AuroraMiddleware/libtommath
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

added libtommath-0.10

Browse Source
This commit is contained in:
Tom St Denis 2003-02-28 16:07:32 +00:00 committed by Steffen Jaeckel
parent 40c00add00
commit fb93a30a25
10 changed files with 954 additions and 98 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

449
bn.c
View File

@ -796,7 +796,7 @@ int mp_mul_2(mp_int *a, mp_int *b)
DECFUNC();
return res;
}
b->used = b->alloc;
++b->used;
/* shift any bit count < DIGIT_BIT */
r = 0;
@ -1017,7 +1017,6 @@ static int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs)
*/
_W = W + ix;
/* the number of digits is limited by their placement. E.g.
we avoid multiplying digits that will end up above the # of
digits of precision requested
@ -2840,11 +2839,404 @@ int mp_reduce(mp_int *x, mp_int *m, mp_int *mu)
return res;
}
/* setups the montgomery reduction stuff */
int mp_montgomery_setup(mp_int *a, mp_digit *mp)
{
mp_int t, tt;
int res;
if ((res = mp_init(&t)) != MP_OKAY) {
return res;
}
if ((res = mp_init(&tt)) != MP_OKAY) {
goto __T;
}
/* tt = b */
tt.dp[0] = 0;
tt.dp[1] = 1;
tt.used = 2;
/* t = m mod b */
t.dp[0] = a->dp[0];
t.used = 1;
/* t = 1/m mod b */
if ((res = mp_invmod(&t, &tt, &t)) != MP_OKAY) {
goto __TT;
}
/* t = -1/m mod b */
*mp = ((mp_digit)1 << ((mp_digit)DIGIT_BIT)) - t.dp[0];
res = MP_OKAY;
__TT: mp_clear(&tt);
__T: mp_clear(&t);
return res;
}
/* computes xR^-1 == x (mod N) via Montgomery Reduction (comba) */
static int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp)
{
int ix, res, olduse;
mp_digit ui;
mp_word W[512];
/* get old used count */
olduse = a->used;
/* grow a as required */
if (a->alloc < m->used*2+1) {
if ((res = mp_grow(a, m->used*2+1)) != MP_OKAY) {
return res;
}
}
/* copy and clear */
for (ix = 0; ix < a->used; ix++) {
W[ix] = a->dp[ix];
}
for (; ix < m->used * 2 + 1; ix++) {
W[ix] = 0;
}
for (ix = 0; ix < m->used; ix++) {
/* ui = ai * m' mod b */
ui = (((mp_digit)(W[ix] & MP_MASK)) * mp) & MP_MASK;
/* a = a + ui * m * b^i */
{
register int iy;
register mp_digit *tmpx;
register mp_word *_W;
/* aliases */
tmpx = m->dp;
_W = W + ix;
for (iy = 0; iy < m->used; iy++) {
*_W++ += ((mp_word)ui) * ((mp_word)*tmpx++);
}
/* now fix carry for W[ix+1] */
W[ix+1] += W[ix] >> ((mp_word)DIGIT_BIT);
W[ix] &= ((mp_word)MP_MASK);
}
}
/* nox fix rest of carries */
for (; ix <= m->used * 2 + 1; ix++) {
W[ix] += (W[ix-1] >> ((mp_word)DIGIT_BIT));
W[ix-1] &= ((mp_word)MP_MASK);
}
/* copy out */
/* A = A/b^n */
for (ix = 0; ix < m->used + 1; ix++) {
a->dp[ix] = W[ix+m->used];
}
a->used = m->used + 1;
/* zero oldused digits */
for (; ix < olduse; ix++) {
a->dp[ix] = 0;
}
mp_clamp(a);
/* if A >= m then A = A - m */
if (mp_cmp_mag(a, m) != MP_LT) {
if ((res = s_mp_sub(a, m, a)) != MP_OKAY) {
return res;
}
}
return MP_OKAY;
}
/* computes xR^-1 == x (mod N) via Montgomery Reduction */
int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp)
{
int ix, res, digs;
mp_digit ui;
REGFUNC("mp_montgomery_reduce");
VERIFY(a);
VERIFY(m);
digs = m->used * 2 + 1;
if ((digs < 512) && digs < (1<<( (CHAR_BIT*sizeof(mp_word)) - (2*DIGIT_BIT)))) {
res = fast_mp_montgomery_reduce(a, m, mp);
DECFUNC();
return res;
}
if (a->alloc < m->used*2+1) {
if ((res = mp_grow(a, m->used*2+1)) != MP_OKAY) {
DECFUNC();
return res;
}
}
a->used = m->used * 2 + 1;
for (ix = 0; ix < m->used; ix++) {
/* ui = ai * m' mod b */
ui = (a->dp[ix] * mp) & MP_MASK;
/* a = a + ui * m * b^i */
{
register int iy;
register mp_digit *tmpx, *tmpy, mu;
register mp_word r;
/* aliases */
tmpx = m->dp;
tmpy = a->dp + ix;
mu = 0;
for (iy = 0; iy < m->used; iy++) {
r = ((mp_word)ui) * ((mp_word)*tmpx++) + ((mp_word)mu) + ((mp_word)*tmpy);
