Pollard's rho algorithm for computing discrete logs.

[u/mdw/catacomb] / rho.c

/* -*-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 -------------------------------------------------*/