[u/mdw/catacomb] / math / rho.c

/* -*-c-*-
 *
 * 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.
 */

/*----- 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 const 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	*
f41f820e	3	* Pollard's rho algorithm for discrete logs
	4	*
	5	* (c) 2000 Straylight/Edgeware
	6	*/
	7
45c0fd36	8	/----- Licensing notice --------------------------------------------------
f41f820e	9	*
	10	* This file is part of Catacomb.
	11	*
	12	* Catacomb is free software; you can redistribute it and/or modify
	13	* it under the terms of the GNU Library General Public License as
	14	* published by the Free Software Foundation; either version 2 of the
	15	* License, or (at your option) any later version.
45c0fd36	16	*
f41f820e	17	* Catacomb is distributed in the hope that it will be useful,
	18	* but WITHOUT ANY WARRANTY; without even the implied warranty of
	19	* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
	20	* GNU Library General Public License for more details.
45c0fd36	21	*
f41f820e	22	* You should have received a copy of the GNU Library General Public
	23	* License along with Catacomb; if not, write to the Free
	24	* Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
	25	* MA 02111-1307, USA.
	26	*/
	27
f41f820e	28	/----- Header files ------------------------------------------------------/
	29
	30	#include "fibrand.h"
	31	#include "mp.h"
	32	#include "mpmont.h"
	33	#include "mprand.h"
	34	#include "rho.h"
	35
	36	/----- Main code ---------------------------------------------------------/
	37
	38	/* --- @rho@ --- *
	39	*
	40	* Arguments: @rho_ctx *cc@ = pointer to the context structure
	41	* @void x, y@ = two (equal) base values (try 1)
	42	* @mp a, b@ = logs of %$x$% (see below)
	43	*
	44	* Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm
	45	* failed. (This is unlikely, though possible.)
	46	*
	47	* Use: Uses Pollard's rho algorithm to compute discrete logs in the
	48	* group %$G$% generated by %$g$%.
	49	*
	50	* The algorithm works by finding a cycle in a pseudo-random
	51	* walk. The function @ops->split@ should return an element
	52	* from %$\{\,0, 1, 2\,\}$% according to its argument, in order
	53	* to determine the walk. At each step in the walk, we know a
	54	* group element %$x \in G$% together with its representation as
	55	* a product of powers of %$g$% and $%a$% (i.e., we know that
	56	* %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
	57	*
	58	* Locating a cycle gives us a collision
	59	*
	60	* %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
	61	*
	62	* Taking logs of both sides (to base %$g$%) gives us that
	63	*
	64	* %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
	65	*
	66	* Good initial values are %$x = y = 1$% (the multiplicative
	67	* identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
	68	* If that doesn't work then start choosing more `interesting'
	69	* values.
	70	*
	71	* Note that the algorithm requires minimal space but
	72	* %$O(\sqrt{n})$% time. Don't do this on large groups,
	73	* particularly if you can find a decent factor base.
	74	*
	75	* Finally, note that this function will free the input values
	76	* when it's finished with them. This probably isn't a great
	77	* problem.
	78	*/
	79
	80	static void step(rho_ctx cc, void x, mp a, mp b)
	81	{
	82	switch (cc->ops->split(x)) {
	83	case 0:
	84	cc->ops->mul(x, cc->g, cc->c);
	85	a = mp_add(a, *a, MP_ONE);
	86	if (MP_CMP(*a, >=, cc->n))
	87	a = mp_sub(a, *a, cc->n);
	88	break;
	89	case 1:
	90	cc->ops->sqr(x, cc->c);
	91	a = mp_lsl(a, *a, 1);
92	if (MP_CMP(*a, >=, cc->n))
93	a = mp_sub(a, *a, cc->n);
94	b = mp_lsl(b, *b, 1);
95	if (MP_CMP(*b, >=, cc->n))
96	b = mp_sub(b, *b, cc->n);
97	break;
98	case 2:
99	cc->ops->mul(x, cc->a, cc->c);
100	b = mp_add(b, *b, MP_ONE);
101	if (MP_CMP(*b, >=, cc->n))
102	b = mp_sub(b, *b, cc->n);
103	break;
104	}
105	}
106
107	mp rho(rho_ctx cc, void x, void y, mp a, mp b)
108	{
109	mp aa = MP_COPY(a), bb = MP_COPY(b);
110	mp *g;
111
112	/* --- Grind through the random walk until we find a collision --- */
113
114	do {
115	step(cc, x, &a, &b);
116	step(cc, y, &aa, &bb);
117	step(cc, y, &aa, &bb);
118	} while (!cc->ops->eq(x, y));
119	cc->ops->drop(x);
120	cc->ops->drop(y);
121
122	/* --- Now sort out the mess --- */
123
124	aa = mp_sub(aa, a, aa);
125	bb = mp_sub(bb, bb, b);
126	g = MP_NEW;
127	mp_gcd(&g, &bb, 0, bb, cc->n);
22bab86c	128	if (!MP_EQ(g, MP_ONE)) {
f41f820e	129	mp_drop(aa);
	130	aa = 0;
	131	} else {
	132	aa = mp_mul(aa, aa, bb);
	133	mp_div(0, &aa, aa, cc->n);
	134	}
	135
	136	/* --- Done --- */
	137
	138	mp_drop(bb);
	139	mp_drop(g);
	140	mp_drop(a);
	141	mp_drop(b);
	142	return (aa);
	143	}
	144
	145	/* --- @rho_prime@ --- *
	146	*
	147	* Arguments: @mp *g@ = generator for the group
	148	* @mp *a@ = value to find the logarithm of
	149	* @mp *n@ = order of the group
	150	* @mp *p@ = prime size of the underlying prime field
	151	*
	152	* Returns: The discrete logarithm %$\log_g a$%.