mu = (r >> ((mp_word)DIGIT_BIT));
*tmpy++ = (r & ((mp_word)MP_MASK));
}
/* propagate carries */
while (mu) {
*tmpy += mu;
mu = (*tmpy>>DIGIT_BIT)&1;
*tmpy++ &= MP_MASK;
}
}
}
/* A = A/b^n */
mp_rshd(a, m->used);
/* if A >= m then A = A - m */
if (mp_cmp_mag(a, m) != MP_LT) {
if ((res = s_mp_sub(a, m, a)) != MP_OKAY) {
DECFUNC();
return res;
}
}
DECFUNC();
return MP_OKAY;
}
/* computes Y == G^X mod P, HAC pp.616, Algorithm 14.85
*
* Uses a left-to-right k-ary sliding window to compute the modular exponentiation.
* The value of k changes based on the size of the exponent.
*
* Uses Montgomery reduction
*/
static int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y)
{
mp_int M[64], res;
mp_digit buf, mp;
int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;
REGFUNC("mp_exptmod_fast");
VERIFY(G);
VERIFY(X);
VERIFY(P);
VERIFY(Y);
/* find window size */
x = mp_count_bits(X);
if (x <= 18) { winsize = 2; }
else if (x <= 84) { winsize = 3; }
else if (x <= 300) { winsize = 4; }
else if (x <= 930) { winsize = 5; }
else { winsize = 6; }
/* init G array */
for (x = 0; x < (1<<winsize); x++) {
if ((err = mp_init_size(&M[x], 1)) != MP_OKAY) {
for (y = 0; y < x; y++) {
mp_clear(&M[y]);
}
DECFUNC();
return err;
}
}
/* now setup montgomery */
if ((err = mp_montgomery_setup(P, &mp)) != MP_OKAY) {
goto __M;
}
/* setup result */
if ((err = mp_init(&res)) != MP_OKAY) {
goto __M;
}
/* now we need R mod m */
mp_set(&res, 1);
if ((err = mp_lshd(&res, P->used)) != MP_OKAY) {
goto __RES;
}
/* res = R mod m */
if ((err = mp_mod(&res, P, &res)) != MP_OKAY) {
goto __RES;
}
/* create M table
*
* The M table contains powers of the input base, e.g. M[x] = G^x mod P
*
* The first half of the table is not computed though accept for M[0] and M[1]
*/
mp_set(&M[0], 1);
if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) {
goto __RES;
}
/* now set M[1] to G * R mod m */
if ((err = mp_mulmod(&M[1], &res, P, &M[1])) != MP_OKAY) {
goto __RES;
}
/* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */
if ((err = mp_copy(&M[1], &M[1<<(winsize-1)])) != MP_OKAY) {
goto __RES;
}
for (x = 0; x < (winsize-1); x++) {
if ((err = mp_sqr(&M[1<<(winsize-1)], &M[1<<(winsize-1)])) != MP_OKAY) {
goto __RES;
}
if ((err = mp_montgomery_reduce(&M[1<<(winsize-1)], P, mp)) != MP_OKAY) {
goto __RES;
}
}
/* create upper table */
for (x = (1<<(winsize-1))+1; x < (1 << winsize); x++) {
if ((err = mp_mul(&M[x-1], &M[1], &M[x])) != MP_OKAY) {
goto __RES;
}
if ((err = mp_montgomery_reduce(&M[x], P, mp)) != MP_OKAY) {
goto __RES;
}
}
/* set initial mode and bit cnt */
mode = 0;
bitcnt = 0;
buf = 0;
digidx = X->used - 1;
bitcpy = bitbuf = 0;
bitcnt = 1;
for (;;) {
/* grab next digit as required */
if (--bitcnt == 0) {
if (digidx == -1) {
break;
}
buf = X->dp[digidx--];
bitcnt = (int)DIGIT_BIT;
}
/* grab the next msb from the exponent */
y = (buf >> (DIGIT_BIT - 1)) & 1;
buf <<= 1;
/* if the bit is zero and mode == 0 then we ignore it
* These represent the leading zero bits before the first 1 bit
* in the exponent. Technically this opt is not required but it
* does lower the # of trivial squaring/reductions used
*/
if (mode == 0 && y == 0) continue;
/* if the bit is zero and mode == 1 then we square */
if (y == 0 && mode == 1) {
if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
goto __RES;
}
if ((err = mp_montgomery_reduce(&res, P, mp)) != MP_OKAY) {
goto __RES;
}
continue;
}
/* else we add it to the window */
bitbuf |= (y<<(winsize-++bitcpy));
mode = 2;
if (bitcpy == winsize) {
/* ok window is filled so square as required and multiply multiply */
/* square first */
for (x = 0; x < winsize; x++) {
if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
goto __RES;
}
