--- /dev/null
+/* -*-c-*-
+ *
+ * $Id: rho.c,v 1.1 2000/07/09 21:32:30 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.
+ */
+
+/*----- Revision history --------------------------------------------------*
+ *
+ * $Log: rho.c,v $
+ * Revision 1.1 2000/07/09 21:32:30 mdw
+ * Pollard's rho algorithm for computing discrete logs.
+ *
+ */
+
+/*----- 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_CMP(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_CMP(*(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 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) {
+ mpmont_factor 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 = g; f[0].exp = aa;
+ f[1].base = 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_CMP(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