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   1 /* -*-c-*-
   2  *
   3  * $Id: rho.c,v 1.4 2004/04/02 01:03:49 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 1.4  2004/04/02 01:03:49  mdw
  34  * Miscellaneous constification.
  35  *
  36  * Revision 1.3  2001/06/16 12:56:38  mdw
  37  * Fixes for interface change to @mpmont_expr@ and @mpmont_mexpr@.
  38  *
  39  * Revision 1.2  2000/10/08 12:11:22  mdw
  40  * Use @MP_EQ@ instead of @MP_CMP@.
  41  *
  42  * Revision 1.1  2000/07/09 21:32:30  mdw
  43  * Pollard's rho algorithm for computing discrete logs.
  44  *
  45  */
  46
  47 /*----- Header files ------------------------------------------------------*/
  48
  49 #include "fibrand.h"
  50 #include "mp.h"
  51 #include "mpmont.h"
  52 #include "mprand.h"
  53 #include "rho.h"
  54
  55 /*----- Main code ---------------------------------------------------------*/
  56
  57 /* --- @rho@ --- *
  58  *
  59  * Arguments:   @rho_ctx *cc@ = pointer to the context structure
  60  *              @void *x, *y@ = two (equal) base values (try 1)
  61  *              @mp *a, *b@ = logs of %$x$% (see below)
  62  *
  63  * Returns:     The discrete logarithm %$\log_g a$%, or null if the algorithm
  64  *              failed.  (This is unlikely, though possible.)
  65  *
  66  * Use:         Uses Pollard's rho algorithm to compute discrete logs in the
  67  *              group %$G$% generated by %$g$%.
  68  *
  69  *              The algorithm works by finding a cycle in a pseudo-random
  70  *              walk.  The function @ops->split@ should return an element
  71  *              from %$\{\,0, 1, 2\,\}$% according to its argument, in order
  72  *              to determine the walk.  At each step in the walk, we know a
  73  *              group element %$x \in G$% together with its representation as
  74  *              a product of powers of %$g$% and $%a$% (i.e., we know that
  75  *              %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
  76  *
  77  *              Locating a cycle gives us a collision
  78  *
  79  *                %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
  80  *
  81  *              Taking logs of both sides (to base %$g$%) gives us that
  82  *
  83  *                %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
  84  *
  85  *              Good initial values are %$x = y = 1$% (the multiplicative
  86  *              identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
  87  *              If that doesn't work then start choosing more `interesting'
  88  *              values.
  89  *
  90  *              Note that the algorithm requires minimal space but
  91  *              %$O(\sqrt{n})$% time.  Don't do this on large groups,
  92  *              particularly if you can find a decent factor base.
  93  *
  94  *              Finally, note that this function will free the input values
  95  *              when it's finished with them.  This probably isn't a great
  96  *              problem.
  97  */
  98
  99 static void step(rho_ctx *cc, void *x, mp **a, mp **b)
 100 {
 101   switch (cc->ops->split(x)) {
 102     case 0:
 103       cc->ops->mul(x, cc->g, cc->c);
 104       *a = mp_add(*a, *a, MP_ONE);
 105       if (MP_CMP(*a, >=, cc->n))
 106         *a = mp_sub(*a, *a, cc->n);
 107       break;
 108     case 1:
 109       cc->ops->sqr(x, cc->c);
 110       *a = mp_lsl(*a, *a, 1);
 111       if (MP_CMP(*a, >=, cc->n))
 112         *a = mp_sub(*a, *a, cc->n);
 113       *b = mp_lsl(*b, *b, 1);
 114       if (MP_CMP(*b, >=, cc->n))
 115         *b = mp_sub(*b, *b, cc->n);
 116       break;
 117     case 2:
 118       cc->ops->mul(x, cc->a, cc->c);
 119       *b = mp_add(*b, *b, MP_ONE);
 120       if (MP_CMP(*b, >=, cc->n))
 121         *b = mp_sub(*b, *b, cc->n);
 122       break;
 123   }
 124 }
 125
 126 mp *rho(rho_ctx *cc, void *x, void *y, mp *a, mp *b)
 127 {
 128   mp *aa = MP_COPY(a), *bb = MP_COPY(b);
 129   mp *g;
 130
 131   /* --- Grind through the random walk until we find a collision --- */
 132
 133   do {
 134     step(cc, x, &a, &b);
 135     step(cc, y, &aa, &bb);
 136     step(cc, y, &aa, &bb);
 137   } while (!cc->ops->eq(x, y));
 138   cc->ops->drop(x);
 139   cc->ops->drop(y);
 140
 141   /* --- Now sort out the mess --- */
 142
 143   aa = mp_sub(aa, a, aa);
 144   bb = mp_sub(bb, bb, b);
 145   g = MP_NEW;
 146   mp_gcd(&g, &bb, 0, bb, cc->n);
 147   if (!MP_EQ(g, MP_ONE)) {
 148     mp_drop(aa);
 149     aa = 0;
 150   } else {
 151     aa = mp_mul(aa, aa, bb);
 152     mp_div(0, &aa, aa, cc->n);
 153   }
 154
 155   /* --- Done --- */
 156
 157   mp_drop(bb);
 158   mp_drop(g);
 159   mp_drop(a);
 160   mp_drop(b);
 161   return (aa);
 162 }
 163
 164 /* --- @rho_prime@ --- *
 165  *
 166  * Arguments:   @mp *g@ = generator for the group
 167  *              @mp *a@ = value to find the logarithm of
 168  *              @mp *n@ = order of the group
 169  *              @mp *p@ = prime size of the underlying prime field
 170  *
 171  * Returns:     The discrete logarithm %$\log_g a$%.