if ((err = mp_montgomery_reduce(&res, P, mp)) != MP_OKAY) {
goto __RES;
}
}
/* then multiply */
if ((err = mp_mul(&res, &M[bitbuf], &res)) != MP_OKAY) {
goto __RES;
}
if ((err = mp_montgomery_reduce(&res, P, mp)) != MP_OKAY) {
goto __RES;
}
/* empty window and reset */
bitcpy = bitbuf = 0;
mode = 1;
}
}
/* if bits remain then square/multiply */
if (mode == 2 && bitcpy > 0) {
/* square then multiply if the bit is set */
for (x = 0; x < bitcpy; x++) {
if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
goto __RES;
}
if ((err = mp_montgomery_reduce(&res, P, mp)) != MP_OKAY) {
goto __RES;
}
bitbuf <<= 1;
if ((bitbuf & (1<<winsize)) != 0) {
/* then multiply */
if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) {
goto __RES;
}
if ((err = mp_montgomery_reduce(&res, P, mp)) != MP_OKAY) {
goto __RES;
}
}
}
}
/* fixup result */
if ((err = mp_montgomery_reduce(&res, P, mp)) != MP_OKAY) {
goto __RES;
}
mp_exch(&res, Y);
err = MP_OKAY;
__RES: mp_clear(&res);
__M :
for (x = 0; x < (1<<winsize); x++) {
mp_clear(&M[x]);
}
DECFUNC();
return err;
}
int mp_exptmod(mp_int *G, mp_int *X, mp_int *P, mp_int *Y)
{
mp_int M[64], res, mu;
@ -2857,6 +3249,13 @@ int mp_exptmod(mp_int *G, mp_int *X, mp_int *P, mp_int *Y)
VERIFY(P);
VERIFY(Y);
/* if the modulus is odd use the fast method */
if (mp_isodd(P) == 1 && P->used > 4 && P->used < MONTGOMERY_EXPT_CUTOFF) {
err = mp_exptmod_fast(G, X, P, Y);
DECFUNC();
return err;
}
/* find window size */
x = mp_count_bits(X);
if (x <= 18) { winsize = 2; }
@ -2918,7 +3317,8 @@ int mp_exptmod(mp_int *G, mp_int *X, mp_int *P, mp_int *Y)
goto __MU;
}
}
/* init result */
/* setup result */
if ((err = mp_init(&res)) != MP_OKAY) {
goto __MU;
}
@ -3009,10 +3409,10 @@ int mp_exptmod(mp_int *G, mp_int *X, mp_int *P, mp_int *Y)
if ((bitbuf & (1<<winsize)) != 0) {
/* then multiply */
if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) {
goto __MU;
goto __RES;
}
if ((err = mp_reduce(&res, P, &mu)) != MP_OKAY) {
goto __MU;
goto __RES;
}
}
}
@ -3060,10 +3460,8 @@ int mp_n_root(mp_int *a, mp_digit b, mp_int *c)
neg = a->sign;
a->sign = MP_ZPOS;
/* t2 = a */
if ((res = mp_copy(a, &t2)) != MP_OKAY) {
goto __T3;
}
/* t2 = 2 */
mp_set(&t2, 2);
do {
/* t1 = t2 */
@ -3098,16 +3496,20 @@ int mp_n_root(mp_int *a, mp_digit b, mp_int *c)
}
} while (mp_cmp(&t1, &t2) != MP_EQ);
/* result can be at most off by one so check */
if ((res = mp_expt_d(&t1, b, &t2)) != MP_OKAY) {
goto __T3;
}
if (mp_cmp(&t2, a) == MP_GT) {
if ((res = mp_sub_d(&t1, 1, &t1)) != MP_OKAY) {
/* result can be off by a few so check */
for (;;) {
if ((res = mp_expt_d(&t1, b, &t2)) != MP_OKAY) {
goto __T3;
}
}
if (mp_cmp(&t2, a) == MP_GT) {
if ((res = mp_sub_d(&t1, 1, &t1)) != MP_OKAY) {
goto __T3;
}
} else {
break;
}
}
/* reset the sign of a first */
a->sign = neg;
@ -3336,13 +3738,14 @@ int mp_to_signed_bin(mp_int *a, unsigned char *b)
/* get the size for an unsigned equivalent */
int mp_unsigned_bin_size(mp_int *a)
{
return (mp_count_bits(a)/8 + ((mp_count_bits(a)&7) != 0 ? 1 : 0));
int size = mp_count_bits(a);
return (size/8 + ((size&7) != 0 ? 1 : 0));
}
/* get the size for an signed equivalent */
int mp_signed_bin_size(mp_int *a)
{
return 1 + (mp_count_bits(a)/8 + ((mp_count_bits(a)&7) != 0 ? 1 : 0));
return 1 + mp_unsigned_bin_size(a);
}
/* read a string [ASCII] in a given radix */
@ -3431,8 +3834,11 @@ int mp_radix_size(mp_int *a, int radix)
mp_int t;
mp_digit d;
digs = 0;
/* special case for binary */
if (radix == 2) {
return mp_count_bits(a) + (a->sign == MP_NEG ? 1 : 0) + 1;
}
if (radix < 2 || radix > 64) {
return 0;
}
@ -3441,6 +3847,7 @@ int mp_radix_size(mp_int *a, int radix)
return 0;
}
digs = 0;
if (t.sign == MP_NEG) {
++digs;
t.sign = MP_ZPOS;

