{subsection.5.4.2}{Division by x}{section.5.4}
\BOOKMARK [1][-]{section.5.5}{Powers of Two}{chapter.5}
\BOOKMARK [2][-]{subsection.5.5.1}{Multiplication by Power of Two}{section.5.5}
\BOOKMARK [2][-]{subsection.5.5.2}{Division by Power of Two}{section.5.5}
\BOOKMARK [2][-]{subsection.5.5.3}{Remainder of Division by Power of Two}{section.5.5}
\BOOKMARK [0][-]{chapter.6}{Multiplication and Squaring}{}
\BOOKMARK [1][-]{section.6.1}{The Multipliers}{chapter.6}
\BOOKMARK [1][-]{section.6.2}{Multiplication}{chapter.6}
\BOOKMARK [2][-]{subsection.6.2.1}{The Baseline Multiplication}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.2}{Faster Multiplication by the ``Comba'' Method}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.3}{Polynomial Basis Multiplication}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.4}{Karatsuba Multiplication}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.5}{Toom-Cook 3-Way Multiplication}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.6}{Signed Multiplication}{section.6.2}
\BOOKMARK [1][-]{section.6.3}{Squaring}{chapter.6}
\BOOKMARK [2][-]{subsection.6.3.1}{The Baseline Squaring Algorithm}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.2}{Faster Squaring by the ``Comba'' Method}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.3}{Polynomial Basis Squaring}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.4}{Karatsuba Squaring}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.5}{Toom-Cook Squaring}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.6}{High Level Squaring}{section.6.3}
\BOOKMARK [0][-]{chapter.7}{Modular Reduction}{}
\BOOKMARK [1][-]{section.7.1}{Basics of Modular Reduction}{chapter.7}
\BOOKMARK [1][-]{section.7.2}{The Barrett Reduction}{chapter.7}
\BOOKMARK [2][-]{subsection.7.2.1}{Fixed Point Arithmetic}{section.7.2}
\BOOKMARK [2][-]{subsection.7.2.2}{Choosing a Radix Point}{section.7.2}
\BOOKMARK [2][-]{subsection.7.2.3}{Trimming the Quotient}{section.7.2}
\BOOKMARK [2][-]{subsection.7.2.4}{Trimming the Residue}{section.7.2}
\BOOKMARK [2][-]{subsection.7.2.5}{The Barrett Algorithm}{section.7.2}
\BOOKMARK [2][-]{subsection.7.2.6}{The Barrett Setup Algorithm}{section.7.2}
\BOOKMARK [1][-]{section.7.3}{The Montgomery Reduction}{chapter.7}
\BOOKMARK [2][-]{subsection.7.3.1}{Digit Based Montgomery Reduction}{section.7.3}
\BOOKMARK [2][-]{subsection.7.3.2}{Baseline Montgomery Reduction}{section.7.3}
\BOOKMARK [2][-]{subsection.7.3.3}{Faster ``Comba'' Montgomery Reduction}{section.7.3}
\BOOKMARK [2][-]{subsection.7.3.4}{Montgomery Setup}{section.7.3}
\BOOKMARK [1][-]{section.7.4}{The Diminished Radix Algorithm}{chapter.7}
\BOOKMARK [2][-]{subsection.7.4.1}{Choice of Moduli}{section.7.4}
\BOOKMARK [2][-]{subsection.7.4.2}{Choice of k}{section.7.4}
\BOOKMARK [2][-]{subsection.7.4.3}{Restricted Diminished Radix Reduction}{section.7.4}
\BOOKMARK [2][-]{subsection.7.4.4}{Unrestricted Diminished Radix Reduction}{section.7.4}
