From 6c48a9b3a634fef0c5b5b5f3b190b4a4c437cf9f Mon Sep 17 00:00:00 2001 From: Tom St Denis Date: Sun, 25 Jan 2004 17:40:21 +0000 Subject: [PATCH] added libtommath-0.29 --- bn.ilg | 6 +- bn.ind | 6 +- bn.pdf | Bin 292890 -> 298166 bytes bn.tex | 44 +- bn_mp_div_3.c | 2 +- bn_mp_div_d.c | 3 +- bn_mp_exteuclid.c | 69 + bn_mp_fwrite.c | 7 +- bn_mp_mul.c | 1 - bn_mp_radix_size.c | 15 +- bn_mp_read_radix.c | 3 +- bn_mp_shrink.c | 2 +- bn_mp_signed_bin_size.c | 3 +- bn_mp_sqr.c | 1 + bn_mp_toradix.c | 3 +- bn_prime_sizes_tab.c | 20 +- bn_s_mp_exptmod.c | 7 +- changes.txt | 15 +- demo/demo.c | 5 +- etc/pprime.c | 8 +- etc/prime.1024 | 414 + etc/prime.512 | 205 + makefile | 12 +- makefile.bcc | 2 +- makefile.cygwin_dll | 2 +- makefile.msvc | 2 +- poster.pdf | Bin 40822 -> 40822 bytes pre_gen/mpi.c | 140 +- tommath.h | 5 +- tommath.out | 254 +- tommath.pdf | 24275 +++++++++++++++++++------------------- tommath.src | 196 +- tommath.tex | 1062 +- 33 files changed, 13855 insertions(+), 12934 deletions(-) create mode 100644 bn_mp_exteuclid.c create mode 100644 etc/prime.1024 create mode 100644 etc/prime.512 diff --git a/bn.ilg b/bn.ilg index d2b5862..89badbc 100644 --- a/bn.ilg +++ b/bn.ilg @@ -1,6 +1,6 @@ This is makeindex, version 2.14 [02-Oct-2002] (kpathsea + Thai support). -Scanning input file bn.idx....done (53 entries accepted, 0 rejected). -Sorting entries....done (317 comparisons). -Generating output file bn.ind....done (56 lines written, 0 warnings). +Scanning input file bn.idx....done (57 entries accepted, 0 rejected). +Sorting entries....done (342 comparisons). +Generating output file bn.ind....done (60 lines written, 0 warnings). Output written in bn.ind. Transcript written in bn.ilg. diff --git a/bn.ind b/bn.ind index da10a3d..451128c 100644 --- a/bn.ind +++ b/bn.ind @@ -14,6 +14,7 @@ \item mp\_error\_to\_string, \hyperpage{6} \item mp\_expt\_d, \hyperpage{31} \item mp\_exptmod, \hyperpage{31} + \item mp\_exteuclid, \hyperpage{39} \item mp\_gcd, \hyperpage{39} \item mp\_grow, \hyperpage{12} \item MP\_GT, \hyperpage{17} @@ -23,7 +24,7 @@ \item mp\_init\_size, \hyperpage{10} \item mp\_int, \hyperpage{6} \item mp\_invmod, \hyperpage{40} - \item mp\_jacobi, \hyperpage{39} + \item mp\_jacobi, \hyperpage{40} \item mp\_lcm, \hyperpage{39} \item mp\_lshd, \hyperpage{23} \item MP\_LT, \hyperpage{17} @@ -43,12 +44,15 @@ \item mp\_prime\_next\_prime, \hyperpage{34} \item mp\_prime\_rabin\_miller\_trials, \hyperpage{34} \item mp\_prime\_random, \hyperpage{35} + \item mp\_radix\_size, \hyperpage{37} + \item mp\_read\_radix, \hyperpage{37} \item mp\_rshd, \hyperpage{23} \item mp\_set, \hyperpage{15} \item mp\_set\_int, \hyperpage{16} \item 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0c1f6a3..bed0b26 100644 --- a/bn.tex +++ b/bn.tex @@ -1001,10 +1001,10 @@ algorithms all which can be called from mp\_sqr(). \section{Tuning Polynomial Basis Routines} Both Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that -the Comba and baseline algorithms use. At $O(n^{1.46})$ and $O(n^{1.58})$ running times respectfully they require -considerably less work. For example, a 10000-digit multiplication would take roughly 692,000 single precision +the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectfully they require +considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor -of 144). +of 138). So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration, @@ -1175,10 +1175,48 @@ there is no skew on the least significant bits. \chapter{Input and Output} \section{ASCII Conversions} +\subsection{To ASCII} +\index{mp\_toradix} +\begin{alltt} +int mp_toradix (mp_int * a, char *str, int radix); +\end{alltt} +This still store ``a'' in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character +to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required +by the conversion before storing any data use the following function. + +\index{mp\_radix\_size} +\begin{alltt} +int mp_radix_size (mp_int * a, int radix, int *size) +\end{alltt} +This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this +function returns an error code and ``size'' will be zero. + +\subsection{From ASCII} +\index{mp\_read\_radix} +\begin{alltt} +int mp_read_radix (mp_int * a, char *str, int radix); +\end{alltt} +This will read the base-``radix'' NUL terminated string from ``str'' into ``a''. It will stop reading when it reads a +character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign +can be used to denote a negative number. + \section{Binary Conversions} \section{Stream Functions} \chapter{Algebraic Functions} +\section{Extended Euclidean Algorithm} +\index{mp\_exteuclid} +\begin{alltt} +int mp_exteuclid(mp_int *a, mp_int *b, + mp_int *U1, mp_int *U2, mp_int *U3); +\end{alltt} + +This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds. + +\begin{equation} +a \cdot U1 + b \cdot U2 = U3 +\end{equation} + \section{Greatest Common Divisor} \index{mp\_gcd} \begin{alltt} diff --git a/bn_mp_div_3.c b/bn_mp_div_3.c index 0c41db6..3bc2bd2 100644 --- a/bn_mp_div_3.c +++ b/bn_mp_div_3.c @@ -41,7 +41,7 @@ mp_div_3 (mp_int * a, mp_int *c, mp_digit * d) t = (w * ((mp_word)b)) >> ((mp_word)DIGIT_BIT); /* now subtract 3 * [w/3] from w, to get the remainder */ - w -= (t << ((mp_word)1)) + t; + w -= t+t+t; /* fixup the remainder as required since * the optimization is not exact. diff --git a/bn_mp_div_d.c b/bn_mp_div_d.c index 244b8da..f8caf83 100644 --- a/bn_mp_div_d.c +++ b/bn_mp_div_d.c @@ -28,8 +28,7 @@ static int s_is_power_of_two(mp_digit b, int *p) } /* single digit division (based on routine from MPI) */ -int -mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) +int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) { mp_int q; mp_word w; diff --git a/bn_mp_exteuclid.c b/bn_mp_exteuclid.c new file mode 100644 index 0000000..6875f72 --- /dev/null +++ b/bn_mp_exteuclid.c @@ -0,0 +1,69 @@ +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +#include + +/* Extended euclidean algorithm of (a, b) produces + a*u1 + b*u2 = u3 + */ +int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) +{ + mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp; + int err; + + if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) { + return err; + } + + /* initialize, (u1,u2,u3) = (1,0,a) */ + mp_set(&u1, 1); + if ((err = mp_copy(a, &u3)) != MP_OKAY) { goto _ERR; } + + /* initialize, (v1,v2,v3) = (0,1,b) */ + mp_set(&v2, 1); + if ((err = mp_copy(b, &v3)) != MP_OKAY) { goto _ERR; } + + /* loop while v3 != 0 */ + while (mp_iszero(&v3) == MP_NO) { + /* q = u3/v3 */ + if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) { goto _ERR; } + + /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ + if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) { goto _ERR; } + + /* (u1,u2,u3) = (v1,v2,v3) */ + if ((err = mp_copy(&v1, &u1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v2, &u2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v3, &u3)) != MP_OKAY) { goto _ERR; } + + /* (v1,v2,v3) = (t1,t2,t3) */ + if ((err = mp_copy(&t1, &v1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t2, &v2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t3, &v3)) != MP_OKAY) { goto _ERR; } + } + + /* copy result out */ + if (U1 != NULL) { mp_exch(U1, &u1); } + if (U2 != NULL) { mp_exch(U2, &u2); } + if (U3 != NULL) { mp_exch(U3, &u3); } + + err = MP_OKAY; +_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); + return err; +} diff --git a/bn_mp_fwrite.c b/bn_mp_fwrite.c index eabfc8c..61acfc3 100644 --- a/bn_mp_fwrite.c +++ b/bn_mp_fwrite.c @@ -19,11 +19,10 @@ int mp_fwrite(mp_int *a, int radix, FILE *stream) char *buf; int err, len, x; - len = mp_radix_size(a, radix); - if (len == 0) { - return MP_VAL; + if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { + return err; } - + buf = XMALLOC (len); if (buf == NULL) { return MP_MEM; diff --git a/bn_mp_mul.c b/bn_mp_mul.c index 859cccf..6f3c491 100644 --- a/bn_mp_mul.c +++ b/bn_mp_mul.c @@ -42,7 +42,6 @@ int mp_mul (mp_int * a, mp_int * b, mp_int * c) } else { res = s_mp_mul (a, b, c); } - } c->sign = neg; return res; diff --git a/bn_mp_radix_size.c b/bn_mp_radix_size.c index 9f888dd..ae2d09a 100644 --- a/bn_mp_radix_size.c +++ b/bn_mp_radix_size.c @@ -15,26 +15,28 @@ #include /* returns size of ASCII reprensentation */ -int -mp_radix_size (mp_int * a, int radix) +int mp_radix_size (mp_int * a, int radix, int *size) { int res, digs; mp_int t; mp_digit d; + *size = 0; + /* special case for binary */ if (radix == 2) { - return mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; + *size = mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; + return MP_OKAY; } /* make sure the radix is in range */ if (radix < 2 || radix > 64) { - return 0; + return MP_VAL; } /* init a copy of the input */ if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return 0; + return res; } /* digs is the digit count */ @@ -57,6 +59,7 @@ mp_radix_size (mp_int * a, int radix) mp_clear (&t); /* return digs + 1, the 1 is for the NULL byte that would be required. */ - return digs + 1; + *size = digs + 1; + return MP_OKAY; } diff --git a/bn_mp_read_radix.c b/bn_mp_read_radix.c index ac87993..653a891 100644 --- a/bn_mp_read_radix.c +++ b/bn_mp_read_radix.c @@ -15,8 +15,7 @@ #include /* read a string [ASCII] in a given radix */ -int -mp_read_radix (mp_int * a, char *str, int radix) +int mp_read_radix (mp_int * a, char *str, int radix) { int y, res, neg; char ch; diff --git a/bn_mp_shrink.c b/bn_mp_shrink.c index f7cb53e..a419b6e 100644 --- a/bn_mp_shrink.c +++ b/bn_mp_shrink.c @@ -18,7 +18,7 @@ int mp_shrink (mp_int * a) { mp_digit *tmp; - if (a->alloc != a->used) { + if (a->alloc != a->used && a->used > 0) { if ((tmp = OPT_CAST XREALLOC (a->dp, sizeof (mp_digit) * a->used)) == NULL) { return MP_MEM; } diff --git a/bn_mp_signed_bin_size.c b/bn_mp_signed_bin_size.c index d392ebc..76d085c 100644 --- a/bn_mp_signed_bin_size.c +++ b/bn_mp_signed_bin_size.c @@ -15,8 +15,7 @@ #include /* get the size for an signed equivalent */ -int -mp_signed_bin_size (mp_int * a) +int mp_signed_bin_size (mp_int * a) { return 1 + mp_unsigned_bin_size (a); } diff --git a/bn_mp_sqr.c b/bn_mp_sqr.c index d354c0b..4857994 100644 --- a/bn_mp_sqr.c +++ b/bn_mp_sqr.c @@ -19,6 +19,7 @@ int mp_sqr (mp_int * a, mp_int * b) { int res; + /* use Toom-Cook? */ if (a->used >= TOOM_SQR_CUTOFF) { res = mp_toom_sqr(a, b); diff --git a/bn_mp_toradix.c b/bn_mp_toradix.c index 359d5f2..c605867 100644 --- a/bn_mp_toradix.c +++ b/bn_mp_toradix.c @@ -15,8 +15,7 @@ #include /* stores a bignum as a ASCII string in a given radix (2..64) */ -int -mp_toradix (mp_int * a, char *str, int radix) +int mp_toradix (mp_int * a, char *str, int radix) { int res, digs; mp_int t; diff --git a/bn_prime_sizes_tab.c b/bn_prime_sizes_tab.c index 05904f1..c6e1dad 100644 --- a/bn_prime_sizes_tab.c +++ b/bn_prime_sizes_tab.c @@ -31,25 +31,7 @@ static const struct { { 1408, 3 }, { 1536, 3 }, { 1664, 3 }, -{ 1792, 2 }, -{ 1920, 2 }, -{ 2048, 2 }, -{ 2176, 2 }, -{ 2304, 2 }, -{ 2432, 2 }, -{ 2560, 2 }, -{ 2688, 2 }, -{ 2816, 2 }, -{ 2944, 2 }, -{ 3072, 2 }, -{ 3200, 2 }, -{ 3328, 2 }, -{ 3456, 2 }, -{ 3584, 2 }, -{ 3712, 2 }, -{ 3840, 1 }, -{ 3968, 1 }, -{ 4096, 1 } }; +{ 1792, 2 } }; /* returns # of RM trials required for a given bit size */ int mp_prime_rabin_miller_trials(int size) diff --git a/bn_s_mp_exptmod.c b/bn_s_mp_exptmod.c index 68c91e7..28c14fd 100644 --- a/bn_s_mp_exptmod.c +++ b/bn_s_mp_exptmod.c @@ -20,8 +20,7 @@ #define TAB_SIZE 256 #endif -int -s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) { mp_int M[TAB_SIZE], res, mu; mp_digit buf; @@ -185,10 +184,10 @@ s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) /* then multiply */ if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { - goto __MU; + goto __RES; } if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { - goto __MU; + goto __RES; } /* empty window and reset */ diff --git a/changes.txt b/changes.txt index 87b3906..3998eb8 100644 --- a/changes.txt +++ b/changes.txt @@ -1,7 +1,18 @@ +Jan 25th, 2004 +v0.29 ++ Note: "Henrik" from the v0.28 changelog refers to Henrik Goldman ;-) + -- Added fix to mp_shrink to prevent a realloc when used == 0 [e.g. realloc zero bytes???] + -- Made the mp_prime_rabin_miller_trials() function internal table smaller and also + set the minimum number of tests to two (sounds a bit safer). + -- Added a mp_exteuclid() which computes the extended euclidean algorithm. + -- Fixed a memory leak in s_mp_exptmod() [called when Barrett reduction is to be used] which would arise + if a multiplication or subsequent reduction failed [would not free the temp result]. + -- Made an API change to mp_radix_size(). It now returns an error code and stores the required size + through an "int star" passed to it. + Dec 24th, 2003 -v0.28 -- Henrik suggested I add casts to the montomgery code [stores into mu...] so compilers wouldn't +v0.28 -- Henrik Goldman suggested I add casts to the montomgery code [stores into mu...] so compilers wouldn't spew [erroneous] diagnostics... fixed. - -- Henrik also spotted two typos. One in mp_radix_size() and another in mp_toradix(). + -- Henrik Goldman also spotted two typos. One in mp_radix_size() and another in mp_toradix(). -- Added fix to mp_shrink() to avoid a memory leak. -- Added mp_prime_random() which requires a callback to make truly random primes of a given nature (idea from chat with Niels Ferguson at Crypto'03) diff --git a/demo/demo.c b/demo/demo.c index 7d78456..089b319 100644 --- a/demo/demo.c +++ b/demo/demo.c @@ -138,15 +138,16 @@ int main(void) /* test mp_div_3 */ #if 0 + mp_set(&d, 3); for (cnt = 0; cnt < 1000000; ) { mp_digit r1, r2; if (!