9
bn.h
View File

@ -84,6 +84,7 @@ typedef int mp_err;
/* you'll have to tune these... */
#define KARATSUBA_MUL_CUTOFF 80 /* Min. number of digits before Karatsuba multiplication is used. */
#define KARATSUBA_SQR_CUTOFF 80 /* Min. number of digits before Karatsuba squaring is used. */
#define MONTGOMERY_EXPT_CUTOFF 40 /* max. number of digits that montgomery reductions will help for */
#define MP_PREC 64 /* default digits of precision */
@ -114,7 +115,7 @@ int mp_shrink(mp_int *a);
#define mp_iszero(a) (((a)->used == 0) ? 1 : 0)
#define mp_iseven(a) (((a)->used == 0 || (((a)->dp[0] & 1) == 0)) ? 1 : 0)
#define mp_isodd(a) (((a)->used > 0 || (((a)->dp[0] & 1) == 1)) ? 1 : 0)
#define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? 1 : 0)
/* set to zero */
void mp_zero(mp_int *a);
@ -256,6 +257,12 @@ int mp_reduce_setup(mp_int *a, mp_int *b);
* compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code].
*/
int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
/* setups the montgomery reduction */
int mp_montgomery_setup(mp_int *a, mp_digit *mp);
/* computes xR^-1 == x (mod N) via Montgomery Reduction */
int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
/* d = a^b (mod c) */
int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);

BIN
bn.pdf
View File

Binary file not shown.

61
bn.tex
View File

@ -1,7 +1,7 @@
\documentclass{article}
\begin{document}
\title{LibTomMath v0.09 \\ A Free Multiple Precision Integer Library}
\title{LibTomMath v0.10 \\ A Free Multiple Precision Integer Library}
\author{Tom St Denis \\ tomstdenis@iahu.ca}
\maketitle
\newpage
@ -279,6 +279,22 @@ int mp_n_root(mp_int *a, mp_digit b, mp_int *c);
/* computes the jacobi c = (a | n) (or Legendre if b is prime) */
int mp_jacobi(mp_int *a, mp_int *n, int *c);
/* used to setup the Barrett reduction for a given modulus b */
int mp_reduce_setup(mp_int *a, mp_int *b);
/* Barrett Reduction, computes a (mod b) with a precomputed value c
*
* Assumes that 0 < a <= b^2, note if 0 > a > -(b^2) then you can merely
* compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code].
*/
int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
/* setups the montgomery reduction */
int mp_montgomery_setup(mp_int *a, mp_digit *mp);
/* computes xR^-1 == x (mod N) via Montgomery Reduction */
int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
/* d = a^b (mod c) */
int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
\end{verbatim}
@ -451,21 +467,38 @@ by $c$. Since the result of the Jacobi function $\left ( {a \over n} \right ) \
natural to store the result in a simple C style \textbf{int}. If $n$ is prime then the Jacobi function produces
the same results as the Legendre function\footnote{Source: Handbook of Applied Cryptography, pp. 73}. This means if
$n$ is prime then $\left ( {a \over n} \right )$ is equal to $1$ if $a$ is a quadratic residue modulo $n$ or $-1$ if
it is not.
it is not.
\subsubsection{mp\_exptmod(mp\_int *a, mp\_int *b, mp\_int *c, mp\_int *d)}
Computes $d = a^b \mbox{ (mod }c\mbox{)}$ using a sliding window $k$-ary exponentiation algorithm. For an $\alpha$-bit
exponent it performs $\alpha$ squarings and at most $\lfloor \alpha/k \rfloor + k$ multiplications. The value of $k$ is
chosen to minimize the number of multiplications required for a given value of $\alpha$. Barrett reductions are used
to reduce the squared or multiplied temporary results modulo $c$. A Barrett reduction requires one division that is
performed only and two half multipliers of $N$ digit numbers resulting in approximation $O((N^2)/2)$ work.
chosen to minimize the number of multiplications required for a given value of $\alpha$. Barrett or Montgomery
reductions are used to reduce the squared or multiplied temporary results modulo $c$.
Let $\gamma = \lfloor \alpha/k \rfloor + k$ represent the total multiplications. The total work of a exponentiation is
therefore, $O(3 \cdot N^2 + (\alpha + \gamma) \cdot ((N^2)/2) + \alpha \cdot ((N^2 + N)/2) + \gamma \cdot N^2)$ which
simplies to $O(3 \cdot N^2 + \gamma N^2 + \alpha (N^2 + (N/2)))$. The bulk of the time is spent in the Barrett
reductions and the squaring algorithms. Since $\gamma < \alpha$ it makes sense to optimize first the Barrett and
squaring routines first. Significant improvements in the future will most likely stem from optimizing these instead
of optimizing the multipliers.
\subsection{Fast Modular Reductions}
\subsubsection{mp\_reduce(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes a Barrett reduction in-place of $a$ modulo $b$ with respect to $c$. In essence it computes
$a \equiv a \mbox{ (mod }b\mbox{)}$ provided $0 \le a \le b^2$. The value of $c$ is precomputed with the
function mp\_reduce\_setup().
The Barrett reduction function has been optimized to use partial multipliers which means compared to MPI it performs
have the number of single precision multipliers (\textit{provided they have the same size digits}). The partial
multipliers (\textit{one of which is shared with mp\_mul}) both have baseline and comba variants. Barrett reduction
can reduce a number modulo a $n-$digit modulus with approximately $2n^2$ single precision multiplications.
\subsubsection{mp\_montgomery\_reduce(mp\_int *a, mp\_int *m, mp\_digit mp)}
Computes a Montgomery reduction in-place of $a$ modulo $b$ with respect to $mp$. If $b$ is some $n-$digit modulus then
$R = \beta^{n+1}$. The result of this function is $aR^{-1} \mbox{ (mod }b\mbox{)}$ provided that $0 \le a \le b^2$.
The value of $mp$ is precomputed with the function mp\_montgomery\_setup().
The Montgomery reduction comes in two variants. A standard baseline and a fast comba method. The baseline routine
is in fact slower than the Barrett reductions, however, the comba routine is much faster. Montomgery reduction can
reduce a number modulo a $n-$digit modulus with approximately $n^2 + n$ single precision multiplications.
Note that the final result of a Montgomery reduction is not just the value reduced modulo $b$. You have to multiply
by $R$ modulo $b$ to get the real result. At first that may not seem like such a worthwhile routine, however, the
exptmod function can be made to take advantage of this such that only one normalization at the end is required.
\section{Timing Analysis}
\subsection{Observed Timings}
@ -503,9 +536,9 @@ Square & 1024 & 18,991 & 5,733 \\
Square & 2048 & 72,126 & 17,621 \\
Square & 4096 & 306,269 & 67,576 \\
\hline
Exptmod & 512 & 32,021,586 & 4,138,354 \\
Exptmod & 768 & 97,595,492 & 9,840,233 \\
Exptmod & 1024 & 223,302,532 & 20,624,553 \\
Exptmod & 512 & 32,021,586 & 3,118,435 \\
Exptmod & 768 & 97,595,492 & 8,493,633 \\
Exptmod & 1024 & 223,302,532 & 17,715,899 \\
Exptmod & 2048 & 1,682,223,369 & 114,936,361 \\
Exptmod & 2560 & 3,268,615,571 & 229,402,426 \\
Exptmod & 3072 & 5,597,240,141 & 367,403,840 \\

