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   1 /* -*-c-*-
   2  *
   3  * $Id: rho.c,v 1.5 2004/04/08 01:36:15 mdw Exp $
   4  *
   5  * Pollard's rho algorithm for discrete logs
   6  *
   7  * (c) 2000 Straylight/Edgeware
   8  */
   9
  10 /*----- Licensing notice --------------------------------------------------*
  11  *
  12  * This file is part of Catacomb.
  13  *
  14  * Catacomb is free software; you can redistribute it and/or modify
  15  * it under the terms of the GNU Library General Public License as
  16  * published by the Free Software Foundation; either version 2 of the
  17  * License, or (at your option) any later version.
  18  *
  19  * Catacomb is distributed in the hope that it will be useful,
  20  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  21  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  22  * GNU Library General Public License for more details.
  23  *
  24  * You should have received a copy of the GNU Library General Public
  25  * License along with Catacomb; if not, write to the Free
  26  * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
  27  * MA 02111-1307, USA.
  28  */
  29
  30 /*----- Header files ------------------------------------------------------*/
  31
  32 #include "fibrand.h"
  33 #include "mp.h"
  34 #include "mpmont.h"
  35 #include "mprand.h"
  36 #include "rho.h"
  37
  38 /*----- Main code ---------------------------------------------------------*/
  39
  40 /* --- @rho@ --- *
  41  *
  42  * Arguments:   @rho_ctx *cc@ = pointer to the context structure
  43  *              @void *x, *y@ = two (equal) base values (try 1)
  44  *              @mp *a, *b@ = logs of %$x$% (see below)
  45  *
  46  * Returns:     The discrete logarithm %$\log_g a$%, or null if the algorithm
  47  *              failed.  (This is unlikely, though possible.)
  48  *
  49  * Use:         Uses Pollard's rho algorithm to compute discrete logs in the
  50  *              group %$G$% generated by %$g$%.
  51  *
  52  *              The algorithm works by finding a cycle in a pseudo-random
  53  *              walk.  The function @ops->split@ should return an element
  54  *              from %$\{\,0, 1, 2\,\}$% according to its argument, in order
  55  *              to determine the walk.  At each step in the walk, we know a
  56  *              group element %$x \in G$% together with its representation as
  57  *              a product of powers of %$g$% and $%a$% (i.e., we know that
  58  *              %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
  59  *
  60  *              Locating a cycle gives us a collision
  61  *
  62  *                %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
  63  *
  64  *              Taking logs of both sides (to base %$g$%) gives us that
  65  *
  66  *                %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
  67  *
  68  *              Good initial values are %$x = y = 1$% (the multiplicative
  69  *              identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
  70  *              If that doesn't work then start choosing more `interesting'
  71  *              values.
  72  *
  73  *              Note that the algorithm requires minimal space but
  74  *              %$O(\sqrt{n})$% time.  Don't do this on large groups,
  75  *              particularly if you can find a decent factor base.
  76  *
  77  *              Finally, note that this function will free the input values
  78  *              when it's finished with them.  This probably isn't a great
  79  *              problem.
  80  */
  81
  82 static void step(rho_ctx *cc, void *x, mp **a, mp **b)
  83 {
  84   switch (cc->ops->split(x)) {
  85     case 0:
  86       cc->ops->mul(x, cc->g, cc->c);
  87       *a = mp_add(*a, *a, MP_ONE);
  88       if (MP_CMP(*a, >=, cc->n))
  89         *a = mp_sub(*a, *a, cc->n);
  90       break;
  91     case 1:
  92       cc->ops->sqr(x, cc->c);
  93       *a = mp_lsl(*a, *a, 1);
  94       if (MP_CMP(*a, >=, cc->n))
  95         *a = mp_sub(*a, *a, cc->n);
  96       *b = mp_lsl(*b, *b, 1);
  97       if (MP_CMP(*b, >=, cc->n))
  98         *b = mp_sub(*b, *b, cc->n);
  99       break;
 100     case 2:
 101       cc->ops->mul(x, cc->a, cc->c);
 102       *b = mp_add(*b, *b, MP_ONE);
 103       if (MP_CMP(*b, >=, cc->n))
 104         *b = mp_sub(*b, *b, cc->n);
 105       break;
 106   }
 107 }
 108
 109 mp *rho(rho_ctx *cc, void *x, void *y, mp *a, mp *b)
 110 {
 111   mp *aa = MP_COPY(a), *bb = MP_COPY(b);
 112   mp *g;
 113
 114   /* --- Grind through the random walk until we find a collision --- */
 115
 116   do {
 117     step(cc, x, &a, &b);
 118     step(cc, y, &aa, &bb);
 119     step(cc, y, &aa, &bb);
 120   } while (!cc->ops->eq(x, y));
 121   cc->ops->drop(x);
 122   cc->ops->drop(y);
 123
 124   /* --- Now sort out the mess --- */
 125
 126   aa = mp_sub(aa, a, aa);
 127   bb = mp_sub(bb, bb, b);
 128   g = MP_NEW;
 129   mp_gcd(&g, &bb, 0, bb, cc->n);
 130   if (!MP_EQ(g, MP_ONE)) {
 131     mp_drop(aa);
 132     aa = 0;
 133   } else {
 134     aa = mp_mul(aa, aa, bb);
 135     mp_div(0, &aa, aa, cc->n);
 136   }
 137
 138   /* --- Done --- */
 139
 140   mp_drop(bb);
 141   mp_drop(g);
 142   mp_drop(a);
 143   mp_drop(b);
 144   return (aa);
 145 }
 146
 147 /* --- @rho_prime@ --- *
 148  *
 149  * Arguments:   @mp *g@ = generator for the group
 150  *              @mp *a@ = value to find the logarithm of
 151  *              @mp *n@ = order of the group
 152  *              @mp *p@ = prime size of the underlying prime field
 153  *
 154  * Returns:     The discrete logarithm %$\log_g a$%.