(++cnt & 127)) printf("%9d\r", cnt); mp_rand(&a, abs(rand()) % 128 + 1); - mp_div_d(&a, 3, &b, &r1); + mp_div(&a, &d, &b, &e); mp_div_3(&a, &c, &r2); - if (mp_cmp(&b, &c) || r1 != r2) { + if (mp_cmp(&b, &c) || mp_cmp_d(&e, r2)) { printf("\n\nmp_div_3 => Failure\n"); } } diff --git a/etc/pprime.c b/etc/pprime.c index 389e1fa..cccb748 100644 --- a/etc/pprime.c +++ b/etc/pprime.c @@ -229,8 +229,8 @@ pprime (int k, int li, mp_int * p, mp_int * q) /* now loop making the single digit */ while (mp_count_bits (&a) < k) { - printf ("prime has %4d bits left\r", k - mp_count_bits (&a)); - fflush (stdout); + fprintf (stderr, "prime has %4d bits left\r", k - mp_count_bits (&a)); + fflush (stderr); top: mp_set (&b, prime_digit ()); @@ -322,7 +322,6 @@ pprime (int k, int li, mp_int * p, mp_int * q) if (ii == li) goto top; -/* { char buf[4096]; @@ -331,10 +330,9 @@ pprime (int k, int li, mp_int * p, mp_int * q) mp_toradix(&a, buf, 10); printf("A == \n%s\n\n", buf); mp_toradix(&b, buf, 10); - printf("B == \n%s\n", buf); + printf("B == \n%s\n\nG == %d\n", buf, bases[ii]); printf("----------------------------------------------------------------\n"); } -*/ /* a = n */ mp_copy (&n, &a); diff --git a/etc/prime.1024 b/etc/prime.1024 new file mode 100644 index 0000000..5636e2d --- /dev/null +++ b/etc/prime.1024 @@ -0,0 +1,414 @@ +Enter # of bits: +Enter number of bases to try (1 to 8): +Certificate of primality for: +36360080703173363 + +A == +89963569 + +B == +202082249 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +4851595597739856136987139 + +A == +36360080703173363 + +B == +66715963 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +19550639734462621430325731591027 + +A == +4851595597739856136987139 + +B == +2014867 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +10409036141344317165691858509923818734539 + +A == +19550639734462621430325731591027 + +B == +266207047 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1049829549988285012736475602118094726647504414203 + +A == +10409036141344317165691858509923818734539 + +B == +50428759 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +77194737385528288387712399596835459931920358844586615003 + +A == +1049829549988285012736475602118094726647504414203 + +B == +36765367 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +35663756695365208574443215955488689578374232732893628896541201763 + +A == +77194737385528288387712399596835459931920358844586615003 + +B == +230998627 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +16711831463502165169495622246023119698415848120292671294127567620396469803 + +A == +35663756695365208574443215955488689578374232732893628896541201763 + +B == +234297127 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +6163534781560285962890718925972249753147470953579266394395432475622345597103528739 + +A == +16711831463502165169495622246023119698415848120292671294127567620396469803 + +B == +184406323 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +814258256205243497704094951432575867360065658372158511036259934640748088306764553488803787 + +A == +6163534781560285962890718925972249753147470953579266394395432475622345597103528739 + +B == +66054487 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +176469695533271657902814176811660357049007467856432383037590673407330246967781451723764079581998187 + +A == +814258256205243497704094951432575867360065658372158511036259934640748088306764553488803787 + +B == +108362239 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +44924492859445516541759485198544012102424796403707253610035148063863073596051272171194806669756971406400419 + +A == +176469695533271657902814176811660357049007467856432383037590673407330246967781451723764079581998187 + +B == +127286707 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +20600996927219343383225424320134474929609459588323857796871086845924186191561749519858600696159932468024710985371059 + +A == +44924492859445516541759485198544012102424796403707253610035148063863073596051272171194806669756971406400419 + +B == +229284691 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +6295696427695493110141186605837397185848992307978456138112526915330347715236378041486547994708748840844217371233735072572979 + +A == +20600996927219343383225424320134474929609459588323857796871086845924186191561749519858600696159932468024710985371059 + +B == +152800771 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +3104984078042317488749073016454213579257792635142218294052134804187631661145261015102617582090263808696699966840735333252107678792123 + +A == +6295696427695493110141186605837397185848992307978456138112526915330347715236378041486547994708748840844217371233735072572979 + +B == +246595759 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +26405175827665701256325699315126705508919255051121452292124404943796947287968603975320562847910946802396632302209435206627913466015741799499 + +A == +3104984078042317488749073016454213579257792635142218294052134804187631661145261015102617582090263808696699966840735333252107678792123 + +B == +4252063 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +11122146237908413610034600609460545703591095894418599759742741406628055069007082998134905595800236452010905900391505454890446585211975124558601770163 + +A == +26405175827665701256325699315126705508919255051121452292124404943796947287968603975320562847910946802396632302209435206627913466015741799499 + +B == +210605419 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1649861642047798890580354082088712649911849362201343649289384923147797960364736011515757482030049342943790127685185806092659832129486307035500638595572396187 + +A == +11122146237908413610034600609460545703591095894418599759742741406628055069007082998134905595800236452010905900391505454890446585211975124558601770163 + +B == +74170111 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +857983367126266717607389719637086684134462613006415859877666235955788392464081914127715967940968197765042399904117392707518175220864852816390004264107201177394565363 + +A == +1649861642047798890580354082088712649911849362201343649289384923147797960364736011515757482030049342943790127685185806092659832129486307035500638595572396187 + +B == +260016763 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +175995909353623703257072120479340610010337144085688850745292031336724691277374210929188442230237711063783727092685448718515661641054886101716698390145283196296702450566161283 + +A == +857983367126266717607389719637086684134462613006415859877666235955788392464081914127715967940968197765042399904117392707518175220864852816390004264107201177394565363 + +B == +102563707 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +48486002551155667224487059713350447239190772068092630563272168418880661006593537218144160068395218642353495339720640699721703003648144463556291315694787862009052641640656933232794283 + +A == +175995909353623703257072120479340610010337144085688850745292031336724691277374210929188442230237711063783727092685448718515661641054886101716698390145283196296702450566161283 + +B == +137747527 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +13156468011529105025061495011938518171328604045212410096476697450506055664012861932372156505805788068791146986282263016790631108386790291275939575123375304599622623328517354163964228279867403 + +A == +48486002551155667224487059713350447239190772068092630563272168418880661006593537218144160068395218642353495339720640699721703003648144463556291315694787862009052641640656933232794283 + +B == +135672847 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +6355194692790533601105154341731997464407930009404822926832136060319955058388106456084549316415200519472481147942263916585428906582726749131479465958107142228236909665306781538860053107680830113869123 + +A == +13156468011529105025061495011938518171328604045212410096476697450506055664012861932372156505805788068791146986282263016790631108386790291275939575123375304599622623328517354163964228279867403 + +B == +241523587 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +3157116676535430302794438027544146642863331358530722860333745617571010460905857862561870488000265751138954271040017454405707755458702044884023184574412221802502351503929935224995314581932097706874819348858083 + +A == +6355194692790533601105154341731997464407930009404822926832136060319955058388106456084549316415200519472481147942263916585428906582726749131479465958107142228236909665306781538860053107680830113869123 + +B == +248388667 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +390533129219992506725320633489467713907837370444962163378727819939092929448752905310115311180032249230394348337568973177802874166228132778126338883671958897238722734394783244237133367055422297736215754829839364158067 + +A == +3157116676535430302794438027544146642863331358530722860333745617571010460905857862561870488000265751138954271040017454405707755458702044884023184574412221802502351503929935224995314581932097706874819348858083 + +B == +61849651 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +48583654555070224891047847050732516652910250240135992225139515777200432486685999462997073444468380434359929499498804723793106565291183220444221080449740542884172281158126259373095216435009661050109711341419005972852770440739 + +A == +390533129219992506725320633489467713907837370444962163378727819939092929448752905310115311180032249230394348337568973177802874166228132778126338883671958897238722734394783244237133367055422297736215754829839364158067 + +B == +62201707 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +25733035251905120039135866524384525138869748427727001128764704499071378939227862068500633813538831598776578372709963673670934388213622433800015759585470542686333039614931682098922935087822950084908715298627996115185849260703525317419 + +A == +48583654555070224891047847050732516652910250240135992225139515777200432486685999462997073444468380434359929499498804723793106565291183220444221080449740542884172281158126259373095216435009661050109711341419005972852770440739 + +B == +264832231 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +2804594464939948901906623499531073917980499195397462605359913717827014360538186518540781517129548650937632008683280555602633122170458773895504894807182664540529077836857897972175530148107545939211339044386106111633510166695386323426241809387 + +A == +25733035251905120039135866524384525138869748427727001128764704499071378939227862068500633813538831598776578372709963673670934388213622433800015759585470542686333039614931682098922935087822950084908715298627996115185849260703525317419 + +B == +54494047 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +738136612083433720096707308165797114449914259256979340471077690416567237592465306112484843530074782721390528773594351482384711900456440808251196845265132086486672447136822046628407467459921823150600138073268385534588238548865012638209515923513516547 + +A == +2804594464939948901906623499531073917980499195397462605359913717827014360538186518540781517129548650937632008683280555602633122170458773895504894807182664540529077836857897972175530148107545939211339044386106111633510166695386323426241809387 + +B == +131594179 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +392847529056126766528615419937165193421166694172790666626558750047057558168124866940509180171236517681470100877687445134633784815352076138790217228749332398026714192707447855731679485746120589851992221508292976900578299504461333767437280988393026452846013683 + +A == +738136612083433720096707308165797114449914259256979340471077690416567237592465306112484843530074782721390528773594351482384711900456440808251196845265132086486672447136822046628407467459921823150600138073268385534588238548865012638209515923513516547 + +B == +266107603 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +168459393231883505975876919268398655632763956627405508859662408056221544310200546265681845397346956580604208064328814319465940958080244889692368602591598503944015835190587740756859842792554282496742843600573336023639256008687581291233481455395123454655488735304365627 + +A == +392847529056126766528615419937165193421166694172790666626558750047057558168124866940509180171236517681470100877687445134633784815352076138790217228749332398026714192707447855731679485746120589851992221508292976900578299504461333767437280988393026452846013683 + +B == +214408111 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +14865774288636941404884923981945833072113667565310054952177860608355263252462409554658728941191929400198053290113492910272458441655458514080123870132092365833472436407455910185221474386718838138135065780840839893113912689594815485706154461164071775481134379794909690501684643 + +A == +168459393231883505975876919268398655632763956627405508859662408056221544310200546265681845397346956580604208064328814319465940958080244889692368602591598503944015835190587740756859842792554282496742843600573336023639256008687581291233481455395123454655488735304365627 + +B == +44122723 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1213301773203241614897109856134894783021668292000023984098824423682568173639394290886185366993108292039068940333907505157813934962357206131450244004178619265868614859794316361031904412926604138893775068853175215502104744339658944443630407632290152772487455298652998368296998719996019 + +A == +14865774288636941404884923981945833072113667565310054952177860608355263252462409554658728941191929400198053290113492910272458441655458514080123870132092365833472436407455910185221474386718838138135065780840839893113912689594815485706154461164071775481134379794909690501684643 + +B == +40808563 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +186935245989515158127969129347464851990429060640910951266513740972248428651109062997368144722015290092846666943896556191257222521203647606911446635194198213436423080005867489516421559330500722264446765608763224572386410155413161172707802334865729654109050873820610813855041667633843601286843 + +A == +1213301773203241614897109856134894783021668292000023984098824423682568173639394290886185366993108292039068940333907505157813934962357206131450244004178619265868614859794316361031904412926604138893775068853175215502104744339658944443630407632290152772487455298652998368296998719996019 + +B == +77035759 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +83142661079751490510739960019112406284111408348732592580459037404394946037094409915127399165633756159385609671956087845517678367844901424617866988187132480585966721962585586730693443536100138246516868613250009028187662080828012497191775172228832247706080044971423654632146928165751885302331924491683 + +A == +186935245989515158127969129347464851990429060640910951266513740972248428651109062997368144722015290092846666943896556191257222521203647606911446635194198213436423080005867489516421559330500722264446765608763224572386410155413161172707802334865729654109050873820610813855041667633843601286843 + +B == +222383587 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +3892354773803809855317742245039794448230625839512638747643814927766738642436392673485997449586432241626440927010641564064764336402368634186618250134234189066179771240232458249806850838490410473462391401438160528157981942499581634732706904411807195259620779379274017704050790865030808501633772117217899534443 + +A == +83142661079751490510739960019112406284111408348732592580459037404394946037094409915127399165633756159385609671956087845517678367844901424617866988187132480585966721962585586730693443536100138246516868613250009028187662080828012497191775172228832247706080044971423654632146928165751885302331924491683 + +B == +23407687 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1663606652988091811284014366560171522582683318514519379924950390627250155440313691226744227787921928894551755219495501365555370027257568506349958010457682898612082048959464465369892842603765280317696116552850664773291371490339084156052244256635115997453399761029567033971998617303988376172539172702246575225837054723 + +A == +3892354773803809855317742245039794448230625839512638747643814927766738642436392673485997449586432241626440927010641564064764336402368634186618250134234189066179771240232458249806850838490410473462391401438160528157981942499581634732706904411807195259620779379274017704050790865030808501633772117217899534443 + +B == +213701827 + +G == 2 +---------------------------------------------------------------- + + +Took 33057 ticks, 1048 bits +P == 1663606652988091811284014366560171522582683318514519379924950390627250155440313691226744227787921928894551755219495501365555370027257568506349958010457682898612082048959464465369892842603765280317696116552850664773291371490339084156052244256635115997453399761029567033971998617303988376172539172702246575225837054723 +Q == 3892354773803809855317742245039794448230625839512638747643814927766738642436392673485997449586432241626440927010641564064764336402368634186618250134234189066179771240232458249806850838490410473462391401438160528157981942499581634732706904411807195259620779379274017704050790865030808501633772117217899534443 diff --git a/etc/prime.512 b/etc/prime.512 new file mode 100644 index 0000000..cb6ec30 --- /dev/null +++ b/etc/prime.512 @@ -0,0 +1,205 @@ +Enter # of bits: +Enter number of bases to try (1 to 8): +Certificate of primality for: +85933926807634727 + +A == +253758023 + +B == +169322581 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +23930198825086241462113799 + +A == +85933926807634727 + +B == +139236037 + +G == 11 +---------------------------------------------------------------- +Certificate of primality for: +6401844647261612602378676572510019 + +A == +23930198825086241462113799 + +B == +133760791 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +269731366027728777712034888684015329354259 + +A == +6401844647261612602378676572510019 + +B == +21066691 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +37942338209025571690075025099189467992329684223707 + +A == +269731366027728777712034888684015329354259 + +B == +70333567 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +15306904714258982484473490774101705363308327436988160248323 + +A == +37942338209025571690075025099189467992329684223707 + +B == +201712723 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1616744757018513392810355191503853040357155275733333124624513530099 + +A == +15306904714258982484473490774101705363308327436988160248323 + +B == +52810963 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +464222094814208047161771036072622485188658077940154689939306386289983787983 + +A == +1616744757018513392810355191503853040357155275733333124624513530099 + +B == +143566909 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +187429931674053784626487560729643601208757374994177258429930699354770049369025096447 + +A == +464222094814208047161771036072622485188658077940154689939306386289983787983 + +B == +201875281 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +100579220846502621074093727119851331775052664444339632682598589456666938521976625305832917563 + +A == +187429931674053784626487560729643601208757374994177258429930699354770049369025096447 + +B == +268311523 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1173616081309758475197022137833792133815753368965945885089720153370737965497134878651384030219765163 + +A == +100579220846502621074093727119851331775052664444339632682598589456666938521976625305832917563 + +B == +5834287 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +191456913489905913185935197655672585713573070349044195411728114905691721186574907738081340754373032735283623 + +A == +1173616081309758475197022137833792133815753368965945885089720153370737965497134878651384030219765163 + +B == +81567097 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +57856530489201750164178576399448868489243874083056587683743345599898489554401618943240901541005080049321706789987519 + +A == +191456913489905913185935197655672585713573070349044195411728114905691721186574907738081340754373032735283623 + +B == +151095433 + +G == 7 +---------------------------------------------------------------- +Certificate of primality for: +13790529750452576698109671710773784949185621244122040804792403407272729038377767162233653248852099545134831722512085881814803 + +A == +57856530489201750164178576399448868489243874083056587683743345599898489554401618943240901541005080049321706789987519 + +B == +119178679 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +7075985989000817742677547821106534174334812111605018857703825637170140040509067704269696198231266351631132464035671858077052876058979 + +A == +13790529750452576698109671710773784949185621244122040804792403407272729038377767162233653248852099545134831722512085881814803 + +B == +256552363 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1227273006232588072907488910282307435921226646895131225407452056677899411162892829564455154080310937471747140942360789623819327234258162420463 + +A == +7075985989000817742677547821106534174334812111605018857703825637170140040509067704269696198231266351631132464035671858077052876058979 + +B == +86720989 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +446764896913554613686067036908702877942872355053329937790398156069936255759889884246832779737114032666318220500106499161852193765380831330106375235763 + +A == +1227273006232588072907488910282307435921226646895131225407452056677899411162892829564455154080310937471747140942360789623819327234258162420463 + +B == +182015287 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +5290203010849586596974953717018896543907195901082056939587768479377028575911127944611236020459652034082251335583308070846379514569838984811187823420951275243 + +A == +446764896913554613686067036908702877942872355053329937790398156069936255759889884246832779737114032666318220500106499161852193765380831330106375235763 + +B == +5920567 + +G == 2 +---------------------------------------------------------------- + + +Took 3454 ticks, 521 bits +P == 5290203010849586596974953717018896543907195901082056939587768479377028575911127944611236020459652034082251335583308070846379514569838984811187823420951275243 +Q == 446764896913554613686067036908702877942872355053329937790398156069936255759889884246832779737114032666318220500106499161852193765380831330106375235763 diff --git a/makefile b/makefile index e2a7f5e..c7e2bc7 100644 --- a/makefile +++ b/makefile @@ -1,12 +1,18 @@ #Makefile for GCC # #Tom St Denis -CFLAGS += -I./ -Wall -W -Wshadow -O3 -funroll-loops +CFLAGS += -I./ -Wall -W -Wshadow + +#for speed +CFLAGS += -O3 -funroll-loops + +#for size +#CFLAGS += -Os #x86 optimizations [should be valid for