5
changes.txt
View File

@ -1,3 +1,8 @@
Jan 9th, 2003
v0.10 -- Pekka Riikonen suggested fixes to the radix conversion code.
-- Added baseline montgomery and comba montgomery reductions, sped up exptmods
[to a point, see bn.h for MONTGOMERY_EXPT_CUTOFF]
Jan 6th, 2003
v0.09 -- Updated the manual to reflect recent changes. :-)
-- Added Jacobi function (mp_jacobi) to supplement the number theory side of the lib

47
demo.c
View File

@ -21,12 +21,15 @@
#define TIMER
extern ulong64 rdtsc(void);
extern void reset(void);
#else
#endif
#ifdef TIMER
#ifndef TIMER_X86
ulong64 _tt;
void reset(void) { _tt = clock(); }
ulong64 rdtsc(void) { return clock() - _tt; }
#endif
#endif
#ifndef DEBUG
int _ifuncs;
@ -82,6 +85,7 @@ int main(void)
mp_int a, b, c, d, e, f;
unsigned long expt_n, add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, gcd_n, lcm_n, inv_n;
int rr;
mp_digit tom;
#ifdef TIMER
int n;
@ -94,29 +98,38 @@ int main(void)
mp_init(&d);
mp_init(&e);
mp_init(&f);
mp_read_radix(&a, "59994534535345535344389423", 10);
mp_read_radix(&b, "49993453555234234565675534", 10);
mp_read_radix(&c, "62398923474472948723847281", 10);
mp_mulmod(&a, &b, &c, &f);
/* setup mont */
mp_montgomery_setup(&c, &tom);
mp_mul(&a, &b, &a);
mp_montgomery_reduce(&a, &c, tom);
mp_montgomery_reduce(&a, &c, tom);
mp_lshd(&a, c.used*2);
mp_mod(&a, &c, &a);
mp_toradix(&a, cmd, 10);
printf("%s\n\n", cmd);
mp_toradix(&f, cmd, 10);
printf("%s\n", cmd);
/* return 0; */
mp_read_radix(&a, "V//////////////////////////////////////////////////////////////////////////////////////", 64);
mp_reduce_setup(&b, &a);
printf("\n\n----\n\n");
mp_toradix(&b, buf, 10);
printf("b == %s\n\n\n", buf);
mp_read_radix(&b, "4982748972349724892742", 10);
mp_sub_d(&a, 1, &c);
#ifdef DEBUG
mp_sqr(&a, &a);mp_sqr(&c, &c);
mp_sqr(&a, &a);mp_sqr(&c, &c);
mp_sqr(&a, &a);mp_sqr(&c, &c);
reset_timings();
#endif
mp_exptmod(&b, &c, &a, &d);
#ifdef DEBUG
dump_timings();
return 0;
#endif
mp_toradix(&d, buf, 10);
printf("b^p-1 == %s\n", buf);
@ -169,7 +182,7 @@ int main(void)
printf("Multiplying %d-bit took %llu cycles\n", mp_count_bits(&a), tt / ((ulong64)100000));
mp_copy(&b, &a);
}
{
char *primes[] = {
"17933601194860113372237070562165128350027320072176844226673287945873370751245439587792371960615073855669274087805055507977323024886880985062002853331424203",