 155  *
 156  * Use:         Computes discrete logarithms in a subgroup of a prime field.
 157  */
 158
 159 static void prime_sqr(void *x, void *c)
 160 {
 161   mp **p = x;
 162   mp *a = *p;
 163   a = mp_sqr(a, a);
 164   a = mpmont_reduce(c, a, a);
 165   *p = a;
 166 }
 167
 168 static void prime_mul(void *x, void *y, void *c)
 169 {
 170   mp **p = x;
 171   mp *a = *p;
 172   a = mpmont_mul(c, a, a, y);
 173   *p = a;
 174 }
 175
 176 static int prime_eq(void *x, void *y)
 177 {
 178   return (MP_EQ(*(mp **)x, *(mp **)y));
 179 }
 180
 181 static int prime_split(void *x)
 182 {
 183   /* --- Notes on the splitting function --- *
 184    *
 185    * The objective is to produce a simple pseudorandom mapping from the
 186    * underlying field \gf{p} to \{\,0, 1, 2\,\}$%.  This is further
 187    * constrained by the fact that we must not have %$1 \mapsto 1$% (since
 188    * otherwise the stepping function above will loop).
 189    *
 190    * The function we choose is very simple: we take the least significant
 191    * word from the integer, add one (to prevent the %$1 \mapsto 1$% property
 192    * described above) and reduce modulo 3.  This is slightly biased against
 193    * the result 2, but this doesn't appear to be relevant.
 194    */
 195
 196   return (((*(mp **)x)->v[0] + 1) % 3);
 197 }
 198
 199 static void prime_drop(void *x)
 200 {
 201   MP_DROP(*(mp **)x);
 202 }
 203
 204 static const rho_ops prime_ops = {
 205   prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
 206 };
 207
 208 mp *rho_prime(mp *g, mp *a, mp *n, mp *p)
 209 {
 210   rho_ctx cc;
 211   grand *r = 0;
 212   mpmont mm;
 213   mp *x, *y;
 214   mp *aa, *bb;
 215   mp *l;
 216
 217   /* --- Initialization --- */
 218
 219   mpmont_create(&mm, p);
 220   cc.ops = &prime_ops;
 221   cc.c = &mm;
 222   cc.n = n;
 223   cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
 224   cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
 225   x = MP_COPY(mm.r);
 226   y = MP_COPY(x);
 227   aa = bb = MP_ZERO;
 228
 229   /* --- The main loop --- */
 230
 231   while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
 232     mp_expfactor f[2];
 233
 234     if (!r)
 235       r = fibrand_create(0);
 236     aa = mprand_range(MP_NEW, n, r, 0);
 237     bb = mprand_range(MP_NEW, n, r, 0);
 238     f[0].base = cc.g; f[0].exp = aa;
 239     f[1].base = cc.a; f[1].exp = bb;
 240     x = mpmont_mexpr(&mm, MP_NEW, f, 2);
 241     y = MP_COPY(x);
 242   }
 243
 244   /* --- Throw everything away now --- */
 245
 246   if (r)
 247     r->ops->destroy(r);
 248   mp_drop(cc.g);
 249   mp_drop(cc.a);
 250   mpmont_destroy(&mm);
 251   return (l);
 252 }
 253
 254 /*----- Test rig ----------------------------------------------------------*/
 255
 256 #ifdef TEST_RIG
 257
 258 #include <stdio.h>
 259
 260 #include "dh.h"
 261
 262 int main(void)
 263 {
 264   dh_param dp;
 265   mp *x, *y;
 266   grand *r = fibrand_create(0);
 267   mpmont mm;
 268   mp *l;
 269   int ok;
 270
 271   fputs("rho: ", stdout);
 272   fflush(stdout);
 273
 274   dh_gen(&dp, 32, 256, 0, r, pgen_evspin, 0);
 275   x = mprand_range(MP_NEW, dp.q, r, 0);
 276   mpmont_create(&mm, dp.p);
 277   y = mpmont_exp(&mm, MP_NEW, dp.g, x);
 278   mpmont_destroy(&mm);
 279   l = rho_prime(dp.g, y, dp.q, dp.p);
 280   if (MP_EQ(x, l)) {
 281     fputs(". ok\n", stdout);
 282     ok = 1;
 283   } else {
 284     fputs("\n*** rho (discrete logs) failed\n", stdout);
 285     ok = 0;
 286   }
 287
 288   mp_drop(l);
 289   mp_drop(x);
 290   mp_drop(y);
 291   r->ops->destroy(r);
 292   dh_paramfree(&dp);
 293   assert(mparena_count(MPARENA_GLOBAL) == 0);
 294
 295   return (ok ? 0 : EXIT_FAILURE);
 296 }
 297
 298 #endif
 299
 300 /*----- That's all, folks -------------------------------------------------*/