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d3409d5e | 1 | /* -*-c-*- |
2 | * | |
d3409d5e | 3 | * Extended GCD calculation |
4 | * | |
5 | * (c) 1999 Straylight/Edgeware | |
6 | */ | |
7 | ||
45c0fd36 | 8 | /*----- Licensing notice --------------------------------------------------* |
d3409d5e | 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 | * |
d3409d5e | 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 | * |
d3409d5e | 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 | ||
d3409d5e | 28 | /*----- Header files ------------------------------------------------------*/ |
29 | ||
30 | #include "mp.h" | |
31 | ||
32 | /*----- Main code ---------------------------------------------------------*/ | |
33 | ||
34 | /* --- @mp_gcd@ --- * | |
35 | * | |
36 | * Arguments: @mp **gcd, **xx, **yy@ = where to write the results | |
37 | * @mp *a, *b@ = sources (must be nonzero) | |
38 | * | |
39 | * Returns: --- | |
40 | * | |
41 | * Use: Calculates @gcd(a, b)@, and two numbers @x@ and @y@ such that | |
42 | * @ax + by = gcd(a, b)@. This is useful for computing modular | |
ef5f4810 | 43 | * inverses. |
d3409d5e | 44 | */ |
45 | ||
46 | void mp_gcd(mp **gcd, mp **xx, mp **yy, mp *a, mp *b) | |
47 | { | |
b2253c70 | 48 | mp *x = MP_ONE, *X = MP_ZERO; |
49 | mp *y = MP_ZERO, *Y = MP_ONE; | |
d3409d5e | 50 | mp *u, *v; |
3d6115d1 | 51 | mp *q = MP_NEW, *t, *spare = MP_NEW; |
ef5f4810 | 52 | unsigned f = 0; |
53 | ||
ceb3f0c0 | 54 | #define f_swap 1u |
55 | #define f_aneg 2u | |
56 | #define f_bneg 4u | |
57 | #define f_ext 8u | |
ef5f4810 | 58 | |
59 | /* --- Sort out some initial flags --- */ | |
d3409d5e | 60 | |
ef5f4810 | 61 | if (xx || yy) |
62 | f |= f_ext; | |
d3409d5e | 63 | |
a69a3efd | 64 | if (MP_NEGP(a)) |
ef5f4810 | 65 | f |= f_aneg; |
a69a3efd | 66 | if (MP_NEGP(b)) |
ef5f4810 | 67 | f |= f_bneg; |
68 | ||
69 | /* --- Ensure that @a@ is larger than @b@ --- * | |
70 | * | |
71 | * Use absolute values here! | |
72 | */ | |
73 | ||
74 | if (MPX_UCMP(a->v, a->vl, <, b->v, b->vl)) { | |
3d6115d1 | 75 | t = a; a = b; b = t; |
ef5f4810 | 76 | f |= f_swap; |
77 | } | |
78 | ||
79 | /* --- Check for zeroness --- */ | |
80 | ||
3d6115d1 | 81 | if (MP_ZEROP(b)) { |
ef5f4810 | 82 | |
83 | /* --- Store %$|a|$% as the GCD --- */ | |
84 | ||
85 | if (gcd) { | |
86 | if (*gcd) MP_DROP(*gcd); | |
87 | a = MP_COPY(a); | |
a69a3efd | 88 | if (MP_NEGP(a)) { |
ef5f4810 | 89 | MP_SPLIT(a); |
90 | a->f &= ~MP_NEG; | |
91 | f |= f_aneg; | |
92 | } | |
93 | *gcd = a; | |
94 | } | |
95 | ||
96 | /* --- Store %$1$% and %$0$% in the appropriate bins --- */ | |
97 | ||
98 | if (f & f_ext) { | |
99 | if (f & f_swap) { | |
3d6115d1 | 100 | mp **tt = xx; xx = yy; yy = tt; |
