+++ /dev/null
-/* -*-c-*-
- *
- * $Id: rho.c,v 1.5 2004/04/08 01:36:15 mdw Exp $
- *
- * Pollard's rho algorithm for discrete logs
- *
- * (c) 2000 Straylight/Edgeware
- */
-
-/*----- Licensing notice --------------------------------------------------*
- *
- * This file is part of Catacomb.
- *
- * Catacomb is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Library General Public License as
- * published by the Free Software Foundation; either version 2 of the
- * License, or (at your option) any later version.
- *
- * Catacomb is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Library General Public License for more details.
- *
- * You should have received a copy of the GNU Library General Public
- * License along with Catacomb; if not, write to the Free
- * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
- * MA 02111-1307, USA.
- */
-
-/*----- Header files ------------------------------------------------------*/
-
-#include "fibrand.h"
-#include "mp.h"
-#include "mpmont.h"
-#include "mprand.h"
-#include "rho.h"
-
-/*----- Main code ---------------------------------------------------------*/
-
-/* --- @rho@ --- *
- *
- * Arguments: @rho_ctx *cc@ = pointer to the context structure
- * @void *x, *y@ = two (equal) base values (try 1)
- * @mp *a, *b@ = logs of %$x$% (see below)
- *
- * Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm
- * failed. (This is unlikely, though possible.)
- *
- * Use: Uses Pollard's rho algorithm to compute discrete logs in the
- * group %$G$% generated by %$g$%.
- *
- * The algorithm works by finding a cycle in a pseudo-random
- * walk. The function @ops->split@ should return an element
- * from %$\{\,0, 1, 2\,\}$% according to its argument, in order
- * to determine the walk. At each step in the walk, we know a
- * group element %$x \in G$% together with its representation as
- * a product of powers of %$g$% and $%a$% (i.e., we know that
- * %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
- *
- * Locating a cycle gives us a collision
- *
- * %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
- *
- * Taking logs of both sides (to base %$g$%) gives us that
- *
- * %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
- *
- * Good initial values are %$x = y = 1$% (the multiplicative
- * identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
- * If that doesn't work then start choosing more `interesting'
- * values.
- *
- * Note that the algorithm requires minimal space but
- * %$O(\sqrt{n})$% time. Don't do this on large groups,
- * particularly if you can find a decent factor base.
- *
- * Finally, note that this function will free the input values
- * when it's finished with them. This probably isn't a great
- * problem.
- */
-
-static void step(rho_ctx *cc, void *x, mp **a, mp **b)
-{
- switch (cc->ops->split(x)) {
- case 0:
- cc->ops->mul(x, cc->g, cc->c);
- *a = mp_add(*a, *a, MP_ONE);
- if (MP_CMP(*a, >=, cc->n))
- *a = mp_sub(*a, *a, cc->n);
- break;
- case 1:
- cc->ops->sqr(x, cc->c);
- *a = mp_lsl(*a, *a, 1);
- if (MP_CMP(*a, >=, cc->n))
- *a = mp_sub(*a, *a, cc->n);
- *b = mp_lsl(*b, *b, 1);
- if (MP_CMP(*b, >=, cc->n))
- *b = mp_sub(*b, *b, cc->n);
- break;
- case 2:
- cc->ops->mul(x, cc->a, cc->c);
- *b = mp_add(*b, *b, MP_ONE);
- if (MP_CMP(*b, >=, cc->n))
- *b = mp_sub(*b, *b, cc->n);
- break;
- }
-}
-
-mp *rho(rho_ctx *cc, void *x, void *y, mp *a, mp *b)
-{
- mp *aa = MP_COPY(a), *bb = MP_COPY(b);
- mp *g;
-
- /* --- Grind through the random walk until we find a collision --- */
-
- do {
- step(cc, x, &a, &b);
- step(cc, y, &aa, &bb);
- step(cc, y, &aa, &bb);
- } while (!cc->ops->eq(x, y));
- cc->ops->drop(x);
- cc->ops->drop(y);
-
- /* --- Now sort out the mess --- */
-
- aa = mp_sub(aa, a, aa);
- bb = mp_sub(bb, bb, b);
- g = MP_NEW;
- mp_gcd(&g, &bb, 0, bb, cc->n);
- if (!MP_EQ(g, MP_ONE)) {
- mp_drop(aa);
- aa = 0;
- } else {
- aa = mp_mul(aa, aa, bb);
- mp_div(0, &aa, aa, cc->n);
- }
-
- /* --- Done --- */
-
- mp_drop(bb);
- mp_drop(g);
- mp_drop(a);
- mp_drop(b);
- return (aa);
-}
-
-/* --- @rho_prime@ --- *
- *
- * Arguments: @mp *g@ = generator for the group
- * @mp *a@ = value to find the logarithm of
- * @mp *n@ = order of the group
- * @mp *p@ = prime size of the underlying prime field
- *
- * Returns: The discrete logarithm %$\log_g a$%.