any GCC install though] CFLAGS += -fomit-frame-pointer -VERSION=0.28 +VERSION=0.29 default: libtommath.a @@ -44,7 +50,7 @@ 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_mp_prime_random.o bn_prime_sizes_tab.o +bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_prime_random.o bn_prime_sizes_tab.o bn_mp_exteuclid.o libtommath.a: $(OBJECTS) $(AR) $(ARFLAGS) libtommath.a $(OBJECTS) diff --git a/makefile.bcc b/makefile.bcc index 4906612..7aba87c 100644 --- a/makefile.bcc +++ b/makefile.bcc @@ -29,7 +29,7 @@ bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \ 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_mp_prime_random.obj bn_prime_sizes_tab.obj +bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_prime_random.obj bn_prime_sizes_tab.obj bn_mp_exteuclid.obj TARGET = libtommath.lib diff --git a/makefile.cygwin_dll b/makefile.cygwin_dll index cf73161..c7cfc13 100644 --- a/makefile.cygwin_dll +++ b/makefile.cygwin_dll @@ -34,7 +34,7 @@ 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_mp_prime_random.o bn_prime_sizes_tab.o +bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_prime_random.o bn_prime_sizes_tab.o bn_mp_exteuclid.o # make a Windows DLL via Cygwin windll: $(OBJECTS) diff --git a/makefile.msvc b/makefile.msvc index eb4b2f1..1789871 100644 --- a/makefile.msvc +++ b/makefile.msvc @@ -28,7 +28,7 @@ bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \ 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_mp_prime_random.obj bn_prime_sizes_tab.obj +bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_prime_random.obj bn_prime_sizes_tab.obj bn_mp_exteuclid.obj library: $(OBJECTS) lib /out:tommath.lib $(OBJECTS) diff --git a/poster.pdf b/poster.pdf index be6cd0110339be16e0e2bafb094a131634078331..dc71d8b9e4ffaa1ccd9a7fcb042cc5c32b6e2952 100644 GIT binary patch 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zsMF6Gyu;=7ieB(n<*DUMSjMaTb}!3-2j!Niqy81f^Te!v)2Fi%V$b_d%M~D7L3}(20Yz0Z0zogcp4mIMV!~z+9^eYAyi@Rv{uTGL9C=$ zSH>9ka-3uyE65_Rs>bQb-m1Pp+Ejh=><|JEXoLl1VH3E+q%r6|?%!~X6d9$D>qPjC z9VYJ-g+#17DfE6Oi2enld{6@XBl!SvD&%H1kUu{miC13d|8FJ%8cp*?mQ&UKs-phi zn&iJ(foZzx|2HEQB5Yhtw}eLEWar@EV&~#+l*k_X*Q0(tLG1r--bTG#Xh=w& KMx#7@$o~WSmw)O2 diff --git a/pre_gen/mpi.c b/pre_gen/mpi.c index 88ada33..52288a9 100644 --- a/pre_gen/mpi.c +++ b/pre_gen/mpi.c @@ -1902,7 +1902,7 @@ mp_div_3 (mp_int * a, mp_int *c, mp_digit * d) t = (w * ((mp_word)b)) >> ((mp_word)DIGIT_BIT); /* now subtract 3 * [w/3] from w, to get the remainder */ - w -= (t << ((mp_word)1)) + t; + w -= t+t+t; /* fixup the remainder as required since * the optimization is not exact. @@ -1966,8 +1966,7 @@ static int s_is_power_of_two(mp_digit b, int *p) } /* single digit division (based on routine from MPI) */ -int -mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) +int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) { mp_int q; mp_word w; @@ -2665,6 +2664,79 @@ __M: /* End: bn_mp_exptmod_fast.c */ +/* Start: bn_mp_exteuclid.c */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +#include + +/* Extended euclidean algorithm of (a, b) produces + a*u1 + b*u2 = u3 + */ +int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) +{ + mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp; + int err; + + if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) { + return err; + } + + /* initialize, (u1,u2,u3) = (1,0,a) */ + mp_set(&u1, 1); + if ((err = mp_copy(a, &u3)) != MP_OKAY) { goto _ERR; } + + /* initialize, (v1,v2,v3) = (0,1,b) */ + mp_set(&v2, 1); + if ((err = mp_copy(b, &v3)) != MP_OKAY) { goto _ERR; } + + /* loop while v3 != 0 */ + while (mp_iszero(&v3) == MP_NO) { + /* q = u3/v3 */ + if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) { goto _ERR; } + + /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ + if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) { goto _ERR; } + + /* (u1,u2,u3) = (v1,v2,v3) */ + if ((err = mp_copy(&v1, &u1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v2, &u2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v3, &u3)) != MP_OKAY) { goto _ERR; } + + /* (v1,v2,v3) = (t1,t2,t3) */ + if ((err = mp_copy(&t1, &v1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t2, &v2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t3, &v3)) != MP_OKAY) { goto _ERR; } + } + + /* copy result out */ + if (U1 != NULL) { mp_exch(U1, &u1); } + if (U2 != NULL) { mp_exch(U2, &u2); } + if (U3 != NULL) { mp_exch(U3, &u3); } + + err = MP_OKAY; +_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); + return err; +} + +/* End: bn_mp_exteuclid.c */ + /* Start: bn_mp_fread.c */ /* LibTomMath, multiple-precision integer library -- Tom St Denis * @@ -2752,11 +2824,10 @@ int mp_fwrite(mp_int *a, int radix, FILE *stream) char *buf; int err, len, x; - len = mp_radix_size(a, radix); - if (len == 0) { - return MP_VAL; + if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { + return err; } - + buf = XMALLOC (len); if (buf == NULL) { return MP_MEM; @@ -4201,7 +4272,6 @@ int mp_mul (mp_int * a, mp_int * b, mp_int * c) } else { res = s_mp_mul (a, b, c); } - } c->sign = neg; return res; @@ -5251,26 +5321,28 @@ error: #include /* returns size of ASCII reprensentation */ -int -mp_radix_size (mp_int * a, int radix) +int mp_radix_size (mp_int * a, int radix, int *size) { int res, digs; mp_int t; mp_digit d; + *size = 0; + /* special case for binary */ if (radix == 2) { - return mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; + *size = mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; + return MP_OKAY; } /* make sure the radix is in range */ if (radix < 2 || radix > 64) { - return 0; + return MP_VAL; } /* init a copy of the input */ if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return 0; + return res; } /* digs is the digit count */ @@ -5293,7 +5365,8 @@ mp_radix_size (mp_int * a, int radix) mp_clear (&t); /* return digs + 1, the 1 is for the NULL byte that would be required. */ - return digs + 1; + *size = digs + 1; + return MP_OKAY; } @@ -5392,8 +5465,7 @@ mp_rand (mp_int * a, int digits) #include /* read a string [ASCII] in a given radix */ -int -mp_read_radix (mp_int * a, char *str, int radix) +int mp_read_radix (mp_int * a, char *str, int radix) { int y, res, neg; char ch; @@ -5989,7 +6061,7 @@ int mp_set_int (mp_int * a, unsigned long b) int mp_shrink (mp_int * a) { mp_digit *tmp; - if (a->alloc != a->used) { + if (a->alloc != a->used && a->used > 0) { if ((tmp = OPT_CAST XREALLOC (a->dp, sizeof (mp_digit) * a->used)) == NULL) { return MP_MEM; } @@ -6019,8 +6091,7 @@ int mp_shrink (mp_int * a) #include /* get the size for an signed equivalent */ -int -mp_signed_bin_size (mp_int * a) +int mp_signed_bin_size (mp_int * a) { return 1 + mp_unsigned_bin_size (a); } @@ -6049,6 +6120,7 @@ int mp_sqr (mp_int * a, mp_int * b) { int res; + /* use Toom-Cook? */ if (a->used >= TOOM_SQR_CUTOFF) { res = mp_toom_sqr(a, b); @@ -6892,8 +6964,7 @@ ERR: #include /* stores a bignum as a ASCII string in a given radix (2..64) */ -int -mp_toradix (mp_int * a, char *str, int radix) +int mp_toradix (mp_int * a, char *str, int radix) { int res, digs; mp_int t; @@ -7086,25 +7157,7 @@ static const struct { { 1408, 3 }, { 1536, 3 }, { 1664, 3 }, -{ 1792, 2 }, -{ 1920, 2 }, -{ 2048, 2 }, -{ 2176, 2 }, -{ 2304, 2 }, -{ 2432, 2 }, -{ 2560, 2 }, -{ 2688, 2 }, -{ 2816, 2 }, -{ 2944, 2 }, -{ 3072, 2 }, -{ 3200, 2 }, -{ 3328, 2 }, -{ 3456, 2 }, -{ 3584, 2 }, -{ 3712, 2 }, -{ 3840, 1 }, -{ 3968, 1 }, -{ 4096, 1 } }; +{ 1792, 2 } }; /* returns # of RM trials required for a given bit size */ int mp_prime_rabin_miller_trials(int size) @@ -7351,8 +7404,7 @@ s_mp_add (mp_int * a, mp_int * b, mp_int * c) #define TAB_SIZE 256 #endif -int -s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) { mp_int M[TAB_SIZE], res, mu; mp_digit buf; @@ -7516,10 +7568,10 @@ s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) /* then multiply */ if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { - goto __MU; + goto __RES; } if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { - goto __MU; + goto __RES; } /* empty window and reset */ diff --git a/tommath.h b/tommath.h index 1af3396..6cc9bb0 100644 --- a/tommath.h +++ b/tommath.h @@ -338,6 +338,9 @@ int mp_invmod(mp_int *a, mp_int *b, mp_int *c); /* c = (a, b) */ int mp_gcd(mp_int *a, mp_int *b, mp_int *c); +/* produces value such that U1*a + U2*b = U3 */ +int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3); + /* c = [a, b] or (a*b)/(a, b) */ int mp_lcm(mp_int *a, mp_int *b, mp_int *c); @@ -466,7 +469,7 @@ int mp_to_signed_bin(mp_int *a, unsigned char *b); int mp_read_radix(mp_int *a, char *str, int radix); int mp_toradix(mp_int *a, char *str, int radix); -int mp_radix_size(mp_int *a, int radix); +int mp_radix_size(mp_int *a, int radix, int *size); int mp_fread(mp_int *a, int radix, FILE *stream); int mp_fwrite(mp_int *a, int radix, FILE *stream); diff --git a/tommath.out b/tommath.out index 7341304..9f62617 100644 --- a/tommath.out +++ b/tommath.out @@ -1,7 +1,7 @@ \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}{What is Multiple Precision Arithmetic?}{section.1.1} +\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1} +\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1} \BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1} \BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1} \BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1} @@ -11,129 +11,129 @@ \BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3} \BOOKMARK [2][-]{subsection.1.3.5}{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}{Maintenance Algorithms}{chapter.3} -\BOOKMARK [2][-]{subsection.3.6.1}{Augmenting an mp\137int's Precision}{section.3.6} -\BOOKMARK [2][-]{subsection.3.6.2}{Initializing Variable Precision mp\137ints}{section.3.6} -\BOOKMARK [2][-]{subsection.3.6.3}{Multiple Integer Initializations and Clearings}{section.3.6} -\BOOKMARK [2][-]{subsection.3.6.4}{Clamping Excess Digits}{section.3.6} -\BOOKMARK [0][-]{chapter.4}{Basic Operations}{} +\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1} +\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5} +\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5} +\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1} +\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6} +\BOOKMARK [0][-]{chapter.2}{Getting Started}{} +\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2} +\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2} +\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2} +\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2} +\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2} +\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2} +\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5} +\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5} +\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2} +\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6} +\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6} +\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6} +\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6} +\BOOKMARK [0][-]{chapter.3}{Basic Operations}{} +\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3} +\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3} +\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2} +\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2} +\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3} +\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3} +\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4} +\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4} +\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3} +\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5} +\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5} +\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3} +\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6} +\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6} +\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{} \BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4} -\BOOKMARK [1][-]{section.4.2}{Assigning Values to mp\137int Structures}{chapter.4} -\BOOKMARK [2][-]{subsection.4.2.1}{Copying an mp\137int}{section.4.2} -\BOOKMARK [2][-]{subsection.4.2.2}{Creating a Clone}{section.4.2} -\BOOKMARK [1][-]{section.4.3}{Zeroing an Integer}{chapter.4} -\BOOKMARK [1][-]{section.4.4}{Sign Manipulation}{chapter.4} -\BOOKMARK [2][-]{subsection.4.4.1}{Absolute Value}{section.4.4} -\BOOKMARK [2][-]{subsection.4.4.2}{Integer Negation}{section.4.4} -\BOOKMARK [1][-]{section.4.5}{Small Constants}{chapter.4} -\BOOKMARK [2][-]{subsection.4.5.1}{Setting Small Constants}{section.4.5} -\BOOKMARK [2][-]{subsection.4.5.2}{Setting Large Constants}{section.4.5} -\BOOKMARK [1][-]{section.4.6}{Comparisons}{chapter.4} -\BOOKMARK [2][-]{subsection.4.6.1}{Unsigned Comparisions}{section.4.6} -\BOOKMARK [2][-]{subsection.4.6.2}{Signed Comparisons}{section.4.6} -\BOOKMARK [0][-]{chapter.5}{Basic Arithmetic}{} -\BOOKMARK [1][-]{section.5.1}{Introduction}{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 [2][-]{subsection.9.1.1}{Quotient Estimation}{section.9.1} -\BOOKMARK [2][-]{subsection.9.1.2}{Normalized Integers}{section.9.1} -\BOOKMARK 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Arithmetic?