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   1 /* -*-c-*-
   2  *
   3  * $Id: rho.c,v 1.2 2000/10/08 12:11:22 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 /*----- Revision history --------------------------------------------------*
  31  *
  32  * $Log: rho.c,v $
  33  * Revision 1.2  2000/10/08 12:11:22  mdw
  34  * Use @MP_EQ@ instead of @MP_CMP@.
  35  *
  36  * Revision 1.1  2000/07/09 21:32:30  mdw
  37  * Pollard's rho algorithm for computing discrete logs.
  38  *
  39  */
  40
  41 /*----- Header files ------------------------------------------------------*/
  42
  43 #include "fibrand.h"
  44 #include "mp.h"
  45 #include "mpmont.h"
  46 #include "mprand.h"
  47 #include "rho.h"
  48
  49 /*----- Main code ---------------------------------------------------------*/
  50
  51 /* --- @rho@ --- *
  52  *
  53  * Arguments:   @rho_ctx *cc@ = pointer to the context structure
  54  *              @void *x, *y@ = two (equal) base values (try 1)
  55  *              @mp *a, *b@ = logs of %$x$% (see below)
  56  *
  57  * Returns:     The discrete logarithm %$\log_g a$%, or null if the algorithm
  58  *              failed.  (This is unlikely, though possible.)
  59  *
  60  * Use:         Uses Pollard's rho algorithm to compute discrete logs in the
  61  *              group %$G$% generated by %$g$%.
  62  *
  63  *              The algorithm works by finding a cycle in a pseudo-random
  64  *              walk.  The function @ops->split@ should return an element
  65  *              from %$\{\,0, 1, 2\,\}$% according to its argument, in order
  66  *              to determine the walk.  At each step in the walk, we know a
  67  *              group element %$x \in G$% together with its representation as
  68  *              a product of powers of %$g$% and $%a$% (i.e., we know that
  69  *              %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
  70  *
  71  *              Locating a cycle gives us a collision
  72  *
  73  *                %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
  74  *
  75  *              Taking logs of both sides (to base %$g$%) gives us that
  76  *
  77  *                %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
  78  *
  79  *              Good initial values are %$x = y = 1$% (the multiplicative
  80  *              identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
  81  *              If that doesn't work then start choosing more `interesting'
  82  *              values.
  83  *
  84  *              Note that the algorithm requires minimal space but
  85  *              %$O(\sqrt{n})$% time.  Don't do this on large groups,
  86  *              particularly if you can find a decent factor base.
  87  *
  88  *              Finally, note that this function will free the input values
  89  *              when it's finished with them.  This probably isn't a great
  90  *              problem.
  91  */
  92
  93 static void step(rho_ctx *cc, void *x, mp **a, mp **b)
  94 {
  95   switch (cc->ops->split(x)) {
  96     case 0:
  97       cc->ops->mul(x, cc->g, cc->c);
  98       *a = mp_add(*a, *a, MP_ONE);
  99       if (MP_CMP(*a, >=, cc->n))
 100         *a = mp_sub(*a, *a, cc->n);
 101       break;
 102     case 1:
 103       cc->ops->sqr(x, cc->c);
 104       *a = mp_lsl(*a, *a, 1);
 105       if (MP_CMP(*a, >=, cc->n))
 106         *a = mp_sub(*a, *a, cc->n);
 107       *b = mp_lsl(*b, *b, 1);
 108       if (MP_CMP(*b, >=, cc->n))
 109         *b = mp_sub(*b, *b, cc->n);
 110       break;
 111     case 2:
 112       cc->ops->mul(x, cc->a, cc->c);
 113       *b = mp_add(*b, *b, MP_ONE);
 114       if (MP_CMP(*b, >=, cc->n))
 115         *b = mp_sub(*b, *b, cc->n);
 116       break;
 117   }
 118 }
 119
 120 mp *rho(rho_ctx *cc, void *x, void *y, mp *a, mp *b)
 121 {
 122   mp *aa = MP_COPY(a), *bb = MP_COPY(b);
 123   mp *g;
 124
 125   /* --- Grind through the random walk until we find a collision --- */
 126
 127   do {
 128     step(cc, x, &a, &b);
 129     step(cc, y, &aa, &bb);
 130     step(cc, y, &aa, &bb);
 131   } while (!cc->ops->eq(x, y));
 132   cc->ops->drop(x);
 133   cc->ops->drop(y);
 134
 135   /* --- Now sort out the mess --- */
 136
 137   aa = mp_sub(aa, a, aa);
 138   bb = mp_sub(bb, bb, b);
 139   g = MP_NEW;
 140   mp_gcd(&g, &bb, 0, bb, cc->n);
 141   if (!MP_EQ(g, MP_ONE)) {
 142     mp_drop(aa);
 143     aa = 0;
 144   } else {
 145     aa = mp_mul(aa, aa, bb);
 146     mp_div(0, &aa, aa, cc->n);
 147   }
 148
 149   /* --- Done --- */
 150
 151   mp_drop(bb);
 152   mp_drop(g);
 153   mp_drop(a);
 154   mp_drop(b);
 155   return (aa);
 156 }
 157
 158 /* --- @rho_prime@ --- *
 159  *
 160  * Arguments:   @mp *g@ = generator for the group
 161  *              @mp *a@ = value to find the logarithm of
 162  *              @mp *n@ = order of the group
 163  *              @mp *p@ = prime size of the underlying prime field
 164  *
 165  * Returns:     The discrete logarithm %$\log_g a$%.
