[u/mdw/catacomb] / rho.c

/* -*-c-*-
 *
 * $Id: rho.c,v 1.3 2001/06/16 12:56:38 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 1.3  2001/06/16 12:56:38  mdw
 * Fixes for interface change to @mpmont_expr@ and @mpmont_mexpr@.
 *
 * Revision 1.2  2000/10/08 12:11:22  mdw
 * Use @MP_EQ@ instead of @MP_CMP@.
 *
 * 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_EQ(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_EQ(*(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) {
    mp_expfactor 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 = cc.g; f[0].exp = aa;
    f[1].base = cc.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_EQ(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 -------------------------------------------------*/
Commit	Line	Data
f41f820e	1	/* --c--
f41f820e	2	*
b0b682aa	3	* $Id: rho.c,v 1.3 2001/06/16 12:56:38 mdw Exp $
f41f820e	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 $
b0b682aa	33	* Revision 1.3 2001/06/16 12:56:38 mdw
	34	* Fixes for interface change to @mpmont_expr@ and @mpmont_mexpr@.
	35	*
22bab86c	36	* Revision 1.2 2000/10/08 12:11:22 mdw
	37	* Use @MP_EQ@ instead of @MP_CMP@.
	38	*
f41f820e	39	* Revision 1.1 2000/07/09 21:32:30 mdw
	40	* Pollard's rho algorithm for computing discrete logs.
	41	*
	42	*/
	43
	44	/----- Header files ------------------------------------------------------/
	45
	46	#include "fibrand.h"
	47	#include "mp.h"
	48	#include "mpmont.h"
	49	#include "mprand.h"
	50	#include "rho.h"
	51
	52	/----- Main code ---------------------------------------------------------/
	53
	54	/* --- @rho@ --- *
	55	*
	56	* Arguments: @rho_ctx *cc@ = pointer to the context structure
	57	* @void x, y@ = two (equal) base values (try 1)
	58	* @mp a, b@ = logs of %$x$% (see below)
	59	*
	60	* Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm
	61	* failed. (This is unlikely, though possible.)
	62	*
	63	* Use: Uses Pollard's rho algorithm to compute discrete logs in the
	64	* group %$G$% generated by %$g$%.
	65	*
	66	* The algorithm works by finding a cycle in a pseudo-random
	67	* walk. The function @ops->split@ should return an element
	68	* from %$\{\,0, 1, 2\,\}$% according to its argument, in order
	69	* to determine the walk. At each step in the walk, we know a
	70	* group element %$x \in G$% together with its representation as
	71	* a product of powers of %$g$% and $%a$% (i.e., we know that
	72	* %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
	73	*
	74	* Locating a cycle gives us a collision
	75	*
	76	* %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
	77	*
	78	* Taking logs of both sides (to base %$g$%) gives us that
	79	*
	80	* %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
	81	*
	82	* Good initial values are %$x = y = 1$% (the multiplicative
	83	* identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
	84	* If that doesn't work then start choosing more `interesting'
	85	* values.
	86	*
	87	* Note that the algorithm requires minimal space but
	88	* %$O(\sqrt{n})$% time. Don't do this on large groups,
	89	* particularly if you can find a decent factor base.
	90	*
	91	* Finally, note that this function will free the input values
	92	* when it's finished with them. This probably isn't a great
	93	* problem.
	94	*/
	95
	96	static void step(rho_ctx cc, void x, mp a, mp b)
	97	{
	98	switch (cc->ops->split(x)) {
	99	case 0:
	100	cc->ops->mul(x, cc->g, cc->c);
	101	a = mp_add(a, *a, MP_ONE);
	102	if (MP_CMP(*a, >=, cc->n))
103	a = mp_sub(a, *a, cc->n);
104	break;
105	case 1:
106	cc->ops->sqr(x, cc->c);
107	a = mp_lsl(a, *a, 1);
108	if (MP_CMP(*a, >=, cc->n))
109	a = mp_sub(a, *a, cc->n);
110	b = mp_lsl(b, *b, 1);
111	if (MP_CMP(*b, >=, cc->n))
112	b = mp_sub(b, *b, cc->n);
113	break;
114	case 2:
115	cc->ops->mul(x, cc->a, cc->c);
116	b = mp_add(b, *b, MP_ONE);
117	if (MP_CMP(*b, >=, cc->n))
118	b = mp_sub(b, *b, cc->n);
119	break;
120	}
121	}
122
123	mp rho(rho_ctx cc, void x, void y, mp a, mp b)
124	{
125	mp aa = MP_COPY(a), bb = MP_COPY(b);
126	mp *g;
127
128	/* --- Grind through the random walk until we find a collision --- */
129
130	do {
131	step(cc, x, &a, &b);
132	step(cc, y, &aa, &bb);
133	step(cc, y, &aa, &bb);
134	} while (!cc->ops->eq(x, y));
135	cc->ops->drop(x);
136	cc->ops->drop(y);
137
138	/* --- Now sort out the mess --- */
139
140	aa = mp_sub(aa, a, aa);
141	bb = mp_sub(bb, bb, b);
142	g = MP_NEW;
143	mp_gcd(&g, &bb, 0, bb, cc->n);
22bab86c	144	if (!MP_EQ(g, MP_ONE)) {
f41f820e	145	mp_drop(aa);
	146	aa = 0;
	147	} else {
	148	aa = mp_mul(aa, aa, bb);
	149	mp_div(0, &aa, aa, cc->n);
	150	}
	151
	152	/* --- Done --- */
	153
	154	mp_drop(bb);
	155	mp_drop(g);
	156	mp_drop(a);
	157	mp_drop(b);
	158	return (aa);
	159	}
	160
	161	/* --- @rho_prime@ --- *
	162	*
	163	* Arguments: @mp *g@ = generator for the group
	164	* @mp *a@ = value to find the logarithm of
	165	* @mp *n@ = order of the group
	166	* @mp *p@ = prime size of the underlying prime field
	167	*
	168	* Returns: The discrete logarithm %$\log_g a$%.
	169	*
	170	* Use: Computes discrete logarithms in a subgroup of a prime field.
	171	*/
	172
	173	static void prime_sqr(void x, void c)
	174	{
	175	mp **p = x;
	176	mp a = p;
	177	a = mp_sqr(a, a);
	178	a = mpmont_reduce(c, a, a);
	179	*p = a;
	180	}
	181
	182	static void prime_mul(void x, void y, void *c)
	183	{
	184	mp **p = x;
	185	mp a = p;
	186	a = mpmont_mul(c, a, a, y);
	187	*p = a;
	188	}
	189
	190	static int prime_eq(void x, void y)
	191	{
22bab86c	192	return (MP_EQ((mp )x, (mp **)y));
f41f820e	193	}
	194
	195	static int prime_split(void *x)
	196	{
	197	/* --- Notes on the splitting function --- *
	198	*
	199	* The objective is to produce a simple pseudorandom mapping from the
	200	* underlying field \gf{p} to \{\,0, 1, 2\,\}$%. This is further
	201	* constrained by the fact that we must not have %$1 \mapsto 1$% (since
	202	* otherwise the stepping function above will loop).
	203	*
	204	* The function we choose is very simple: we take the least significant
	205	* word from the integer, add one (to prevent the %$1 \mapsto 1$% property
	206	* described above) and reduce modulo 3. This is slightly biased against
	207	* the result 2, but this doesn't appear to be relevant.
	208	*/
	209
	210	return ((((mp *)x)->v[0] + 1) % 3);
	211	}
	212
	213	static void prime_drop(void *x)
	214	{
	215	MP_DROP((mp *)x);
	216	}
	217
	218	static rho_ops prime_ops = {
	219	prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
	220	};
	221
	222	mp rho_prime(mp g, mp a, mp n, mp *p)
	223	{
	224	rho_ctx cc;
	225	grand *r = 0;
	226	mpmont mm;
	227	mp x, y;
	228	mp aa, bb;
	229	mp *l;
	230
	231	/* --- Initialization --- */
	232
	233	mpmont_create(&mm, p);
	234	cc.ops = &prime_ops;
	235	cc.c = &mm;
	236	cc.n = n;
	237	cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
	238	cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
	239	x = MP_COPY(mm.r);
	240	y = MP_COPY(x);
	241	aa = bb = MP_ZERO;
	242
	243	/* --- The main loop --- */
	244
	245	while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
b0b682aa	246	mp_expfactor f[2];
f41f820e	247
	248	if (!r)
	249	r = fibrand_create(0);
	250	aa = mprand_range(MP_NEW, n, r, 0);
	251	bb = mprand_range(MP_NEW, n, r, 0);
b0b682aa	252	f[0].base = cc.g; f[0].exp = aa;
b0b682aa	253	f[1].base = cc.a; f[1].exp = bb;
f41f820e	254	x = mpmont_mexpr(&mm, MP_NEW, f, 2);
	255	y = MP_COPY(x);
	256	}
	257
	258	/* --- Throw everything away now --- */
	259
	260	if (r)
	261	r->ops->destroy(r);
	262	mp_drop(cc.g);
	263	mp_drop(cc.a);
	264	mpmont_destroy(&mm);
	265	return (l);
	266	}
	267
	268	/----- Test rig ----------------------------------------------------------/
	269
	270	#ifdef TEST_RIG
	271
	272	#include <stdio.h>
	273
	274	#include "dh.h"
	275
	276	int main(void)
	277	{
	278	dh_param dp;
	279	mp x, y;
	280	grand *r = fibrand_create(0);
	281	mpmont mm;
	282	mp *l;
	283	int ok;
	284
	285	fputs("rho: ", stdout);
	286	fflush(stdout);
	287
	288	dh_gen(&dp, 32, 256, 0, r, pgen_evspin, 0);
	289	x = mprand_range(MP_NEW, dp.q, r, 0);
	290	mpmont_create(&mm, dp.p);
	291	y = mpmont_exp(&mm, MP_NEW, dp.g, x);
	292	mpmont_destroy(&mm);
	293	l = rho_prime(dp.g, y, dp.q, dp.p);
22bab86c	294	if (MP_EQ(x, l)) {
f41f820e	295	fputs(". ok\n", stdout);
	296	ok = 1;
	297	} else {
	298	fputs("\n*** rho (discrete logs) failed\n", stdout);
	299	ok = 0;
	300	}
	301
	302	mp_drop(l);
	303	mp_drop(x);
	304	mp_drop(y);
	305	r->ops->destroy(r);
	306	dh_paramfree(&dp);
	307	assert(mparena_count(MPARENA_GLOBAL) == 0);
	308
	309	return (ok ? 0 : EXIT_FAILURE);
	310	}
	311
	312	#endif
	313
	314	/----- That's all, folks -------------------------------------------------/