104
etc/pprime.c
View File

@ -85,10 +85,11 @@ static mp_digit prime_digit()
}
/* makes a prime of at least k bits */
int pprime(int k, mp_int *p, mp_int *q)
int pprime(int k, int li, mp_int *p, mp_int *q)
{
mp_int a, b, c, n, x, y, z, v;
int res;
int res, ii;
static const mp_digit bases[] = { 2, 3, 5, 7, 11, 13, 17, 19 };
/* single digit ? */
if (k <= (int)DIGIT_BIT) {
@ -167,55 +168,60 @@ int pprime(int k, mp_int *p, mp_int *q)
if (mp_cmp_d(&y, 1) != MP_EQ) goto top;
/* now try base x=2 */
mp_set(&x, 2);
/* now try base x=bases[ii] */
for (ii = 0; ii < li; ii++) {
mp_set(&x, bases[ii]);
/* compute x^a mod n */
if ((res = mp_exptmod(&x, &a, &n, &y)) != MP_OKAY) { /* y = x^a mod n */
goto __Z;
}
/* compute x^a mod n */
if ((res = mp_exptmod(&x, &a, &n, &y)) != MP_OKAY) { /* y = x^a mod n */
goto __Z;
}
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) continue;
/* now x^2a mod n */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2a mod n */
goto __Z;
}
/* now x^2a mod n */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2a mod n */
goto __Z;
}
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
if (mp_cmp_d(&y, 1) == MP_EQ) continue;
/* compute x^b mod n */
if ((res = mp_exptmod(&x, &b, &n, &y)) != MP_OKAY) { /* y = x^b mod n */
goto __Z;
}
/* compute x^b mod n */
if ((res = mp_exptmod(&x, &b, &n, &y)) != MP_OKAY) { /* y = x^b mod n */
goto __Z;
}
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) continue;
/* now x^2b mod n */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2b mod n */
goto __Z;
}
/* now x^2b mod n */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2b mod n */
goto __Z;
}
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
if (mp_cmp_d(&y, 1) == MP_EQ) continue;
/* compute x^c mod n == x^ab mod n */
if ((res = mp_exptmod(&x, &c, &n, &y)) != MP_OKAY) { /* y = x^ab mod n */
goto __Z;
}
/* compute x^c mod n == x^ab mod n */
if ((res = mp_exptmod(&x, &c, &n, &y)) != MP_OKAY) { /* y = x^ab mod n */
goto __Z;
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) continue;
/* now compute (x^c mod n)^2 */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2ab mod n */
goto __Z;
}
/* y should be 1 */
if (mp_cmp_d(&y, 1) != MP_EQ) continue;
break;
}
/* if y == 1 loop */
if (mp_cmp_d(&y, 1) == MP_EQ) goto top;
/* now compute (x^c mod n)^2 */
if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2ab mod n */
goto __Z;
}
/* y should be 1 */
if (mp_cmp_d(&y, 1) != MP_EQ) goto top;
/* no bases worked? */
if (ii == li) goto top;
/*
{
@ -232,10 +238,14 @@ int pprime(int k, mp_int *p, mp_int *q)
*/
/* a = n */
mp_copy(&n, &a);
}
}
/* get q to be the order of the large prime subgroup */
mp_sub_d(&n, 1, q);
mp_div_2(q, q);
mp_div(q, &b, q, NULL);
mp_exch(&n, p);
mp_exch(&b, q);
res = MP_OKAY;
__Z: mp_clear(&z);
@ -254,19 +264,25 @@ int main(void)
{
mp_int p, q;
char buf[4096];
int k;
int k, li;
clock_t t1;
srand(time(NULL));
printf("Enter # of bits: \n");
scanf("%d", &k);
fgets(buf, sizeof(buf), stdin);
sscanf(buf, "%d", &k);
printf("Enter number of bases to try (1 to 8):\n");
fgets(buf, sizeof(buf), stdin);
sscanf(buf, "%d", &li);
mp_init(&p);
mp_init(&q);
t1 = clock();
pprime(k, &p, &q);
pprime(k, li, &p, &q);
t1 = clock() - t1;
printf("\n\nTook %ld ticks, %d bits\n", t1, mp_count_bits(&p));

