3 * $Id: rho.c,v 1.2 2000/10/08 12:11:22 mdw Exp $
5 * Pollard's rho algorithm for discrete logs
7 * (c) 2000 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
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.
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.
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,
30 /*----- Revision history --------------------------------------------------*
33 * Revision 1.2 2000/10/08 12:11:22 mdw
34 * Use @MP_EQ@ instead of @MP_CMP@.
36 * Revision 1.1 2000/07/09 21:32:30 mdw
37 * Pollard's rho algorithm for computing discrete logs.
41 /*----- Header files ------------------------------------------------------*/
49 /*----- Main code ---------------------------------------------------------*/
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)
57 * Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm
58 * failed. (This is unlikely, though possible.)
60 * Use: Uses Pollard's rho algorithm to compute discrete logs in the
61 * group %$G$% generated by %$g$%.
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 %$\alpha$%, %$\beta$%).
71 * Locating a cycle gives us a collision
73 * %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
75 * Taking logs of both sides (to base %$g$%) gives us that
77 * %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
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'
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.
88 * Finally, note that this function will free the input values
89 * when it's finished with them. This probably isn't a great
93 static void step(rho_ctx *cc, void *x, mp **a, mp **b)
95 switch (cc->ops->split(x)) {
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);
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);
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);
120 mp *rho(rho_ctx *cc, void *x, void *y, mp *a, mp *b)
122 mp *aa = MP_COPY(a), *bb = MP_COPY(b);
125 /* --- Grind through the random walk until we find a collision --- */
129 step(cc, y, &aa, &bb);
130 step(cc, y, &aa, &bb);
131 } while (!cc->ops->eq(x, y));
135 /* --- Now sort out the mess --- */
137 aa = mp_sub(aa, a, aa);
138 bb = mp_sub(bb, bb, b);
140 mp_gcd(&g, &bb, 0, bb, cc->n);
141 if (!MP_EQ(g, MP_ONE)) {
145 aa = mp_mul(aa, aa, bb);
146 mp_div(0, &aa, aa, cc->n);
158 /* --- @rho_prime@ --- *
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
165 * Returns: The discrete logarithm %$\log_g a$%.
167 * Use: Computes discrete logarithms in a subgroup of a prime field.
170 static void prime_sqr(void *x, void *c)
175 a = mpmont_reduce(c, a, a);
179 static void prime_mul(void *x, void *y, void *c)
183 a = mpmont_mul(c, a, a, y);
187 static int prime_eq(void *x, void *y)
189 return (MP_EQ(*(mp **)x, *(mp **)y));
192 static int prime_split(void *x)
194 /* --- Notes on the splitting function --- *
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).
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.
207 return (((*(mp **)x)->v[0] + 1) % 3);
210 static void prime_drop(void *x)
215 static rho_ops prime_ops = {
216 prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
219 mp *rho_prime(mp *g, mp *a, mp *n, mp *p)
228 /* --- Initialization --- */
230 mpmont_create(&mm, p);
234 cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
235 cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
240 /* --- The main loop --- */
242 while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
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);
255 /* --- Throw everything away now --- */
265 /*----- Test rig ----------------------------------------------------------*/
277 grand *r = fibrand_create(0);
282 fputs("rho: ", stdout);
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);
290 l = rho_prime(dp.g, y, dp.q, dp.p);
292 fputs(". ok\n", stdout);
295 fputs("\n*** rho (discrete logs) failed\n", stdout);
304 assert(mparena_count(MPARENA_GLOBAL) == 0);
306 return (ok ? 0 : EXIT_FAILURE);
311 /*----- That's all, folks -------------------------------------------------*/