3 * $Id: rho.c,v 1.5 2004/04/08 01:36:15 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 /*----- Header files ------------------------------------------------------*/
38 /*----- Main code ---------------------------------------------------------*/
42 * Arguments: @rho_ctx *cc@ = pointer to the context structure
43 * @void *x, *y@ = two (equal) base values (try 1)
44 * @mp *a, *b@ = logs of %$x$% (see below)
46 * Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm
47 * failed. (This is unlikely, though possible.)
49 * Use: Uses Pollard's rho algorithm to compute discrete logs in the
50 * group %$G$% generated by %$g$%.
52 * The algorithm works by finding a cycle in a pseudo-random
53 * walk. The function @ops->split@ should return an element
54 * from %$\{\,0, 1, 2\,\}$% according to its argument, in order
55 * to determine the walk. At each step in the walk, we know a
56 * group element %$x \in G$% together with its representation as
57 * a product of powers of %$g$% and $%a$% (i.e., we know that
58 * %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
60 * Locating a cycle gives us a collision
62 * %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
64 * Taking logs of both sides (to base %$g$%) gives us that
66 * %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
68 * Good initial values are %$x = y = 1$% (the multiplicative
69 * identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
70 * If that doesn't work then start choosing more `interesting'
73 * Note that the algorithm requires minimal space but
74 * %$O(\sqrt{n})$% time. Don't do this on large groups,
75 * particularly if you can find a decent factor base.
77 * Finally, note that this function will free the input values
78 * when it's finished with them. This probably isn't a great
82 static void step(rho_ctx
*cc
, void *x
, mp
**a
, mp
**b
)
84 switch (cc
->ops
->split(x
)) {
86 cc
->ops
->mul(x
, cc
->g
, cc
->c
);
87 *a
= mp_add(*a
, *a
, MP_ONE
);
88 if (MP_CMP(*a
, >=, cc
->n
))
89 *a
= mp_sub(*a
, *a
, cc
->n
);
92 cc
->ops
->sqr(x
, cc
->c
);
93 *a
= mp_lsl(*a
, *a
, 1);
94 if (MP_CMP(*a
, >=, cc
->n
))
95 *a
= mp_sub(*a
, *a
, cc
->n
);
96 *b
= mp_lsl(*b
, *b
, 1);
97 if (MP_CMP(*b
, >=, cc
->n
))
98 *b
= mp_sub(*b
, *b
, cc
->n
);
101 cc
->ops
->mul(x
, cc
->a
, cc
->c
);
102 *b
= mp_add(*b
, *b
, MP_ONE
);
103 if (MP_CMP(*b
, >=, cc
->n
))
104 *b
= mp_sub(*b
, *b
, cc
->n
);
109 mp
*rho(rho_ctx
*cc
, void *x
, void *y
, mp
*a
, mp
*b
)
111 mp
*aa
= MP_COPY(a
), *bb
= MP_COPY(b
);
114 /* --- Grind through the random walk until we find a collision --- */
118 step(cc
, y
, &aa
, &bb
);
119 step(cc
, y
, &aa
, &bb
);
120 } while (!cc
->ops
->eq(x
, y
));
124 /* --- Now sort out the mess --- */
126 aa
= mp_sub(aa
, a
, aa
);
127 bb
= mp_sub(bb
, bb
, b
);
129 mp_gcd(&g
, &bb
, 0, bb
, cc
->n
);
130 if (!MP_EQ(g
, MP_ONE
)) {
134 aa
= mp_mul(aa
, aa
, bb
);
135 mp_div(0, &aa
, aa
, cc
->n
);
147 /* --- @rho_prime@ --- *
149 * Arguments: @mp *g@ = generator for the group
150 * @mp *a@ = value to find the logarithm of
151 * @mp *n@ = order of the group
152 * @mp *p@ = prime size of the underlying prime field
154 * Returns: The discrete logarithm %$\log_g a$%.
156 * Use: Computes discrete logarithms in a subgroup of a prime field.
159 static void prime_sqr(void *x
, void *c
)
164 a
= mpmont_reduce(c
, a
, a
);
168 static void prime_mul(void *x
, void *y
, void *c
)
172 a
= mpmont_mul(c
, a
, a
, y
);
176 static int prime_eq(void *x
, void *y
)
178 return (MP_EQ(*(mp
**)x
, *(mp
**)y
));
181 static int prime_split(void *x
)
183 /* --- Notes on the splitting function --- *
185 * The objective is to produce a simple pseudorandom mapping from the
186 * underlying field \gf{p} to \{\,0, 1, 2\,\}$%. This is further
187 * constrained by the fact that we must not have %$1 \mapsto 1$% (since
188 * otherwise the stepping function above will loop).
190 * The function we choose is very simple: we take the least significant
191 * word from the integer, add one (to prevent the %$1 \mapsto 1$% property
192 * described above) and reduce modulo 3. This is slightly biased against
193 * the result 2, but this doesn't appear to be relevant.
196 return (((*(mp
**)x
)->v
[0] + 1) % 3);
199 static void prime_drop(void *x
)
204 static const rho_ops prime_ops
= {
205 prime_sqr
, prime_mul
, prime_eq
, prime_split
, prime_drop
208 mp
*rho_prime(mp
*g
, mp
*a
, mp
*n
, mp
*p
)
217 /* --- Initialization --- */
219 mpmont_create(&mm
, p
);
223 cc
.g
= mpmont_mul(&mm
, MP_NEW
, g
, mm
.r2
);
224 cc
.a
= mpmont_mul(&mm
, MP_NEW
, a
, mm
.r2
);
229 /* --- The main loop --- */
231 while ((l
= rho(&cc
, &x
, &y
, aa
, bb
)) == 0) {
235 r
= fibrand_create(0);
236 aa
= mprand_range(MP_NEW
, n
, r
, 0);
237 bb
= mprand_range(MP_NEW
, n
, r
, 0);
238 f
[0].base
= cc
.g
; f
[0].exp
= aa
;
239 f
[1].base
= cc
.a
; f
[1].exp
= bb
;
240 x
= mpmont_mexpr(&mm
, MP_NEW
, f
, 2);
244 /* --- Throw everything away now --- */
254 /*----- Test rig ----------------------------------------------------------*/
266 grand
*r
= fibrand_create(0);
271 fputs("rho: ", stdout
);
274 dh_gen(&dp
, 32, 256, 0, r
, pgen_evspin
, 0);
275 x
= mprand_range(MP_NEW
, dp
.q
, r
, 0);
276 mpmont_create(&mm
, dp
.p
);
277 y
= mpmont_exp(&mm
, MP_NEW
, dp
.g
, x
);
279 l
= rho_prime(dp
.g
, y
, dp
.q
, dp
.p
);
281 fputs(". ok\n", stdout
);
284 fputs("\n*** rho (discrete logs) failed\n", stdout
);
293 assert(mparena_count(MPARENA_GLOBAL
) == 0);
295 return (ok ?
0 : EXIT_FAILURE
);
300 /*----- That's all, folks -------------------------------------------------*/