2
makefile
View File

@ -1,7 +1,7 @@
CC = gcc
CFLAGS += -Wall -W -Wshadow -ansi -O3 -fomit-frame-pointer -funroll-loops
VERSION=0.09
VERSION=0.10
default: test

302
poly.c Normal file
View File

@ -0,0 +1,302 @@
/* LibTomMath, multiple-precision integer library -- Tom St Denis
*
* LibTomMath is library that provides for multiple-precision
* integer arithmetic as well as number theoretic functionality.
*
* This file "poly.c" provides GF(p^k) functionality on top of the
* libtommath library.
*
* The library is 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://libtommath.iahu.ca
*/
#include "poly.h"
#undef MIN
#define MIN(x,y) ((x)<(y)?(x):(y))
#undef MAX
#define MAX(x,y) ((x)>(y)?(x):(y))
static void s_free(mp_poly *a)
{
int k;
for (k = 0; k < a->alloc; k++) {
mp_clear(&(a->co[k]));
}
}
static int s_setup(mp_poly *a, int low, int high)
{
int res, k, j;
for (k = low; k < high; k++) {
if ((res = mp_init(&(a->co[k]))) != MP_OKAY) {
for (j = low; j < k; j++) {
mp_clear(&(a->co[j]));
}
return MP_MEM;
}
}
return MP_OKAY;
}
int mp_poly_init(mp_poly *a, mp_int *cha)
{
return mp_poly_init_size(a, cha, MP_POLY_PREC);
}
/* init a poly of a given (size) degree */
int mp_poly_init_size(mp_poly *a, mp_int *cha, int size)
{
int res;
/* allocate array of mp_ints for coefficients */
a->co = malloc(size * sizeof(mp_int));
if (a->co == NULL) {
return MP_MEM;
}
a->used = 0;
a->alloc = size;
/* now init the range */
if ((res = s_setup(a, 0, size)) != MP_OKAY) {
free(a->co);
return res;
}
/* copy characteristic */
if ((res = mp_init_copy(&(a->cha), cha)) != MP_OKAY) {
s_free(a);
free(a->co);
return res;
}
/* return ok at this point */
return MP_OKAY;
}
/* grow the size of a poly */
static int mp_poly_grow(mp_poly *a, int size)
{
int res;
if (size > a->alloc) {
/* resize the array of coefficients */
a->co = realloc(a->co, sizeof(mp_int) * size);
if (a->co == NULL) {
return MP_MEM;
}
/* now setup the coefficients */
if ((res = s_setup(a, a->alloc, a->alloc + size)) != MP_OKAY) {
return res;
}
a->alloc += size;
}
return MP_OKAY;
}
/* copy, b = a */
int mp_poly_copy(mp_poly *a, mp_poly *b)
{
int res, k;
/* resize b */
if ((res = mp_poly_grow(b, a->used)) != MP_OKAY) {
return res;
}
/* now copy the used part */
b->used = a->used;
/* now the cha */
if ((res = mp_copy(&(a->cha), &(b->cha))) != MP_OKAY) {
return res;
}
/* now all the coefficients */
for (k = 0; k < b->used; k++) {
if ((res = mp_copy(&(a->co[k]), &(b->co[k]))) != MP_OKAY) {
return res;
}
}
/* now zero the top */
for (k = b->used; k < b->alloc; k++) {
mp_zero(&(b->co[k]));
}
return MP_OKAY;
}
/* init from a copy, a = b */
int mp_poly_init_copy(mp_poly *a, mp_poly *b)
{
int res;
if ((res = mp_poly_init(a, &(b->cha))) != MP_OKAY) {
return res;
}
return mp_poly_copy(b, a);
}
/* free a poly from ram */
void mp_poly_clear(mp_poly *a)
{
s_free(a);
mp_clear(&(a->cha));
free(a->co);
a->co = NULL;
a->used = a->alloc = 0;
}
/* exchange two polys */
void mp_poly_exch(mp_poly *a, mp_poly *b)
{
mp_poly t;
t = *a; *a = *b; *b = t;
}
/* clamp the # of used digits */
static void mp_poly_clamp(mp_poly *a)
{
while (a->used > 0 && mp_cmp_d(&(a->co[a->used]), 0) == MP_EQ) --(a->used);
}
/* add two polynomials, c(x) = a(x) + b(x) */
int mp_poly_add(mp_poly *a, mp_poly *b, mp_poly *c)
{
mp_poly t, *x, *y;
int res, k;
/* ensure char's are the same */
if (mp_cmp(&(a->cha), &(b->cha)) != MP_EQ) {
return MP_VAL;
}
/* now figure out the sizes such that x is the
largest degree poly and y is less or equal in degree
*/
if (a->used > b->used) {
x = a;
y = b;
} else {
x = b;
y = a;
}
/* now init the result to be a copy of the largest */
if ((res = mp_poly_init_copy(&t, x)) != MP_OKAY) {
return res;
}
/* now add the coeffcients of the smaller one */
for (k = 0; k < y->used; k++) {
if ((res = mp_addmod(&(a->co[k]), &(b->co[k]), &(a->cha), &(t.co[k]))) != MP_OKAY) {
goto __T;
}
}
mp_poly_clamp(&t);
mp_poly_exch(&t, c);
res = MP_OKAY;
__T: mp_poly_clear(&t);
return res;
}
/* subtracts two polynomials, c(x) = a(x) - b(x) */
int mp_poly_sub(mp_poly *a, mp_poly *b, mp_poly *c)
{
mp_poly t, *x, *y;
int res, k;
/* ensure char's are the same */
if (mp_cmp(&(a->cha), &(b->cha)) != MP_EQ) {
return MP_VAL;
}
/* now figure out the sizes such that x is the
largest degree poly and y is less or equal in degree
*/
if (a->used > b->used) {
x = a;
y = b;
} else {
x = b;
y = a;
}
/* now init the result to be a copy of the largest */
if ((res = mp_poly_init_copy(&t, x)) != MP_OKAY) {
return res;
}
/* now add the coeffcients of the smaller one */
for (k = 0; k < y->used; k++) {
if ((res = mp_submod(&(a->co[k]), &(b->co[k]), &(a->cha), &(t.co[k]))) != MP_OKAY) {
goto __T;
}
}
mp_poly_clamp(&t);
mp_poly_exch(&t, c);
res = MP_OKAY;
__T: mp_poly_clear(&t);
return res;
}
/* multiplies two polynomials, c(x) = a(x) * b(x) */
int mp_poly_mul(mp_poly *a, mp_poly *b, mp_poly *c)
{
mp_poly t;
mp_int tt;
int res, pa, pb, ix, iy;
/* ensure char's are the same */
if (mp_cmp(&(a->cha), &(b->cha)) != MP_EQ) {
return MP_VAL;
}
/* degrees of a and b */
pa = a->used;
pb = b->used;
/* now init the temp polynomial to be of degree pa+pb */
if ((res = mp_poly_init_size(&t, &(a->cha), pa+pb)) != MP_OKAY) {
return res;
}
/* now init our temp int */
if ((res = mp_init(&tt)) != MP_OKAY) {
goto __T;
}
/* now loop through all the digits */
for (ix = 0; ix < pa; ix++) {
for (iy = 0; iy < pb; iy++) {
if ((res = mp_mul(&(a->co[ix]), &(b->co[iy]), &tt)) != MP_OKAY) {
goto __TT;
}
if ((res = mp_addmod(&tt, &(t.co[ix+iy]), &(a->cha), &(t.co[ix+iy]))) != MP_OKAY) {
goto __TT;
}
}
}
mp_poly_clamp(&t);
mp_poly_exch(&t, c);
res = MP_OKAY;
__TT: mp_clear(&tt);
__T: mp_poly_clear(&t);
return res;
}