ef5f4810 | 101 | } |
102 | if (xx) { | |
103 | if (*xx) MP_DROP(*xx); | |
b2253c70 | 104 | if (MP_EQ(a, MP_ZERO)) |
ef5f4810 | 105 | *xx = MP_ZERO; |
106 | else if (f & f_aneg) | |
107 | *xx = MP_MONE; | |
108 | else | |
109 | *xx = MP_ONE; | |
110 | } | |
111 | if (yy) { | |
112 | if (*yy) MP_DROP(*yy); | |
113 | *yy = MP_ZERO; | |
114 | } | |
115 | } | |
116 | return; | |
d3409d5e | 117 | } |
118 | ||
cf76bcbb | 119 | /* --- Force the signs on the arguments and take copies --- */ |
d3409d5e | 120 | |
121 | a = MP_COPY(a); | |
122 | b = MP_COPY(b); | |
123 | ||
ef5f4810 | 124 | MP_SPLIT(a); a->f &= ~MP_NEG; |
125 | MP_SPLIT(b); b->f &= ~MP_NEG; | |
126 | ||
d3409d5e | 127 | u = MP_COPY(a); |
128 | v = MP_COPY(b); | |
129 | ||
cf76bcbb MW |
130 | /* --- Main extended Euclidean algorithm --- */ |
131 | ||
a69a3efd | 132 | while (!MP_ZEROP(v)) { |
b2253c70 | 133 | mp_div(&q, &u, u, v); |
134 | if (f & f_ext) { | |
3d6115d1 | 135 | t = mp_mul(spare, X, q); |
b2253c70 | 136 | t = mp_sub(t, x, t); |
3d6115d1 MW |
137 | spare = x; x = X; X = t; |
138 | t = mp_mul(spare, Y, q); | |
b2253c70 | 139 | t = mp_sub(t, y, t); |
3d6115d1 | 140 | spare = y; y = Y; Y = t; |
d3409d5e | 141 | } |
b2253c70 | 142 | t = u; u = v; v = t; |
d3409d5e | 143 | } |
144 | ||
3d6115d1 | 145 | MP_DROP(q); if (spare) MP_DROP(spare); |
ef5f4810 | 146 | if (!gcd) |
b2253c70 | 147 | MP_DROP(u); |
ef5f4810 | 148 | else { |
149 | if (*gcd) MP_DROP(*gcd); | |
b2253c70 | 150 | u->f &= ~MP_NEG; |
151 | *gcd = u; | |
ef5f4810 | 152 | } |
d3409d5e | 153 | |
154 | /* --- Perform a little normalization --- * | |
155 | * | |
156 | * Ensure that the coefficient returned is positive, if there is only one. | |
ef5f4810 | 157 | * If there are two, favour @y@. Of course, if the original arguments were |
158 | * negative then I'll need to twiddle their signs as well. | |
d3409d5e | 159 | */ |
160 | ||
ef5f4810 | 161 | if (f & f_ext) { |
162 | ||
163 | /* --- If @a@ and @b@ got swapped, swap the coefficients back --- */ | |
164 | ||
165 | if (f & f_swap) { | |
3d6115d1 | 166 | t = x; x = y; y = t; |
da83a561 | 167 | t = a; a = b; b = t; |
d3409d5e | 168 | } |
ef5f4810 | 169 | |
170 | /* --- Sort out the signs --- * | |
171 | * | |
172 | * Note that %$ax + by = a(x - b) + b(y + a)$%. | |
c9110b35 | 173 | * |
174 | * This is currently bodgy. It needs sorting out at some time. | |
ef5f4810 | 175 | */ |
176 | ||
d3409d5e | 177 | if (yy) { |
a69a3efd | 178 | if (MP_NEGP(y)) { |
c9110b35 | 179 | do { |
180 | y = mp_add(y, y, a); | |
181 | x = mp_sub(x, x, b); | |
a69a3efd | 182 | } while (MP_NEGP(y)); |
c9110b35 | 183 | } else { |
184 | while (MP_CMP(y, >=, a)) { | |
185 | y = mp_sub(y, y, a); | |