\BOOKMARK [1][-]{section.7.5}{Algorithm Comparison}{chapter.7}
\BOOKMARK [0][-]{chapter.8}{Exponentiation}{}
\BOOKMARK [1][-]{section.8.1}{Exponentiation Basics}{chapter.8}
\BOOKMARK [2][-]{subsection.8.1.1}{Single Digit Exponentiation}{section.8.1}
\BOOKMARK [1][-]{section.8.2}{k-ary Exponentiation}{chapter.8}
\BOOKMARK [2][-]{subsection.8.2.1}{Optimal Values of k}{section.8.2}
\BOOKMARK [2][-]{subsection.8.2.2}{Sliding-Window Exponentiation}{section.8.2}
\BOOKMARK [1][-]{section.8.3}{Modular Exponentiation}{chapter.8}
\BOOKMARK [2][-]{subsection.8.3.1}{Barrett Modular Exponentiation}{section.8.3}
\BOOKMARK [1][-]{section.8.4}{Quick Power of Two}{chapter.8}
\BOOKMARK [0][-]{chapter.9}{Higher Level Algorithms}{}
\BOOKMARK [1][-]{section.9.1}{Integer Division with Remainder}{chapter.9}
\BOOKMARK [1][-]{section.9.2}{Single Digit Helpers}{chapter.9}
\BOOKMARK [2][-]{subsection.9.2.1}{Single Digit Addition}{section.9.2}
\BOOKMARK [2][-]{subsection.9.2.2}{Single Digit Subtraction}{section.9.2}
\BOOKMARK [2][-]{subsection.9.2.3}{Single Digit Multiplication}{section.9.2}
\BOOKMARK [2][-]{subsection.9.2.4}{Single Digit Division}{section.9.2}
\BOOKMARK [2][-]{subsection.9.2.5}{Single Digit Modulo}{section.9.2}
\BOOKMARK [2][-]{subsection.9.2.6}{Single Digit Root Extraction}{section.9.2}
\BOOKMARK [1][-]{section.9.3}{Random Number Generation}{chapter.9}
\BOOKMARK [1][-]{section.9.4}{Formatted Output}{chapter.9}
\BOOKMARK [2][-]{subsection.9.4.1}{Getting The Output Size}{section.9.4}
\BOOKMARK [2][-]{subsection.9.4.2}{Generating Radix-n Output}{section.9.4}
\BOOKMARK [2][-]{subsection.9.4.3}{Reading Radix-n Input}{section.9.4}
\BOOKMARK [1][-]{section.9.5}{Unformatted Output}{chapter.9}
\BOOKMARK [2][-]{subsection.9.5.1}{Getting The Output Size}{section.9.5}
\BOOKMARK [2][-]{subsection.9.5.2}{Generating Output}{section.9.5}
\BOOKMARK [2][-]{subsection.9.5.3}{Reading Input}{section.9.5}
\BOOKMARK [0][-]{chapter.10}{Number Theoretic Algorithms}{}
\BOOKMARK [1][-]{section.10.1}{Greatest Common Divisor}{chapter.10}
\BOOKMARK [1][-]{section.10.2}{Least Common Multiple}{chapter.10}
\BOOKMARK [1][-]{section.10.3}{Jacobi Symbol Computation}{chapter.10}
\BOOKMARK [1][-]{section.10.4}{Modular Inverse}{chapter.10}
\BOOKMARK [2][-]{subsection.10.4.1}{General Case}{section.10.4}
\BOOKMARK [2][-]{subsection.10.4.2}{Odd Moduli}{section.10.4}
\BOOKMARK [1][-]{section.10.5}{Primality Tests}{chapter.10}
\BOOKMARK [2][-]{subsection.10.5.1}{Trial Division}{section.10.5}
\BOOKMARK [2][-]{subsection.10.5.2}{The Fermat Test}{section.10.5}
\BOOKMARK [2][-]{subsection.10.5.3}{The Miller-Rabin Test}{section.10.5}
\BOOKMARK [2][-]{subsection.10.5.4}{Primality Test in a Bottle}{section.10.5}
\BOOKMARK [2][-]{subsection.10.5.5}{The Next Prime}{section.10.5}
\BOOKMARK [1][-]{section.10.6}{Root Extraction}{chapter.10}
\BOOKMARK [0][-]{appendix*.16}{Appendix}{}

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