 172  *
 173  * Use:         Computes discrete logarithms in a subgroup of a prime field.
 174  */
 175
 176 static void prime_sqr(void *x, void *c)
 177 {
 178   mp **p = x;
 179   mp *a = *p;
 180   a = mp_sqr(a, a);
 181   a = mpmont_reduce(c, a, a);
 182   *p = a;
 183 }
 184
 185 static void prime_mul(void *x, void *y, void *c)
 186 {
 187   mp **p = x;
 188   mp *a = *p;
 189   a = mpmont_mul(c, a, a, y);
 190   *p = a;
 191 }
 192
 193 static int prime_eq(void *x, void *y)
 194 {
 195   return (MP_EQ(*(mp **)x, *(mp **)y));
 196 }
 197
 198 static int prime_split(void *x)
 199 {
 200   /* --- Notes on the splitting function --- *
 201    *
 202    * The objective is to produce a simple pseudorandom mapping from the
 203    * underlying field \gf{p} to \{\,0, 1, 2\,\}$%.  This is further
 204    * constrained by the fact that we must not have %$1 \mapsto 1$% (since
 205    * otherwise the stepping function above will loop).
 206    *
 207    * The function we choose is very simple: we take the least significant
 208    * word from the integer, add one (to prevent the %$1 \mapsto 1$% property
 209    * described above) and reduce modulo 3.  This is slightly biased against
 210    * the result 2, but this doesn't appear to be relevant.
 211    */
 212
 213   return (((*(mp **)x)->v[0] + 1) % 3);
 214 }
 215
 216 static void prime_drop(void *x)
 217 {
 218   MP_DROP(*(mp **)x);
 219 }
 220
 221 static const rho_ops prime_ops = {
 222   prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
 223 };
 224
 225 mp *rho_prime(mp *g, mp *a, mp *n, mp *p)
 226 {
 227   rho_ctx cc;
 228   grand *r = 0;
 229   mpmont mm;
 230   mp *x, *y;
 231   mp *aa, *bb;
 232   mp *l;
 233
 234   /* --- Initialization --- */
 235
 236   mpmont_create(&mm, p);
 237   cc.ops = &prime_ops;
 238   cc.c = &mm;
 239   cc.n = n;
 240   cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
 241   cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
 242   x = MP_COPY(mm.r);
 243   y = MP_COPY(x);
 244   aa = bb = MP_ZERO;
 245
 246   /* --- The main loop --- */
 247
 248   while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
 249     mp_expfactor f[2];
 250
 251     if (!r)
 252       r = fibrand_create(0);
 253     aa = mprand_range(MP_NEW, n, r, 0);
 254     bb = mprand_range(MP_NEW, n, r, 0);
 255     f[0].base = cc.g; f[0].exp = aa;
 256     f[1].base = cc.a; f[1].exp = bb;
 257     x = mpmont_mexpr(&mm, MP_NEW, f, 2);
 258     y = MP_COPY(x);
 259   }
 260
 261   /* --- Throw everything away now --- */
 262
 263   if (r)
 264     r->ops->destroy(r);
 265   mp_drop(cc.g);
 266   mp_drop(cc.a);
 267   mpmont_destroy(&mm);
 268   return (l);
 269 }
 270
 271 /*----- Test rig ----------------------------------------------------------*/
 272
 273 #ifdef TEST_RIG
 274
 275 #include <stdio.h>
 276
 277 #include "dh.h"
 278
 279 int main(void)
 280 {
 281   dh_param dp;
 282   mp *x, *y;
 283   grand *r = fibrand_create(0);
 284   mpmont mm;
 285   mp *l;
 286   int ok;
 287
 288   fputs("rho: ", stdout);
 289   fflush(stdout);
 290
 291   dh_gen(&dp, 32, 256, 0, r, pgen_evspin, 0);
 292   x = mprand_range(MP_NEW, dp.q, r, 0);
 293   mpmont_create(&mm, dp.p);
 294   y = mpmont_exp(&mm, MP_NEW, dp.g, x);
 295   mpmont_destroy(&mm);
 296   l = rho_prime(dp.g, y, dp.q, dp.p);
 297   if (MP_EQ(x, l)) {
 298     fputs(". ok\n", stdout);
 299     ok = 1;
 300   } else {
 301     fputs("\n*** rho (discrete logs) failed\n", stdout);
 302     ok = 0;
 303   }
 304
 305   mp_drop(l);
 306   mp_drop(x);
 307   mp_drop(y);
 308   r->ops->destroy(r);
 309   dh_paramfree(&dp);
 310   assert(mparena_count(MPARENA_GLOBAL) == 0);
 311
 312   return (ok ? 0 : EXIT_FAILURE);
 313 }
 314
 315 #endif
 316
 317 /*----- That's all, folks -------------------------------------------------*/