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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: rho.c,v 1.2 2000/10/08 12:11:22 mdw Exp $ |
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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 $ |
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33 | * Revision 1.2 2000/10/08 12:11:22 mdw |
34 | * Use @MP_EQ@ instead of @MP_CMP@. |
35 | * |
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36 | * Revision 1.1 2000/07/09 21:32:30 mdw |
37 | * Pollard's rho algorithm for computing discrete logs. |
38 | * |
39 | */ |
40 | |
41 | /*----- Header files ------------------------------------------------------*/ |
42 | |
43 | #include "fibrand.h" |
44 | #include "mp.h" |
45 | #include "mpmont.h" |
46 | #include "mprand.h" |
47 | #include "rho.h" |
48 | |
49 | /*----- Main code ---------------------------------------------------------*/ |
50 | |
51 | /* --- @rho@ --- * |
52 | * |
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) |
56 | * |
57 | * Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm |
58 | * failed. (This is unlikely, though possible.) |
59 | * |
60 | * Use: Uses Pollard's rho algorithm to compute discrete logs in the |
61 | * group %$G$% generated by %$g$%. |
62 | * |
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 %$\alpha$%, %$\beta$%). |
70 | * |
71 | * Locating a cycle gives us a collision |
72 | * |
73 | * %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$% |
74 | * |
75 | * Taking logs of both sides (to base %$g$%) gives us that |
76 | * |
77 | * %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$% |
78 | * |
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' |
82 | * values. |
83 | * |
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. |
87 | * |
88 | * Finally, note that this function will free the input values |
89 | * when it's finished with them. This probably isn't a great |
90 | * problem. |
91 | */ |
92 | |
93 | static void step(rho_ctx *cc, void *x, mp **a, mp **b) |
94 | { |
95 | switch (cc->ops->split(x)) { |
96 | case 0: |
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); |
101 | break; |
102 | case 1: |
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); |
110 | break; |
111 | case 2: |
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); |
116 | break; |
117 | } |
118 | } |
119 | |
120 | mp *rho(rho_ctx *cc, void *x, void *y, mp *a, mp *b) |
121 | { |
122 | mp *aa = MP_COPY(a), *bb = MP_COPY(b); |
123 | mp *g; |
124 | |
125 | /* --- Grind through the random walk until we find a collision --- */ |
126 | |
127 | do { |
128 | step(cc, x, &a, &b); |
129 | step(cc, y, &aa, &bb); |
130 | step(cc, y, &aa, &bb); |
131 | } while (!cc->ops->eq(x, y)); |
132 | cc->ops->drop(x); |
133 | cc->ops->drop(y); |
134 | |
135 | /* --- Now sort out the mess --- */ |
136 | |
137 | aa = mp_sub(aa, a, aa); |
138 | bb = mp_sub(bb, bb, b); |
139 | g = MP_NEW; |
140 | mp_gcd(&g, &bb, 0, bb, cc->n); |
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141 | if (!MP_EQ(g, MP_ONE)) { |
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142 | mp_drop(aa); |
143 | aa = 0; |
144 | } else { |
145 | aa = mp_mul(aa, aa, bb); |
146 | mp_div(0, &aa, aa, cc->n); |
147 | } |
148 | |
149 | /* --- Done --- */ |
150 | |
151 | mp_drop(bb); |
152 | mp_drop(g); |
153 | mp_drop(a); |
154 | mp_drop(b); |
155 | return (aa); |
156 | } |
157 | |
158 | /* --- @rho_prime@ --- * |
159 | * |
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 |