73
poly.h Normal file
View File

@ -0,0 +1,73 @@
/* LibTomMath, multiple-precision integer library -- Tom St Denis
*
* LibTomMath is library that provides for multiple-precision
* integer arithmetic as well as number theoretic functionality.
*
* This file "poly.h" provides GF(p^k) functionality on top of the
* libtommath library.
*
* The library is 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://libtommath.iahu.ca
*/
#ifndef POLY_H_
#define POLY_H_
#include "bn.h"
/* a mp_poly is basically a derived "class" of a mp_int
* it uses the same technique of growing arrays via
* used/alloc parameters except the base unit or "digit"
* is in fact a mp_int. These hold the coefficients
* of the polynomial
*/
typedef struct {
int used, /* coefficients used */
alloc; /* coefficients allocated (and initialized) */
mp_int *co, /* coefficients */
cha; /* characteristic */
} mp_poly;
#define MP_POLY_PREC 16 /* default coefficients allocated */
/* init a poly */
int mp_poly_init(mp_poly *a, mp_int *cha);
/* init a poly of a given (size) degree */
int mp_poly_init_size(mp_poly *a, mp_int *cha, int size);
/* copy, b = a */
int mp_poly_copy(mp_poly *a, mp_poly *b);
/* init from a copy, a = b */
int mp_poly_init_copy(mp_poly *a, mp_poly *b);
/* free a poly from ram */
void mp_poly_clear(mp_poly *a);
/* exchange two polys */
void mp_poly_exch(mp_poly *a, mp_poly *b);
/* ---> Basic Arithmetic <--- */
/* add two polynomials, c(x) = a(x) + b(x) */
int mp_poly_add(mp_poly *a, mp_poly *b, mp_poly *c);
/* subtracts two polynomials, c(x) = a(x) - b(x) */
int mp_poly_sub(mp_poly *a, mp_poly *b, mp_poly *c);
/* multiplies two polynomials, c(x) = a(x) * b(x) */
int mp_poly_mul(mp_poly *a, mp_poly *b, mp_poly *c);
#endif