	153	*
	154	* Use: Computes discrete logarithms in a subgroup of a prime field.
	155	*/
	156
	157	static void prime_sqr(void x, void c)
	158	{
	159	mp **p = x;
	160	mp a = p;
	161	a = mp_sqr(a, a);
	162	a = mpmont_reduce(c, a, a);
	163	*p = a;
	164	}
	165
	166	static void prime_mul(void x, void y, void *c)
	167	{
	168	mp **p = x;
	169	mp a = p;
	170	a = mpmont_mul(c, a, a, y);
	171	*p = a;
	172	}
	173
	174	static int prime_eq(void x, void y)
	175	{
22bab86c	176	return (MP_EQ((mp )x, (mp **)y));
f41f820e	177	}
	178
	179	static int prime_split(void *x)
	180	{
	181	/* --- Notes on the splitting function --- *
	182	*
	183	* The objective is to produce a simple pseudorandom mapping from the
	184	* underlying field \gf{p} to \{\,0, 1, 2\,\}$%. This is further
	185	* constrained by the fact that we must not have %$1 \mapsto 1$% (since
	186	* otherwise the stepping function above will loop).
	187	*
	188	* The function we choose is very simple: we take the least significant
	189	* word from the integer, add one (to prevent the %$1 \mapsto 1$% property
	190	* described above) and reduce modulo 3. This is slightly biased against
	191	* the result 2, but this doesn't appear to be relevant.
	192	*/
	193
	194	return ((((mp *)x)->v[0] + 1) % 3);
	195	}
	196
	197	static void prime_drop(void *x)
	198	{
	199	MP_DROP((mp *)x);
	200	}
	201
4e66da02	202	static const rho_ops prime_ops = {
f41f820e	203	prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
	204	};
	205
	206	mp rho_prime(mp g, mp a, mp n, mp *p)
	207	{
	208	rho_ctx cc;
	209	grand *r = 0;
	210	mpmont mm;
	211	mp x, y;
	212	mp aa, bb;
	213	mp *l;
	214
	215	/* --- Initialization --- */
	216
	217	mpmont_create(&mm, p);
	218	cc.ops = &prime_ops;
	219	cc.c = &mm;
	220	cc.n = n;
	221	cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
	222	cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
	223	x = MP_COPY(mm.r);
	224	y = MP_COPY(x);
	225	aa = bb = MP_ZERO;
	226
	227	/* --- The main loop --- */
	228
	229	while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
b0b682aa	230	mp_expfactor f[2];
f41f820e	231
	232	if (!r)
	233	r = fibrand_create(0);
	234	aa = mprand_range(MP_NEW, n, r, 0);
	235	bb = mprand_range(MP_NEW, n, r, 0);
b0b682aa	236	f[0].base = cc.g; f[0].exp = aa;
b0b682aa	237	f[1].base = cc.a; f[1].exp = bb;
f41f820e	238	x = mpmont_mexpr(&mm, MP_NEW, f, 2);
	239	y = MP_COPY(x);
	240	}
	241
	242	/* --- Throw everything away now --- */
	243
	244	if (r)
	245	r->ops->destroy(r);
	246	mp_drop(cc.g);
	247	mp_drop(cc.a);
	248	mpmont_destroy(&mm);
	249	return (l);
	250	}
	251
	252	/----- Test rig ----------------------------------------------------------/
	253
	254	#ifdef TEST_RIG
	255
	256	#include <stdio.h>
	257
	258	#include "dh.h"
	259
	260	int main(void)
	261	{
	262	dh_param dp;
	263	mp x, y;
	264	grand *r = fibrand_create(0);
	265	mpmont mm;
	266	mp *l;
	267	int ok;
	268
	269	fputs("rho: ", stdout);
	270	fflush(stdout);
	271
	272	dh_gen(&dp, 32, 256, 0, r, pgen_evspin, 0);
	273	x = mprand_range(MP_NEW, dp.q, r, 0);
	274	mpmont_create(&mm, dp.p);
	275	y = mpmont_exp(&mm, MP_NEW, dp.g, x);
	276	mpmont_destroy(&mm);
	277	l = rho_prime(dp.g, y, dp.q, dp.p);
22bab86c	278	if (MP_EQ(x, l)) {
f41f820e	279	fputs(". ok\n", stdout);
	280	ok = 1;
	281	} else {
	282	fputs("\n*** rho (discrete logs) failed\n", stdout);
	283	ok = 0;
	284	}
	285
	286	mp_drop(l);
	287	mp_drop(x);
	288	mp_drop(y);
	289	r->ops->destroy(r);
	290	dh_paramfree(&dp);
	291	assert(mparena_count(MPARENA_GLOBAL) == 0);
	292
	293	return (ok ? 0 : EXIT_FAILURE);
	294	}
	295
	296	#endif
	297
	298	/----- That's all, folks -------------------------------------------------/