186 | x = mp_add(x, x, b); | |
187 | } | |
d3409d5e | 188 | } |
c9110b35 | 189 | } else { |
a69a3efd | 190 | if (MP_NEGP(x)) { |
c9110b35 | 191 | do |
192 | x = mp_add(x, x, b); | |
a69a3efd | 193 | while (MP_NEGP(x)); |
c9110b35 | 194 | } else { |
195 | while (MP_CMP(x, >=, b)) | |
196 | x = mp_sub(x, x, b); | |
197 | } | |
198 | } | |
d3409d5e | 199 | |
ef5f4810 | 200 | /* --- Twiddle the signs --- */ |
201 | ||
202 | if (f & f_aneg) | |
203 | x->f ^= MP_NEG; | |
204 | if (f & f_bneg) | |
205 | y->f ^= MP_NEG; | |
206 | ||
207 | /* --- Store the results --- */ | |
208 | ||
209 | if (!xx) | |
210 | MP_DROP(x); | |
211 | else { | |
212 | if (*xx) MP_DROP(*xx); | |
213 | *xx = x; | |
214 | } | |
215 | ||
216 | if (!yy) | |
217 | MP_DROP(y); | |
218 | else { | |
219 | if (*yy) MP_DROP(*yy); | |
220 | *yy = y; | |
221 | } | |
d3409d5e | 222 | } |
223 | ||
b2253c70 | 224 | MP_DROP(v); |
d3409d5e | 225 | MP_DROP(X); MP_DROP(Y); |
226 | MP_DROP(a); MP_DROP(b); | |
227 | } | |
228 | ||
b817bfc6 | 229 | /* -- @mp_modinv@ --- * |
230 | * | |
231 | * Arguments: @mp *d@ = destination | |
232 | * @mp *x@ = argument | |
233 | * @mp *p@ = modulus | |
234 | * | |
235 | * Returns: The inverse %$x^{-1} \bmod p$%. | |
236 | * | |
237 | * Use: Computes a modular inverse. An assertion fails if %$p$% | |
238 | * has no inverse. | |
239 | */ | |
240 | ||
241 | mp *mp_modinv(mp *d, mp *x, mp *p) | |
242 | { | |
243 | mp *g = MP_NEW; | |
244 | mp_gcd(&g, 0, &d, p, x); | |
245 | assert(MP_EQ(g, MP_ONE)); | |
246 | mp_drop(g); | |
247 | return (d); | |
248 | } | |
249 | ||
d3409d5e | 250 | /*----- Test rig ----------------------------------------------------------*/ |
251 | ||
252 | #ifdef TEST_RIG | |
253 | ||
7c404803 MW |
254 | static int modinv(dstr *v) |
255 | { | |
256 | int ok = 1; | |
257 | mp *x = *(mp **)v[0].buf; | |
258 | mp *m = *(mp **)v[1].buf; | |
259 | mp *r = *(mp **)v[2].buf; | |
260 | ||
261 | mp *y = mp_modinv(MP_NEW, x, m); | |
262 | if (!MP_EQ(y, r)) { | |
263 | fputs("\n*** mp_modinv failed", stderr); | |
45c0fd36 MW |
264 | fputs("\nx = ", stderr); mp_writefile(x, stderr, 10); |
265 | fputs("\nm = ", stderr); mp_writefile(m, stderr, 10); | |
7c404803 MW |
266 | fputs("\nexpect = ", stderr); mp_writefile(r, stderr, 10); |
267 | fputs("\nresult = ", stderr); mp_writefile(y, stderr, 10); | |
268 | ok = 0; | |
269 | } | |
270 | MP_DROP(x); MP_DROP(m); MP_DROP(r); MP_DROP(y); | |
271 | assert(mparena_count(MPARENA_GLOBAL) == 0); | |
45c0fd36 | 272 | return (ok); |
7c404803 MW |
273 | } |
274 | ||
d3409d5e | 275 | static int gcd(dstr *v) |
276 | { | |
277 | int ok = 1; | |
278 | mp *a = *(mp **)v[0].buf; | |
279 | mp *b = *(mp **)v[1].buf; | |