164 | * |
165 | * Returns: The discrete logarithm %$\log_g a$%. |
166 | * |
167 | * Use: Computes discrete logarithms in a subgroup of a prime field. |
168 | */ |
169 | |
170 | static void prime_sqr(void *x, void *c) |
171 | { |
172 | mp **p = x; |
173 | mp *a = *p; |
174 | a = mp_sqr(a, a); |
175 | a = mpmont_reduce(c, a, a); |
176 | *p = a; |
177 | } |
178 | |
179 | static void prime_mul(void *x, void *y, void *c) |
180 | { |
181 | mp **p = x; |
182 | mp *a = *p; |
183 | a = mpmont_mul(c, a, a, y); |
184 | *p = a; |
185 | } |
186 | |
187 | static int prime_eq(void *x, void *y) |
188 | { |
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189 | return (MP_EQ(*(mp **)x, *(mp **)y)); |
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190 | } |
191 | |
192 | static int prime_split(void *x) |
193 | { |
194 | /* --- Notes on the splitting function --- * |
195 | * |
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). |
200 | * |
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. |
205 | */ |
206 | |
207 | return (((*(mp **)x)->v[0] + 1) % 3); |
208 | } |
209 | |
210 | static void prime_drop(void *x) |
211 | { |
212 | MP_DROP(*(mp **)x); |
213 | } |
214 | |
215 | static rho_ops prime_ops = { |
216 | prime_sqr, prime_mul, prime_eq, prime_split, prime_drop |
217 | }; |
218 | |
219 | mp *rho_prime(mp *g, mp *a, mp *n, mp *p) |
220 | { |
221 | rho_ctx cc; |
222 | grand *r = 0; |
223 | mpmont mm; |
224 | mp *x, *y; |
225 | mp *aa, *bb; |
226 | mp *l; |
227 | |
228 | /* --- Initialization --- */ |
229 | |
230 | mpmont_create(&mm, p); |
231 | cc.ops = &prime_ops; |
232 | cc.c = &mm; |
233 | cc.n = n; |
234 | cc.g = mpmont_mul(&mm, MP_NEW, g, mm.r2); |
235 | cc.a = mpmont_mul(&mm, MP_NEW, a, mm.r2); |
236 | x = MP_COPY(mm.r); |
237 | y = MP_COPY(x); |
238 | aa = bb = MP_ZERO; |
239 | |
240 | /* --- The main loop --- */ |
241 | |
242 | while ((l = rho(&cc, &x, &y, aa, bb)) == 0) { |
243 | mpmont_factor f[2]; |
244 | |
245 | if (!r) |
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); |
252 | y = MP_COPY(x); |
253 | } |
254 | |
255 | /* --- Throw everything away now --- */ |
256 | |
257 | if (r) |
258 | r->ops->destroy(r); |
259 | mp_drop(cc.g); |
260 | mp_drop(cc.a); |
261 | mpmont_destroy(&mm); |
262 | return (l); |
263 | } |
264 | |
265 | /*----- Test rig ----------------------------------------------------------*/ |
266 | |
267 | #ifdef TEST_RIG |
268 | |
269 | #include <stdio.h> |
270 | |
271 | #include "dh.h" |
272 | |
273 | int main(void) |
274 | { |
275 | dh_param dp; |
276 | mp *x, *y; |
277 | grand *r = fibrand_create(0); |
278 | mpmont mm; |
279 | mp *l; |
280 | int ok; |
281 | |
282 | fputs("rho: ", stdout); |
283 | fflush(stdout); |
284 | |
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); |
289 | mpmont_destroy(&mm); |
290 | l = rho_prime(dp.g, y, dp.q, dp.p); |
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291 | if (MP_EQ(x, l)) { |
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292 | fputs(". ok\n", stdout); |
293 | ok = 1; |
294 | } else { |
295 | fputs("\n*** rho (discrete logs) failed\n", stdout); |
296 | ok = 0; |
297 | } |
298 | |
299 | mp_drop(l); |
300 | mp_drop(x); |
301 | mp_drop(y); |
302 | r->ops->destroy(r); |
303 | dh_paramfree(&dp); |
304 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
305 | |
306 | return (ok ? 0 : EXIT_FAILURE); |
307 | } |
308 | |
309 | #endif |
310 | |
311 | /*----- That's all, folks -------------------------------------------------*/ |