} +When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively +raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can +reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. +Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple +precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} + of algorithms can be designed to accomodate them. + +By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in +the decimal system with fixed precision $6 \cdot 7 = 2$. + +Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in +schools to manually add, subtract, multiply and divide. + \subsection{The Need for Multiple Precision Arithmetic} The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at greater than $10^{309}$. However, modern programming languages such as ISO C \ref{ISOC} and -Java \ref{JAVA} only provide instrinsic support for integers which are relatively small and are single precision. +typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and +Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. \begin{figure}[!here] \begin{center} @@ -227,9 +242,9 @@ The largest data type guaranteed to be provided by the ISO C programming language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored on the average desktop computer, rendering any protocol based on the algorithm insecure. Multiple -precision algorithms solve this very problem by extending the range of representable integers while using single precision -data types. +trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, +rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by +extending the range of representable integers while using single precision data types. Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in @@ -246,27 +261,16 @@ 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. -\subsection{What is Multiple Precision Arithmetic?} -At the heart of all multiple precision integer operations are the ``long-hand'' algorithms taught to children in grade -school. For example, to multiply $1,234$ by $981$ the student is not taught to memorize the times table for -$1,234$. Instead, they are taught how to long-multiply one digit at a time. That is to multiply each column using -simple single digit multiplications, line up the partial results, and add the resulting products by column. The -representation that most are familiar with is known as decimal or more formally as radix-10. A radix-$n$ representation -simply means there are $n$ possible values per digit. For example, binary would be a radix-2 representation. - -In essence computer based multiple precision arithmetic is very much the same. In most cases the same algorithms -which seem instinctive are the basis of computer based algorithms. The most notable difference is the usage -of a binary friendly radix, that is, to use a radix of the form $2^k$ where $k$ is typically the size of a computer -machine register\footnote{For example, with an x86 based processor $k$ could be $32$ while on an Alpha it would likely -be $64$.}. +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} \index{precision} The benefit of multiple precision representations over single or fixed precision representations is that -often no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of resolution to be precisely represented. -A multiple precision algorithm would augment the precision of the destination to accomodate the result while a single -precision system would truncate excess bits to maintain a fixed level of precision. +no precision is lost while representing the result of an operation which requires excess precision. For example, +the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple +precision algorithm would augment the precision of the destination to accomodate the result while a single precision system +would truncate excess bits to maintain a fixed level of precision. It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum @@ -279,11 +283,12 @@ Multiple precision algorithms have the most overhead of any style of arithmetic. overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. +without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to +be written and tested once. \section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement multiple precision algorithms. That is -to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' +The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. +That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} give considerably detailed explanations of the theoretical aspects of algorithms and often very little information regarding the practical implementation aspects. @@ -309,6 +314,9 @@ The algorithms that are presented will always include at least one ``pseudo-code by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same algorithm in other programming languages as the reader sees fit. +This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing +the reader how the algorithms fit together as well as where to start on various taskings. + \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 @@ -368,41 +376,36 @@ The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an inte itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When a value is presented as a -fraction such as $5 \over 2$ a real value division is implied. +$5/2 = 2$ 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 -of the integer. For example, $\vert \vert 123 \vert \vert = 3$. +of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. \subsection{Work Effort} -\index{big-O} -To measure the efficiency of the specified algorithms, a modified big-O notation is used. In this system all +\index{big-Oh} +To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. That is a single precision addition, multiplication and division are assumed to take the same time to complete. While this is generally not true in practice, it will simplify the discussions considerably. Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal multiplication requires $O(n^2)$ work while a squaring requires -$O({{n^2 + n}\over 2})$ work. In standard big-O notation these would both be said to be equivalent to $O(n^2)$. However, +the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a +baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these +would both be said to be equivalent to $O(n^2)$. However, in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a result small constant factors in the work effort will make an observable difference in algorithm efficiency. -Throughout the discussions various ``work levels'' will be discussed. The term work level shall refer to -the complexity of an algorithm with respect to its time requirements. For example, -$O(1)$, $O(n)$, $O(n^2)$, ..., $O(n^k)$ are various possible work levels that will be of concern in this text. Any -sequence of operations said to be at the $O(n^k)$ work level will often be nested $k-$deep within loops and are performed -$n^k$ times. - -Operations which are deeply nested within algorithms will have a higher big-O rating and be the target of the most -optimizatons. For example, in integer multiplication, by moving the carry propagation from the innermost -$O(n^2)$ nesting to the $O(n)$ nesting level the algorithm becomes vastly more -efficient\footnote{This is known as Comba multiplication.}. +All of the algorithms presented in this text have a polynomial time work level. That is, of the form +$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how +various optimizations will help pay off in the long run. \section{Exercises} -Within the more advanced chapters a section will be set aside to give the reader some challenging exercises. These -exercises are not designed to be prize winning problems, but instead to be thought provoking. Wherever possible the -problems are forward minded, stating problems that will be answered in subsequent chapters. The reader is encouraged to -finish the exercises as they appear to get a better understanding of the subject material. +Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to +the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought +provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent +chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the +subject material. That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular are encouraged to verify they can answer the problems correctly before moving on. @@ -410,52 +413,60 @@ are encouraged to verify they can answer the problems correctly before moving on Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the -scoring. +scoring system used. -\begin{tabular}{cl} -$\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ - & minutes to solve. Usually does not involve much computer time \\ - & to solve. \\ - & \\ -$\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|l|} +\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ + & minutes to solve. Usually does not involve much computer time \\ + & to solve. \\ +\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ & time usage. Usually requires a program to be written to \\ & solve the problem. \\ - & \\ -$\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ +\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ & of work. Usually involves trivial research and development of \\ & new theory from the perspective of a student. \\ - & \\ -$\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ +\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ & of work and research, the solution to which will demonstrate \\ & a higher mastery of the subject matter. \\ - & \\ -$\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ +\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ & novice to solve. Solutions to these problems will demonstrate a \\ & complete mastery of the given subject. \\ - & \\ +\hline \end{tabular} +\end{small} +\end{center} +\caption{Exercise Scoring System} +\end{figure} -Essentially problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also -designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. +Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or +devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level +are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These +two levels are essentially entry level questions. -Problems at the third level are meant to be a bit more difficult. Often the answer is fairly obvious but arriving at an exacting solution -requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of -another algorithm. +Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often +fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always +involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can +answer these questions will feel comfortable with the concepts behind the topic at hand. -Problems at the fourth level are meant to be even more difficult as well as involve some research. The reader will most -likely not know the answer right away, nor will the text provide the exact details of the answer until a subsequent -chapter. Problems at the fifth level are meant to be the hardest problems relative to all the other problems in the -chapter. People who can correctly answer fifth level problems have a mastery of the subject matter at hand. +Problems at the fourth level are meant to be similar to those of the level three questions except they will require +additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide +the exact details of the answer until a subsequent chapter. + +Problems at the fifth level are meant to be the hardest +problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a +mastery of the subject matter at hand. Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader is encouraged to answer the follow-up problems and try to draw the relevance of problems. -\chapter{Introduction to LibTomMath} +\section{Introduction to LibTomMath} -\section{What is LibTomMath?} -LibTomMath is a free and open source multiple precision library written in portable ISO C. By portable it is -meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on +\subsection{What is LibTomMath?} +LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it +is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on any given platform. The library has been successfully tested under numerous operating systems including Unix\footnote{All of these @@ -463,7 +474,7 @@ trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such as public key cryptosystems and still maintain a relatively small footprint. -\section{Goals of LibTomMath} +\subsection{Goals of LibTomMath} Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the @@ -479,13 +490,13 @@ based on the magnitude of the inputs and the configuration of the library. Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen as the template because it is -another library that fits in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the -function names and argument passing conventions, LibTomMath has been written from scratch by Tom St Denis. +MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits +in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument +passing conventions, it has been written from scratch by Tom St Denis. The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. +integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. \section{Choice of LibTomMath} LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but @@ -500,15 +511,13 @@ developer can more readily discern the true intent of a given section of source what conditional code will be used. The code base of LibTomMath is also well organized. Each function is in its own separate source code file -which allows the reader to find a given function very quickly. When compiled with GCC for the x86 processor the entire -library is a mere 87,760 bytes ($116,182$ bytes for ARMv4 processors). This includes every single function -LibTomMath provides from basic arithmetic to various number theoretic functions such as modular exponentiation, various -reduction algorithms and Jacobi symbol computation. +which allows the reader to find a given function very quickly. On average there are about $76$ lines of code per source +file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing +very hard. GMP has many conditional code segments which also hinder tracing. -By comparison MPI, which has fewer functions than LibTomMath, compiled with the same conditions occupied 45,429 bytes -($54,536$ for ARMv4). GMP which has a rather large collection of functions with the default configuration on an -x86 Athlon is 2,950,688 bytes. Note that while LibTomMath has fewer functions than GMP it has been used as the sole basis -for several public key cryptosystems without having to seek additional outside functions to supplement the library. +When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $66$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} + which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about +$50$KiB) but LibTomMath is also much faster and more complete than MPI. \subsection{API Simplicity} LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build @@ -519,6 +528,11 @@ which is an extremely valuable benefit for the student and developer alike. The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to illegible short hand. LibTomMath does not share this characteristic. +The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors +are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In +effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely +undersireable in many situations. + \subsection{Optimizations} While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP @@ -2320,6 +2334,7 @@ This technique led to the discovery of polynomial basis algorithms (\textit{good \section{Multiplication} \subsection{The Baseline Multiplication} +\label{sec:basemult} \index{baseline multiplication} Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision @@ -2985,6 +3000,7 @@ The implementation is rather simplistic and is not particularly noteworthy. Lin operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. \section{Squaring} +\label{sec:basesquare} Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications @@ -6261,6 +6277,12 @@ OpenSSL Cryptographic Toolkit, \url{http://openssl.org} \bibitem[17]{LIP} Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} +\bibitem[18]{ISOC} +JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' + +\bibitem[19]{JAVA} +The Sun Java Website, \url{http://java.sun.com/} + \end{thebibliography} \input{tommath.ind} diff --git a/tommath.tex b/tommath.tex index ef2e648..488c871 100644 --- a/tommath.tex +++ b/tommath.tex @@ -201,12 +201,27 @@ Greg Rose, Sydney, Australia, June 2003. \pagestyle{headings} \chapter{Introduction} \section{Multiple Precision Arithmetic} + +\subsection{What is Multiple Precision Arithmetic?} +When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively +raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can +reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. +Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple +precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} + of algorithms can be designed to accomodate them. + +By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in +the decimal system with fixed precision $6 \cdot 7 = 2$. + +Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in +schools to manually add, subtract, multiply and divide. + \subsection{The Need for Multiple Precision Arithmetic} The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at greater than $10^{309}$. However, modern programming languages such as ISO C \ref{ISOC} and -Java \ref{JAVA} only provide instrinsic support for integers which are relatively small and are single precision. +typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and +Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. \begin{figure}[!here] \begin{center} @@ -227,9 +242,9 @@ The largest data type guaranteed to be provided by the ISO C programming language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored on the average desktop computer, rendering any protocol based on the algorithm insecure. Multiple -precision algorithms solve this very problem by extending the range of representable integers while using single precision -data types. +trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, +rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by +extending the range of representable integers while using single precision data types. Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in @@ -246,27 +261,16 @@ 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. -\subsection{What is Multiple Precision Arithmetic?} -At the heart of all multiple precision integer operations are the ``long-hand'' algorithms taught to children in grade -school. For example, to multiply $1,234$ by $981$ the student is not taught to memorize the times table for -$1,234$. Instead, they are taught how to long-multiply one digit at a time. That is to multiply each column using -simple single digit multiplications, line up the partial results, and add the resulting products by column. The -representation that most are familiar with is known as decimal or more formally as radix-10. A radix-$n$ representation -simply means there are $n$ possible values per digit. For example, binary would be a radix-2 representation. - -In essence computer based multiple precision arithmetic is very much the same. In most cases the same algorithms -which seem instinctive are the basis of computer based algorithms. The most notable difference is the usage -of a binary friendly radix, that is, to use a radix of the form $2^k$ where $k$ is typically the size of a computer -machine register\footnote{For example, with an x86 based processor $k$ could be $32$ while on an Alpha it would likely -be $64$.}. +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} \index{precision} The benefit of multiple precision representations over single or fixed precision representations is that -often no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of resolution to be precisely represented. -A multiple precision algorithm would augment the precision of the destination to accomodate the result while a single -precision system would truncate excess bits to maintain a fixed level of precision. +no precision is lost while representing the result of an operation which requires excess precision. For example, +the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple +precision algorithm would augment the precision of the destination to accomodate the result while a single precision system +would truncate excess bits to maintain a fixed level of precision. It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum @@ -279,11 +283,12 @@ Multiple precision algorithms have the most overhead of any style of arithmetic. overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. +without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to +be written and tested once. \section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement multiple precision algorithms. That is -to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' +The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. +That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} give considerably detailed explanations of the theoretical aspects of algorithms and often very little information regarding the practical implementation aspects. @@ -309,6 +314,9 @@ The algorithms that are presented will always include at least one ``pseudo-code by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same algorithm in other programming languages as the reader sees fit. +This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing +the reader how the algorithms fit together as well as where to start on various taskings. + \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 @@ -368,41 +376,36 @@ The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an inte itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When a value is presented as a -fraction such as $5 \over 2$ a real value division is implied. +$5/2 = 2$ 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 -of the integer. For example, $\vert \vert 123 \vert \vert = 3$. +of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. \subsection{Work Effort} -\index{big-O} -To measure the efficiency of the specified algorithms, a modified big-O notation is used. In this system all +\index{big-Oh} +To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. That is a single precision addition, multiplication and division are assumed to take the same time to complete. While this is generally not true in practice, it will simplify the discussions considerably. Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal multiplication requires $O(n^2)$ work while a squaring requires -$O({{n^2 + n}\over 2})$ work. In standard big-O notation these would both be said to be equivalent to $O(n^2)$. However, +the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a +baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these +would both be said to be equivalent to $O(n^2)$. However, in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a result small constant factors in the work effort will make an observable difference in algorithm efficiency. -Throughout the discussions various ``work levels'' will be discussed. The term work level shall refer to -the complexity of an algorithm with respect to its time requirements. For example, -$O(1)$, $O(n)$, $O(n^2)$, ..., $O(n^k)$ are various possible work levels that will be of concern in this text. Any -sequence of operations said to be at the $O(n^k)$ work level will often be nested $k-$deep within loops and are performed -$n^k$ times. - -Operations which are deeply nested within algorithms will have a higher big-O rating and be the target of the most -optimizatons. For example, in integer multiplication, by moving the carry propagation from the innermost -$O(n^2)$ nesting to the $O(n)$ nesting level the algorithm becomes vastly more -efficient\footnote{This is known as Comba multiplication.}. +All of the algorithms presented in this text have a polynomial time work level. That is, of the form +$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how +various optimizations will help pay off in the long run. \section{Exercises} -Within the more advanced chapters a section will be set aside to give the reader some challenging exercises. These -exercises are not designed to be prize winning problems, but instead to be thought provoking. Wherever possible the -problems are forward minded, stating problems that will be answered in subsequent chapters. The reader is encouraged to -finish the exercises as they appear to get a better understanding of the subject material. +Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to +the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought +provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent +chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the +subject material. That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular are encouraged to verify they can answer the problems correctly before moving on. @@ -410,52 +413,60 @@ are encouraged to verify they can answer the problems correctly before moving on Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the -scoring. +scoring system used. -\begin{tabular}{cl} -$\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ - & minutes to solve. Usually does not involve much computer time \\ - & to solve. \\ - & \\ -$\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|l|} +\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ + & minutes to solve. Usually does not involve much computer time \\ + & to solve. \\ +\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ & time usage. Usually requires a program to be written to \\ & solve the problem. \\ - & \\ -$\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ +\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ & of work. Usually involves trivial research and development of \\ & new theory from the perspective of a student. \\ - & \\ -$\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ +\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ & of work and research, the solution to which will demonstrate \\ & a higher mastery of the subject matter. \\ - & \\ -$\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ +\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ & novice to solve. Solutions to these problems will demonstrate a \\ & complete mastery of the given subject. \\ - & \\ +\hline \end{tabular} +\end{small} +\end{center} +\caption{Exercise Scoring System} +\end{figure} -Essentially problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also -designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. +Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or +devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level +are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These +two levels are essentially entry level questions. -Problems at the third level are meant to be a bit more difficult. Often the answer is fairly obvious but arriving at an exacting solution -requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of -another algorithm. +Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often +fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always +involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can +answer these questions will feel comfortable with the concepts behind the topic at hand. -Problems at the fourth level are meant to be even more difficult as well as involve some research. The reader will most -likely not know the answer right away, nor will the text provide the exact details of the answer until a subsequent -chapter. Problems at the fifth level are meant to be the hardest problems relative to all the other problems in the -chapter. People who can correctly answer fifth level problems have a mastery of the subject matter at hand. +Problems at the fourth level are meant to be similar to those of the level three questions except they will require +additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide +the exact details of the answer until a subsequent chapter. + +Problems at the fifth level are meant to be the hardest +problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a +mastery of the subject matter at hand. Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader is encouraged to answer the follow-up problems and try to draw the relevance of problems. -\chapter{Introduction to LibTomMath} +\section{Introduction to LibTomMath} -\section{What is LibTomMath?} -LibTomMath is a free and open source multiple precision library written in portable ISO C. By portable it is -meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on +\subsection{What is LibTomMath?} +LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it +is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on any given platform. The library has been successfully tested under numerous operating systems including Unix\footnote{All of these @@ -463,7 +474,7 @@ trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such as public key cryptosystems and still maintain a relatively small footprint. -\section{Goals of LibTomMath} +\subsection{Goals of LibTomMath} Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the @@ -479,13 +490,13 @@ based on the magnitude of the inputs and the configuration of the library. Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen as the template because it is -another library that fits in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the -function names and argument passing conventions, LibTomMath has been written from scratch by Tom St Denis. +MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits +in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument +passing conventions, it has been written from scratch by Tom St Denis. The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. +integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. \section{Choice of LibTomMath} LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but @@ -500,15 +511,13 @@ developer can more readily discern the true intent of a given section of source what conditional code will be used. The code base of LibTomMath is also well organized. Each function is in its own separate source code file -which allows the reader to find a given function very quickly. When compiled with GCC for the x86 processor the entire -library is a mere 87,760 bytes ($116,182$ bytes for ARMv4 processors). This includes every single function -LibTomMath provides from basic arithmetic to various number theoretic functions such as modular exponentiation, various -reduction algorithms and Jacobi symbol computation. +which allows the reader to find a given function very quickly. On average there are about $76$ lines of code per source +file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing +very hard. GMP has many conditional code segments which also hinder tracing. -By comparison MPI, which has fewer functions than LibTomMath, compiled with the same conditions occupied 45,429 bytes -($54,536$ for ARMv4). GMP which has a rather large collection of functions with the default configuration on an -x86 Athlon is 2,950,688 bytes. Note that while LibTomMath has fewer functions than GMP it has been used as the sole basis -for several public key cryptosystems without having to seek additional outside functions to supplement the library. +When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $66$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} + which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about +$50$KiB) but LibTomMath is also much faster and more complete than MPI. \subsection{API Simplicity} LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build @@ -519,6 +528,11 @@ which is an extremely valuable benefit for the student and developer alike. The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to illegible short hand. LibTomMath does not share this characteristic. +The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors +are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In +effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely +undersireable in many situations. + \subsection{Optimizations} While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP @@ -2613,7 +2627,7 @@ is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{u Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus -the previous carry. Recall from section 5.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with +the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with forwarding the carry to the next iteration. Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. @@ -2787,7 +2801,7 @@ least significant bit not the most significant bit. \end{small} \section{Polynomial Basis Operations} -Recall from section 5.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as +Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer division and Karatsuba multiplication. @@ -3414,6 +3428,7 @@ This technique led to the discovery of polynomial basis algorithms (\textit{good \section{Multiplication} \subsection{The Baseline Multiplication} +\label{sec:basemult} \index{baseline multiplication} Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision @@ -3426,9 +3441,9 @@ facet of this algorithm, is that it has been modified to only produce a certain modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. -Recall from sub-section 5.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to +Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The -constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 6.2.2 for more information}). +constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}). \newpage\begin{figure}[!here] \begin{small} @@ -4718,11 +4733,10 @@ s\_mp\_mul\_digs will clear it. 042 \} else \{ 043 res = s_mp_mul (a, b, c); 044 \} -045 -046 \} -047 c->sign = neg; -048 return res; -049 \} +045 \} +046 c->sign = neg; +047 return res; +048 \} \end{alltt} \end{small} @@ -4730,6 +4744,7 @@ The implementation is rather simplistic and is not particularly noteworthy. Lin operator from the C programming language. Line 40 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. \section{Squaring} +\label{sec:basesquare} Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications @@ -5374,25 +5389,26 @@ neither of the polynomial basis algorithms should be used then either the Comba 019 mp_sqr (mp_int * a, mp_int * b) 020 \{ 021 int res; -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 \} +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 \} \end{alltt} \end{small} @@ -5401,7 +5417,7 @@ neither of the polynomial basis algorithms should be used then either the Comba $\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ & that have different number of digits in Karatsuba multiplication. \\ & \\ -$\left [ 3 \right ] $ & In section 6.3 the fact that every column of a squaring is made up \\ +$\left [ 3 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ & of double products and at most one square is stated. Prove this statement. \\ & \\ $\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\ @@ -6005,7 +6021,7 @@ for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will ex advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on -the size of the input. This algorithm is discussed in sub-section 7.3.3. +the size of the input. This algorithm is discussed in sub-section 6.3.3. Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and @@ -7402,7 +7418,7 @@ $d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of f modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using -one of the algorithms presented in chapter seven. +one of the algorithms presented in chapter six. Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The @@ -7692,230 +7708,229 @@ a Left-to-Right algorithm is used to process the remaining few bits. 020 #define TAB_SIZE 256 021 #endif 022 -023 int -024 s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) -025 \{ -026 mp_int M[TAB_SIZE], res, mu; -027 mp_digit buf; -028 int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; -029 -030 /* find window size */ -031 x = mp_count_bits (X); -032 if (x <= 7) \{ -033 winsize = 2; -034 \} else if (x <= 36) \{ -035 winsize = 3; -036 \} else if (x <= 140) \{ -037 winsize = 4; -038 \} else if (x <= 450) \{ -039 winsize = 5; -040 \} else if (x <= 1303) \{ -041 winsize = 6; -042 \} else if (x <= 3529) \{ -043 winsize = 7; -044 \} else \{ -045 winsize = 8; -046 \} -047 -048 #ifdef MP_LOW_MEM -049 if (winsize > 5) \{ -050 winsize = 5; -051 \} -052 #endif -053 -054 /* init M array */ -055 /* init first cell */ -056 if ((err = mp_init(&M[1])) != MP_OKAY) \{ -057 return err; -058 \} -059 -060 /* now init the second half of the array */ -061 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ -062 if ((err = mp_init(&M[x])) != MP_OKAY) \{ -063 for (y = 1<<(winsize-1); y < x; y++) \{ -064 mp_clear (&M[y]); -065 \} -066 mp_clear(&M[1]); -067 return err; -068 \} -069 \} -070 -071 /* create mu, used for Barrett reduction */ -072 if ((err = mp_init (&mu)) != MP_OKAY) \{ -073 goto __M; -074 \} -075 if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{ -076 goto __MU; -077 \} -078 -079 /* create M table -080 * -081 * The M table contains powers of the base, -082 * e.g. M[x] = G**x mod P -083 * -084 * The first half of the table is not -085 * computed though accept for M[0] and M[1] -086 */ -087 if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{ -088 goto __MU; -089 \} -090 -091 /* compute the value at M[1<<(winsize-1)] by squaring -092 * M[1] (winsize-1) times -093 */ -094 if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{ -095 goto __MU; -096 \} -097 -098 for (x = 0; x < (winsize - 1); x++) \{ -099 if ((err = mp_sqr (&M[1 << (winsize - 1)], -100 &M[1 << (winsize - 1)])) != MP_OKAY) \{ -101 goto __MU; -102 \} -103 if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{ -104 goto __MU; -105 \} -106 \} -107 -108 /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) -109 * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) -110 */ -111 for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{ -112 if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{ -113 goto __MU; -114 \} -115 if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) \{ -116 goto __MU; -117 \} -118 \} -119 -120 /* setup result */ -121 if ((err = mp_init (&res)) != MP_OKAY) \{ -122 goto __MU; -123 \} -124 mp_set (&res, 1); -125 -126 /* set initial mode and bit cnt */ -127 mode = 0; -128 bitcnt = 1; -129 buf = 0; -130 digidx = X->used - 1; -131 bitcpy = 0; -132 bitbuf = 0; -133 -134 for (;;) \{ -135 /* grab next digit as required */ -136 if (--bitcnt == 0) \{ -137 /* if digidx == -1 we are out of digits */ -138 if (digidx == -1) \{ -139 break; -140 \} -141 /* read next digit and reset the bitcnt */ -142 buf = X->dp[digidx--]; -143 bitcnt = (int) DIGIT_BIT; -144 \} -145 -146 /* grab the next msb from the exponent */ -147 y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; -148 buf <<= (mp_digit)1; -149 -150 /* if the bit is zero and mode == 0 then we ignore it -151 * These represent the leading zero bits before the first 1 bit -152 * in the exponent. Technically this opt is not required but it -153 * does lower the # of trivial squaring/reductions used -154 */ -155 if (mode == 0 && y == 0) \{ -156 continue; -157 \} -158 -159 /* if the bit is zero and mode == 1 then we square */ -160 if (mode == 1 && y == 0) \{ -161 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -162 goto __RES; -163 \} -164 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -165 goto __RES; -166 \} -167 continue; -168 \} -169 -170 /* else we add it to the window */ -171 bitbuf |= (y << (winsize - ++bitcpy)); -172 mode = 2; -173 -174 if (bitcpy == winsize) \{ -175 /* ok window is filled so square as required and multiply */ -176 /* square first */ -177 for (x = 0; x < winsize; x++) \{ -178 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -179 goto __RES; -180 \} -181 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -182 goto __RES; -183 \} -184 \} -185 -186 /* then multiply */ -187 if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{ -188 goto __MU; -189 \} -190 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -191 goto __MU; -192 \} -193 -194 /* empty window and reset */ -195 bitcpy = 0; -196 bitbuf = 0; -197 mode = 1; -198 \} -199 \} -200 -201 /* if bits remain then square/multiply */ -202 if (mode == 2 && bitcpy > 0) \{ -203 /* square then multiply if the bit is set */ -204 for (x = 0; x < bitcpy; x++) \{ -205 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -206 goto __RES; -207 \} -208 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -209 goto __RES; -210 \} -211 -212 bitbuf <<= 1; -213 if ((bitbuf & (1 << winsize)) != 0) \{ -214 /* then multiply */ -215 if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{ -216 goto __RES; -217 \} -218 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -219 goto __RES; -220 \} -221 \} -222 \} -223 \} -224 -225 mp_exch (&res, Y); -226 err = MP_OKAY; -227 __RES:mp_clear (&res); -228 __MU:mp_clear (&mu); -229 __M: -230 mp_clear(&M[1]); -231 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ -232 mp_clear (&M[x]); -233 \} -234 return err; -235 \} +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 \} \end{alltt} \end{small} -Lines 21 through 40 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted +Lines 31 through 41 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 32 the value of $x$ is already known to be greater than $140$. +on line 33 the value of $x$ is already known to be greater than $140$. -The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits. This logic is used to ensure +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 table of precomputed powers of $G$ remains relatively small. -The for loop on line 61 initializes the $M$ array while lines 62 and 75 compute the value of $\mu$ required for +The for loop on line 60 initializes the $M$ array while lines 61 and 74 compute the value of $\mu$ required for Barrett reduction. -- More later. @@ -8751,86 +8766,85 @@ from chapter seven. 028 \} 029 030 /* single digit division (based on routine from MPI) */ -031 int -032 mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) -033 \{ -034 mp_int q; -035 mp_word w; -036 mp_digit t; -037 int res, ix; -038 -039 /* cannot divide by zero */ -040 if (b == 0) \{ -041 return MP_VAL; -042 \} -043 -044 /* quick outs */ -045 if (b == 1 || mp_iszero(a) == 1) \{ -046 if (d != NULL) \{ -047 *d = 0; -048 \} -049 if (c != NULL) \{ -050 return mp_copy(a, c); -051 \} -052 return MP_OKAY; -053 \} -054 -055 /* power of two ? */ -056 if (s_is_power_of_two(b, &ix) == 1) \{ -057 if (d != NULL) \{ -058 *d = a->dp[0] & ((1<used)) != MP_OKAY) \{ -073 return res; -074 \} -075 -076 q.used = a->used; -077 q.sign = a->sign; -078 w = 0; -079 for (ix = a->used - 1; ix >= 0; ix--) \{ -080 w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); -081 -082 if (w >= b) \{ -083 t = (mp_digit)(w / b); -084 w -= ((mp_word)t) * ((mp_word)b); -085 \} else \{ -086 t = 0; -087 \} -088 q.dp[ix] = (mp_digit)t; -089 \} -090 -091 if (d != NULL) \{ -092 *d = (mp_digit)w; -093 \} -094 -095 if (c != NULL) \{ -096 mp_clamp(&q); -097 mp_exch(&q, c); -098 \} -099 mp_clear(&q); -100 -101 return res; -102 \} -103 +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<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 \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 44 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 43 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. @@ -9168,66 +9182,65 @@ as part of larger input without any significant problem. \begin{alltt} 016 017 /* read a string [ASCII] in a given radix */ -018 int -019 mp_read_radix (mp_int * a, char *str, int radix) -020 \{ -021 int y, res, neg; -022 char ch; -023 -024 /* make sure the radix is ok */ -025 if (radix < 2 || radix > 64) \{ -026 return MP_VAL; -027 \} -028 -029 /* if the leading digit is a -030 * minus set the sign to negative. -031 */ -032 if (*str == '-') \{ -033 ++str; -034 neg = MP_NEG; -035 \} else \{ -036 neg = MP_ZPOS; -037 \} -038 -039 /* set the integer to the default of zero */ -040 mp_zero (a); -041 -042 /* process each digit of the string */ -043 while (*str) \{ -044 /* if the radix < 36 the conversion is case insensitive -045 * this allows numbers like 1AB and 1ab to represent the same value -046 * [e.g. in hex] -047 */ -048 ch = (char) ((radix < 36) ? toupper (*str) : *str); -049 for (y = 0; y < 64; y++) \{ -050 if (ch == mp_s_rmap[y]) \{ -051 break; -052 \} -053 \} -054 -055 /* if the char was found in the map -056 * and is less than the given radix add it -057 * to the number, otherwise exit the loop. -058 */ -059 if (y < radix) \{ -060 if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) \{ -061 return res; -062 \} -063 if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) \{ -064 return res; -065 \} -066 \} else \{ -067 break; -068 \} -069 ++str; -070 \} -071 -072 /* set the sign only if a != 0 */ -073 if (mp_iszero(a) != 1) \{ -074 a->sign = neg; -075 \} -076 return MP_OKAY; -077 \} +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 \} \end{alltt} \end{small} @@ -9294,59 +9307,58 @@ are required instead of a series of $n \times k$ divisions. One design flaw of \begin{alltt} 016 017 /* stores a bignum as a ASCII string in a given radix (2..64) */ -018 int -019 mp_toradix (mp_int * a, char *str, int radix) -020 \{ -021 int res, digs; -022 mp_int t; -023 mp_digit d; -024 char *_s = str; -025 -026 /* check range of the radix */ -027 if (radix < 2 || radix > 64) \{ -028 return MP_VAL; -029 \} -030 -031 /* quick out if its zero */ -032 if (mp_iszero(a) == 1) \{ -033 *str++ = '0'; -034 *str = '\symbol{92}0'; -035 return MP_OKAY; -036 \} -037 -038 if ((res = mp_init_copy (&t, a)) != MP_OKAY) \{ -039 return res; -040 \} -041 -042 /* if it is negative output a - */ -043 if (t.sign == MP_NEG) \{ -044 ++_s; -045 *str++ = '-'; -046 t.sign = MP_ZPOS; -047 \} -048 -049 digs = 0; -050 while (mp_iszero (&t) == 0) \{ -051 if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) \{ -052 mp_clear (&t); -053 return res; -054 \} -055 *str++ = mp_s_rmap[d]; -056 ++digs; -057 \} -058 -059 /* reverse the digits of the string. In this case _s points -060 * to the first digit [exluding the sign] of the number] -061 */ -062 bn_reverse ((unsigned char *)_s, digs); -063 -064 /* append a NULL so the string is properly terminated */ -065 *str = '\symbol{92}0'; -066 -067 mp_clear (&t); -068 return MP_OKAY; -069 \} -070 +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 \end{alltt} \end{small} @@ -10676,6 +10688,12 @@ OpenSSL Cryptographic Toolkit, \url{http://openssl.org} \bibitem[17]{LIP} Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} +\bibitem[18]{ISOC} +JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' + +\bibitem[19]{JAVA} +The Sun Java Website, \url{http://java.sun.com/} + \end{thebibliography} \input{tommath.ind}

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