280 | mp *g = *(mp **)v[2].buf; | |
281 | mp *x = *(mp **)v[3].buf; | |
282 | mp *y = *(mp **)v[4].buf; | |
283 | ||
ef5f4810 | 284 | mp *gg = MP_NEW, *xx = MP_NEW, *yy = MP_NEW; |
d3409d5e | 285 | mp_gcd(&gg, &xx, &yy, a, b); |
b2253c70 | 286 | if (!MP_EQ(x, xx)) { |
d3409d5e | 287 | fputs("\n*** mp_gcd(x) failed", stderr); |
45c0fd36 MW |
288 | fputs("\na = ", stderr); mp_writefile(a, stderr, 10); |
289 | fputs("\nb = ", stderr); mp_writefile(b, stderr, 10); | |
d3409d5e | 290 | fputs("\nexpect = ", stderr); mp_writefile(x, stderr, 10); |
291 | fputs("\nresult = ", stderr); mp_writefile(xx, stderr, 10); | |
292 | fputc('\n', stderr); | |
293 | ok = 0; | |
294 | } | |
b2253c70 | 295 | if (!MP_EQ(y, yy)) { |
d3409d5e | 296 | fputs("\n*** mp_gcd(y) failed", stderr); |
45c0fd36 MW |
297 | fputs("\na = ", stderr); mp_writefile(a, stderr, 10); |
298 | fputs("\nb = ", stderr); mp_writefile(b, stderr, 10); | |
d3409d5e | 299 | fputs("\nexpect = ", stderr); mp_writefile(y, stderr, 10); |
300 | fputs("\nresult = ", stderr); mp_writefile(yy, stderr, 10); | |
301 | fputc('\n', stderr); | |
302 | ok = 0; | |
303 | } | |
304 | ||
305 | if (!ok) { | |
306 | mp *ax = mp_mul(MP_NEW, a, xx); | |
307 | mp *by = mp_mul(MP_NEW, b, yy); | |
308 | ax = mp_add(ax, ax, by); | |
b2253c70 | 309 | if (MP_EQ(ax, gg)) |
d3409d5e | 310 | fputs("\n*** (Alternative result found.)\n", stderr); |
311 | MP_DROP(ax); | |
312 | MP_DROP(by); | |
313 | } | |
314 | ||
b2253c70 | 315 | if (!MP_EQ(g, gg)) { |
d3409d5e | 316 | fputs("\n*** mp_gcd(gcd) failed", stderr); |
45c0fd36 MW |
317 | fputs("\na = ", stderr); mp_writefile(a, stderr, 10); |
318 | fputs("\nb = ", stderr); mp_writefile(b, stderr, 10); | |
d3409d5e | 319 | fputs("\nexpect = ", stderr); mp_writefile(g, stderr, 10); |
320 | fputs("\nresult = ", stderr); mp_writefile(gg, stderr, 10); | |
321 | fputc('\n', stderr); | |
322 | ok = 0; | |
323 | } | |
324 | MP_DROP(a); MP_DROP(b); MP_DROP(g); MP_DROP(x); MP_DROP(y); | |
325 | MP_DROP(gg); MP_DROP(xx); MP_DROP(yy); | |
ef5f4810 | 326 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
d3409d5e | 327 | return (ok); |
328 | } | |
329 | ||
330 | static test_chunk tests[] = { | |
331 | { "gcd", gcd, { &type_mp, &type_mp, &type_mp, &type_mp, &type_mp, 0 } }, | |
7c404803 | 332 | { "modinv", modinv, { &type_mp, &type_mp, &type_mp, 0 } }, |
d3409d5e | 333 | { 0, 0, { 0 } } |
334 | }; | |
335 | ||
336 | int main(int argc, char *argv[]) | |
337 | { | |
338 | sub_init(); | |
0f00dc4c | 339 | test_run(argc, argv, tests, SRCDIR "/t/mp"); |
d3409d5e | 340 | return (0); |
341 | } | |
342 | ||
343 | #endif | |
344 | ||
345 | /*----- That's all, folks -------------------------------------------------*/ |