 166  *
 167  * Use:         Computes discrete logarithms in a subgroup of a prime field.
 168  */
 169
 170 static void prime_sqr(void *x, void *c)
 171 {
 172   mp **p = x;
 173   mp *a = *p;
 174   a = mp_sqr(a, a);
 175   a = mpmont_reduce(c, a, a);
 176   *p = a;
 177 }
 178
 179 static void prime_mul(void *x, void *y, void *c)
 180 {
 181   mp **p = x;
 182   mp *a = *p;
 183   a = mpmont_mul(c, a, a, y);
 184   *p = a;
 185 }
 186
 187 static int prime_eq(void *x, void *y)
 188 {
 189   return (MP_EQ(*(mp **)x, *(mp **)y));
 190 }
 191
 192 static int prime_split(void *x)
 193 {
 194   /* --- Notes on the splitting function --- *
 195    *
 196    * The objective is to produce a simple pseudorandom mapping from the
 197    * underlying field \gf{p} to \{\,0, 1, 2\,\}$%.  This is further
 198    * constrained by the fact that we must not have %$1 \mapsto 1$% (since
 199    * otherwise the stepping function above will loop).
 200    *
 201    * The function we choose is very simple: we take the least significant
 202    * word from the integer, add one (to prevent the %$1 \mapsto 1$% property
 203    * described above) and reduce modulo 3.  This is slightly biased against
 204    * the result 2, but this doesn't appear to be relevant.
 205    */
 206
 207   return (((*(mp **)x)->v[0] + 1) % 3);
 208 }
 209
 210 static void prime_drop(void *x)
 211 {
 212   MP_DROP(*(mp **)x);
 213 }
 214
 215 static rho_ops prime_ops = {
 216   prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
 217 };
 218
 219 mp *rho_prime(mp *g, mp *a, mp *n, mp *p)
 220 {
 221   rho_ctx cc;
 222   grand *r = 0;
 223   mpmont mm;
 224   mp *x, *y;
 225   mp *aa, *bb;
 226   mp *l;
 227
 228   /* --- Initialization --- */
 229
 230   mpmont_create(&mm, p);
 231   cc.ops = &prime_ops;
 232   cc.c = &mm;
 233   cc.n = n;
 234   cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
 235   cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
 236   x = MP_COPY(mm.r);
 237   y = MP_COPY(x);
 238   aa = bb = MP_ZERO;
 239
 240   /* --- The main loop --- */
 241
 242   while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
 243     mpmont_factor f[2];
 244
 245     if (!r)
 246       r = fibrand_create(0);
 247     aa = mprand_range(MP_NEW, n, r, 0);
 248     bb = mprand_range(MP_NEW, n, r, 0);
 249     f[0].base = g; f[0].exp = aa;
 250     f[1].base = a; f[1].exp = bb;
 251     x = mpmont_mexpr(&mm, MP_NEW, f, 2);
 252     y = MP_COPY(x);
 253   }
 254
 255   /* --- Throw everything away now --- */
 256
 257   if (r)
 258     r->ops->destroy(r);
 259   mp_drop(cc.g);
 260   mp_drop(cc.a);
 261   mpmont_destroy(&mm);
 262   return (l);
 263 }
 264
 265 /*----- Test rig ----------------------------------------------------------*/
 266
 267 #ifdef TEST_RIG
 268
 269 #include <stdio.h>
 270
 271 #include "dh.h"
 272
 273 int main(void)
 274 {
 275   dh_param dp;
 276   mp *x, *y;
 277   grand *r = fibrand_create(0);
 278   mpmont mm;
 279   mp *l;
 280   int ok;
 281
 282   fputs("rho: ", stdout);
 283   fflush(stdout);
 284
 285   dh_gen(&dp, 32, 256, 0, r, pgen_evspin, 0);
 286   x = mprand_range(MP_NEW, dp.q, r, 0);
 287   mpmont_create(&mm, dp.p);
 288   y = mpmont_exp(&mm, MP_NEW, dp.g, x);
 289   mpmont_destroy(&mm);
 290   l = rho_prime(dp.g, y, dp.q, dp.p);
 291   if (MP_EQ(x, l)) {
 292     fputs(". ok\n", stdout);
 293     ok = 1;
 294   } else {
 295     fputs("\n*** rho (discrete logs) failed\n", stdout);
 296     ok = 0;
 297   }
 298
 299   mp_drop(l);
 300   mp_drop(x);
 301   mp_drop(y);
 302   r->ops->destroy(r);
 303   dh_paramfree(&dp);
 304   assert(mparena_count(MPARENA_GLOBAL) == 0);
 305
 306   return (ok ? 0 : EXIT_FAILURE);
 307 }
 308
 309 #endif
 310
 311 /*----- That's all, folks -------------------------------------------------*/