3 * $Id: rho.c,v 1.4 2004/04/02 01:03:49 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
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19 * Catacomb is distributed in the hope that it will be useful,
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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 1.4 2004/04/02 01:03:49 mdw
34 * Miscellaneous constification.
36 * Revision 1.3 2001/06/16 12:56:38 mdw
37 * Fixes for interface change to @mpmont_expr@ and @mpmont_mexpr@.
39 * Revision 1.2 2000/10/08 12:11:22 mdw
40 * Use @MP_EQ@ instead of @MP_CMP@.
42 * Revision 1.1 2000/07/09 21:32:30 mdw
43 * Pollard's rho algorithm for computing discrete logs.
47 /*----- Header files ------------------------------------------------------*/
55 /*----- Main code ---------------------------------------------------------*/
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)
63 * Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm
64 * failed. (This is unlikely, though possible.)
66 * Use: Uses Pollard's rho algorithm to compute discrete logs in the
67 * group %$G$% generated by %$g$%.
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 %$\alpha$%, %$\beta$%).
77 * Locating a cycle gives us a collision
79 * %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
81 * Taking logs of both sides (to base %$g$%) gives us that
83 * %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
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'
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.
94 * Finally, note that this function will free the input values
95 * when it's finished with them. This probably isn't a great
99 static void step(rho_ctx *cc, void *x, mp **a, mp **b)
101 switch (cc->ops->split(x)) {
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);
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);
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);
126 mp *rho(rho_ctx *cc, void *x, void *y, mp *a, mp *b)
128 mp *aa = MP_COPY(a), *bb = MP_COPY(b);
131 /* --- Grind through the random walk until we find a collision --- */
135 step(cc, y, &aa, &bb);
136 step(cc, y, &aa, &bb);
137 } while (!cc->ops->eq(x, y));
141 /* --- Now sort out the mess --- */
143 aa = mp_sub(aa, a, aa);
144 bb = mp_sub(bb, bb, b);
146 mp_gcd(&g, &bb, 0, bb, cc->n);
147 if (!MP_EQ(g, MP_ONE)) {
151 aa = mp_mul(aa, aa, bb);
152 mp_div(0, &aa, aa, cc->n);
164 /* --- @rho_prime@ --- *
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
171 * Returns: The discrete logarithm %$\log_g a$%.
173 * Use: Computes discrete logarithms in a subgroup of a prime field.
176 static void prime_sqr(void *x, void *c)
181 a = mpmont_reduce(c, a, a);
185 static void prime_mul(void *x, void *y, void *c)
189 a = mpmont_mul(c, a, a, y);
193 static int prime_eq(void *x, void *y)
195 return (MP_EQ(*(mp **)x, *(mp **)y));
198 static int prime_split(void *x)
200 /* --- Notes on the splitting function --- *
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).
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.
213 return (((*(mp **)x)->v[0] + 1) % 3);
216 static void prime_drop(void *x)
221 static const rho_ops prime_ops = {
222 prime_sqr, prime_mul, prime_eq, prime_split, prime_drop
225 mp *rho_prime(mp *g, mp *a, mp *n, mp *p)
234 /* --- Initialization --- */
236 mpmont_create(&mm, p);
240 cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2);
241 cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2);
246 /* --- The main loop --- */
248 while ((l = rho(&cc, &x, &y, aa, bb)) == 0) {
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);
261 /* --- Throw everything away now --- */
271 /*----- Test rig ----------------------------------------------------------*/
283 grand *r = fibrand_create(0);
288 fputs("rho: ", stdout);
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);
296 l = rho_prime(dp.g, y, dp.q, dp.p);
298 fputs(". ok\n", stdout);
301 fputs("\n*** rho (discrete logs) failed\n", stdout);
310 assert(mparena_count(MPARENA_GLOBAL) == 0);
312 return (ok ? 0 : EXIT_FAILURE);
317 /*----- That's all, folks -------------------------------------------------*/