- *
- * Use: Computes discrete logarithms in a subgroup of a prime field.
- */
-
-static void prime_sqr(void *x, void *c)
-{
- mp **p = x;
- mp *a = *p;
- a = mp_sqr(a, a);
- a = mpmont_reduce(c, a, a);
- *p = a;
-}
-
-static void prime_mul(void *x, void *y, void *c)
-{
- mp **p = x;
- mp *a = *p;
- a = mpmont_mul(c, a, a, y);
- *p = a;
-}
-
-static int prime_eq(void *x, void *y)
-{
- return (MP_EQ(*(mp **)x, *(mp **)y));
-}
-
-static int prime_split(void *x)
-{
- /* --- Notes on the splitting function --- *
- *
- * The objective is to produce a simple pseudorandom mapping from the
- * underlying field \gf{p} to \{\,0, 1, 2\,\}$%. This is further
- * constrained by the fact that we must not have %$1 \mapsto 1$% (since
- * otherwise the stepping function above will loop).
- *
- * The function we choose is very simple: we take the least significant
- * word from the integer, add one (to prevent the %$1 \mapsto 1$% property
- * described above) and reduce modulo 3. This is slightly biased against
- * the result 2, but this doesn't appear to be relevant.
- */
-
- return (((*(mp **)x)->v[0] + 1) % 3);
-}
-
-static void prime_drop(void *x)
-{
- MP_DROP(*(mp **)x);
-}
-
-static const rho_ops prime_ops = {
- prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
-};
-
-mp *rho_prime(mp *g, mp *a, mp *n, mp *p)
-{
- rho_ctx cc;
- grand *r = 0;
- mpmont mm;
- mp *x, *y;
- mp *aa, *bb;
- mp *l;
-
- /* --- Initialization --- */
-
- mpmont_create(&mm, p);
- cc.ops = &prime_ops;
- cc.c = &mm;
- cc.n = n;
- cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
- cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
- x = MP_COPY(mm.r);
- y = MP_COPY(x);
- aa = bb = MP_ZERO;
-
- /* --- The main loop --- */
-
- while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
- mp_expfactor f[2];
-
- if (!r)
- r = fibrand_create(0);
- aa = mprand_range(MP_NEW, n, r, 0);
- bb = mprand_range(MP_NEW, n, r, 0);
- f[0].base = cc.g; f[0].exp = aa;
- f[1].base = cc.a; f[1].exp = bb;
- x = mpmont_mexpr(&mm, MP_NEW, f, 2);
- y = MP_COPY(x);
- }
-
- /* --- Throw everything away now --- */
-
- if (r)
- r->ops->destroy(r);
- mp_drop(cc.g);
- mp_drop(cc.a);
- mpmont_destroy(&mm);
- return (l);
-}
-
-/*----- Test rig ----------------------------------------------------------*/
-
-#ifdef TEST_RIG
-
-#include <stdio.h>
-
-#include "dh.h"
-
-int main(void)
-{
- dh_param dp;
- mp *x, *y;
- grand *r = fibrand_create(0);
- mpmont mm;
- mp *l;
- int ok;
-
- fputs("rho: ", stdout);
- fflush(stdout);
-
- dh_gen(&dp, 32, 256, 0, r, pgen_evspin, 0);
- x = mprand_range(MP_NEW, dp.q, r, 0);
- mpmont_create(&mm, dp.p);
- y = mpmont_exp(&mm, MP_NEW, dp.g, x);
- mpmont_destroy(&mm);
- l = rho_prime(dp.g, y, dp.q, dp.p);
- if (MP_EQ(x, l)) {
- fputs(". ok\n", stdout);
- ok = 1;
- } else {
- fputs("\n*** rho (discrete logs) failed\n", stdout);
- ok = 0;
- }
-
- mp_drop(l);
- mp_drop(x);
- mp_drop(y);
- r->ops->destroy(r);
- dh_paramfree(&dp);
- assert(mparena_count(MPARENA_GLOBAL) == 0);
-
- return (ok ? 0 : EXIT_FAILURE);
-}
-
-#endif
-
-/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff