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Add cyclic group abstraction, with test code. Separate off exponentation
[u/mdw/catacomb] / ec-prime.c
CommitLineData
b0ab12e6 1/* -*-c-*-
2 *
34e4f738 3 * $Id: ec-prime.c,v 1.9 2004/04/01 12:50:09 mdw Exp $
b0ab12e6 4 *
5 * Elliptic curves over prime fields
6 *
7 * (c) 2001 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: ec-prime.c,v $
34e4f738 33 * Revision 1.9 2004/04/01 12:50:09 mdw
34 * Add cyclic group abstraction, with test code. Separate off exponentation
35 * functions for better static linking. Fix a buttload of bugs on the way.
36 * Generally ensure that negative exponents do inversion correctly. Add
37 * table of standard prime-field subgroups. (Binary field subgroups are
38 * currently unimplemented but easy to add if anyone ever finds a good one.)
39 *
432c4e18 40 * Revision 1.8 2004/03/27 17:54:11 mdw
41 * Standard curves and curve checking.
42 *
f46efa79 43 * Revision 1.7 2004/03/27 00:04:46 mdw
44 * Implement efficient reduction for pleasant-looking primes.
45 *
bc985cef 46 * Revision 1.6 2004/03/23 15:19:32 mdw
47 * Test elliptic curves more thoroughly.
48 *
391faf42 49 * Revision 1.5 2004/03/22 02:19:10 mdw
50 * Rationalise the sliding-window threshold. Drop guarantee that right
51 * arguments to EC @add@ are canonical, and fix up projective implementations
52 * to cope.
53 *
c3caa2fa 54 * Revision 1.4 2004/03/21 22:52:06 mdw
55 * Merge and close elliptic curve branch.
56 *
ceb3f0c0 57 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
58 * Elliptic curves on binary fields work.
59 *
8823192f 60 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
61 * Projective coordinates for prime curves
62 *
dbfee00a 63 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
64 * Simple (non-projective) curves over prime fields now seem to work.
65 *
41cb1beb 66 * Revision 1.3 2003/05/15 23:25:59 mdw
67 * Make elliptic curve stuff build.
68 *
b085fd91 69 * Revision 1.2 2002/01/13 13:48:44 mdw
70 * Further progress.
71 *
b0ab12e6 72 * Revision 1.1 2001/04/29 18:12:33 mdw
73 * Prototype version.
74 *
75 */
76
77/*----- Header files ------------------------------------------------------*/
78
41cb1beb 79#include <mLib/sub.h>
80
b0ab12e6 81#include "ec.h"
82
dbfee00a 83/*----- Simple prime curves -----------------------------------------------*/
b0ab12e6 84
8823192f 85static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
41cb1beb 86
87static ec *ecneg(ec_curve *c, ec *d, const ec *p)
b085fd91 88{
89 EC_COPY(d, p);
ceb3f0c0 90 if (d->y)
91 d->y = F_NEG(c->f, d->y, d->y);
b085fd91 92 return (d);
93}
94
8823192f 95static ec *ecfind(ec_curve *c, ec *d, mp *x)
96{
97 mp *p, *q;
8823192f 98 field *f = c->f;
99
100 q = F_SQR(f, MP_NEW, x);
101 p = F_MUL(f, MP_NEW, x, q);
432c4e18 102 q = F_MUL(f, q, x, c->a);
8823192f 103 p = F_ADD(f, p, p, q);
432c4e18 104 p = F_ADD(f, p, p, c->b);
8823192f 105 MP_DROP(q);
106 p = F_SQRT(f, p, p);
107 if (!p)
108 return (0);
109 EC_DESTROY(d);
110 d->x = MP_COPY(x);
111 d->y = p;
112 d->z = MP_COPY(f->one);
b085fd91 113 return (d);
114}
115
116static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
b0ab12e6 117{
b085fd91 118 if (EC_ATINF(a))
119 EC_SETINF(d);
8823192f 120 else if (F_ZEROP(c->f, a->y))
b085fd91 121 EC_COPY(d, a);
122 else {
123 field *f = c->f;
b085fd91 124 mp *lambda;
125 mp *dy, *dx;
126
8823192f 127 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
128 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
129 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
432c4e18 130 dx = F_ADD(f, dx, dx, c->a); /* %$3 x^2 + A$% */
8823192f 131 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
132 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
b085fd91 133
8823192f 134 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
135 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
136 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
137 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
138 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
139 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
b0ab12e6 140
b085fd91 141 EC_DESTROY(d);
142 d->x = dx;
143 d->y = dy;
144 d->z = 0;
145 MP_DROP(lambda);
146 }
147 return (d);
148}
149
8823192f 150static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
151{
152 if (EC_ATINF(a))
153 EC_SETINF(d);
154 else if (F_ZEROP(c->f, a->y))
155 EC_COPY(d, a);
156 else {
157 field *f = c->f;
8823192f 158 mp *p, *q, *m, *s, *dx, *dy, *dz;
159
160 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
161 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
432c4e18 162 p = F_MUL(f, p, q, c->a); /* %$A z^4$% */
8823192f 163 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
164 m = F_TPL(f, m, m); /* %$3 x^2$% */
165 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
166
167 q = F_DBL(f, q, a->y); /* %$2 y$% */
168 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
169
170 p = F_SQR(f, p, q); /* %$4 y^2$% */
171 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
172 q = F_SQR(f, q, p); /* %$16 y^4$% */
173 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
174
175 p = F_DBL(f, p, s); /* %$2 s$% */
176 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
177 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
178
179 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
180 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
181 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
182
183 EC_DESTROY(d);
184 d->x = dx;
185 d->y = dy;
186 d->z = dz;
187 MP_DROP(m);
188 MP_DROP(q);
189 MP_DROP(s);
190 }
191 return (d);
192}
193
194static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
195{
196 if (EC_ATINF(a))
197 EC_SETINF(d);
198 else if (F_ZEROP(c->f, a->y))
199 EC_COPY(d, a);
200 else {
201 field *f = c->f;
202 mp *p, *q, *m, *s, *dx, *dy, *dz;
203
204 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
205 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
206 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
207 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
208 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
209
210 q = F_DBL(f, q, a->y); /* %$2 y$% */
211 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
212
213 p = F_SQR(f, p, q); /* %$4 y^2$% */
214 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
215 q = F_SQR(f, q, p); /* %$16 y^4$% */
216 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
217
218 p = F_DBL(f, p, s); /* %$2 s$% */
219 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
220 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
221
222 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
223 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
224 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
225
226 EC_DESTROY(d);
227 d->x = dx;
228 d->y = dy;
229 d->z = dz;
230 MP_DROP(m);
231 MP_DROP(q);
232 MP_DROP(s);
233 }
234 return (d);
235}
236
b085fd91 237static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
238{
b0ab12e6 239 if (a == b)
240 ecdbl(c, d, a);
241 else if (EC_ATINF(a))
242 EC_COPY(d, b);
243 else if (EC_ATINF(b))
244 EC_COPY(d, a);
b085fd91 245 else {
246 field *f = c->f;
247 mp *lambda;
248 mp *dy, *dx;
249
250 if (!MP_EQ(a->x, b->x)) {
8823192f 251 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
252 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
253 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
b085fd91 254 lambda = F_MUL(f, MP_NEW, dy, dx);
8823192f 255 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
256 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
b0ab12e6 257 EC_SETINF(d);
b085fd91 258 return (d);
259 } else {
8823192f 260 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
261 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
432c4e18 262 dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */
8823192f 263 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
264 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
41cb1beb 265 lambda = F_MUL(f, MP_NEW, dx, dy);
8823192f 266 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
b085fd91 267 }
268
8823192f 269 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
270 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
271 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
272 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
273 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
ceb3f0c0 274 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
b0ab12e6 275
b085fd91 276 EC_DESTROY(d);
277 d->x = dx;
278 d->y = dy;
279 d->z = 0;
280 MP_DROP(lambda);
b0ab12e6 281 }
b085fd91 282 return (d);
b0ab12e6 283}
284
8823192f 285static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
286{
287 if (a == b)
288 c->ops->dbl(c, d, a);
289 else if (EC_ATINF(a))
290 EC_COPY(d, b);
291 else if (EC_ATINF(b))
292 EC_COPY(d, a);
293 else {
294 field *f = c->f;
391faf42 295 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
8823192f 296
297 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
298 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
299 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
300 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
301
391faf42 302 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
303 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
304 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
305 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
306
307 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
308 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
8823192f 309 if (F_ZEROP(f, w)) {
ceb3f0c0 310 MP_DROP(w);
311 MP_DROP(u);
312 MP_DROP(s);
391faf42 313 MP_DROP(uu);
314 MP_DROP(ss);
8823192f 315 if (F_ZEROP(f, r)) {
8823192f 316 MP_DROP(r);
8823192f 317 return (c->ops->dbl(c, d, a));
318 } else {
8823192f 319 MP_DROP(r);
8823192f 320 EC_SETINF(d);
321 return (d);
322 }
323 }
391faf42 324 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
325 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
8823192f 326
391faf42 327 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
328 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
8823192f 329
391faf42 330 p = F_SQR(f, uu, w); /* %$w^2$% */
8823192f 331 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
332 u = F_MUL(f, u, p, w); /* %$w^3$% */
333 p = F_MUL(f, p, u, s); /* %$m w^3$% */
334
335 dx = F_SQR(f, u, r); /* %$r^2$% */
336 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
337
338 s = F_DBL(f, s, dx); /* %$2 x'$% */
339 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
340 dy = F_MUL(f, s, q, r); /* %$v r$% */
341 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
342 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
343
344 EC_DESTROY(d);
345 d->x = dx;
346 d->y = dy;
347 d->z = dz;
348 MP_DROP(p);
349 MP_DROP(q);
350 MP_DROP(r);
351 MP_DROP(w);
352 }
353 return (d);
354}
355
356static int eccheck(ec_curve *c, const ec *p)
357{
8823192f 358 field *f = c->f;
34e4f738 359 mp *l, *x, *r;
8823192f 360 int rc;
34e4f738 361 if (EC_ATINF(p)) return (0);
362 l = F_SQR(f, MP_NEW, p->y);
363 x = F_SQR(f, MP_NEW, p->x);
364 r = F_MUL(f, MP_NEW, x, p->x);
432c4e18 365 x = F_MUL(f, x, c->a, p->x);
8823192f 366 r = F_ADD(f, r, r, x);
432c4e18 367 r = F_ADD(f, r, r, c->b);
8823192f 368 rc = MP_EQ(l, r) ? 0 : -1;
369 mp_drop(l);
370 mp_drop(x);
371 mp_drop(r);
372 return (rc);
373}
374
375static int ecprojcheck(ec_curve *c, const ec *p)
376{
377 ec t = EC_INIT;
378 int rc;
379
380 c->ops->fix(c, &t, p);
381 rc = eccheck(c, &t);
382 EC_DESTROY(&t);
383 return (rc);
384}
385
41cb1beb 386static void ecdestroy(ec_curve *c)
387{
432c4e18 388 MP_DROP(c->a);
389 MP_DROP(c->b);
390 DESTROY(c);
41cb1beb 391}
392
393/* --- @ec_prime@, @ec_primeproj@ --- *
394 *
dbfee00a 395 * Arguments: @field *f@ = the underlying field for this elliptic curve
41cb1beb 396 * @mp *a, *b@ = the coefficients for this curve
397 *
398 * Returns: A pointer to the curve.
399 *
400 * Use: Creates a curve structure for an elliptic curve defined over
401 * a prime field. The @primeproj@ variant uses projective
402 * coordinates, which can be a win.
403 */
404
405extern ec_curve *ec_prime(field *f, mp *a, mp *b)
406{
432c4e18 407 ec_curve *c = CREATE(ec_curve);
408 c->ops = &ec_primeops;
409 c->f = f;
410 c->a = F_IN(f, MP_NEW, a);
411 c->b = F_IN(f, MP_NEW, b);
412 return (c);
41cb1beb 413}
414
8823192f 415extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
416{
432c4e18 417 ec_curve *c = CREATE(ec_curve);
8823192f 418 mp *ax;
419
420 ax = mp_add(MP_NEW, a, MP_THREE);
421 ax = F_IN(f, ax, ax);
422 if (F_ZEROP(f, ax))
432c4e18 423 c->ops = &ec_primeprojxops;
8823192f 424 else
432c4e18 425 c->ops = &ec_primeprojops;
8823192f 426 MP_DROP(ax);
432c4e18 427 c->f = f;
428 c->a = F_IN(f, MP_NEW, a);
429 c->b = F_IN(f, MP_NEW, b);
430 return (c);
41cb1beb 431}
432
433static const ec_ops ec_primeops = {
34e4f738 434 ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
bc985cef 435 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
8823192f 436};
437
438static const ec_ops ec_primeprojops = {
34e4f738 439 ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
bc985cef 440 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
8823192f 441};
442
443static const ec_ops ec_primeprojxops = {
34e4f738 444 ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
bc985cef 445 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
41cb1beb 446};
447
448/*----- Test rig ----------------------------------------------------------*/
449
450#ifdef TEST_RIG
451
452#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
453
ceb3f0c0 454int main(int argc, char *argv[])
41cb1beb 455{
456 field *f;
457 ec_curve *c;
458 ec g = EC_INIT, d = EC_INIT;
459 mp *p, *a, *b, *r;
ceb3f0c0 460 int i, n = argc == 1 ? 1 : atoi(argv[1]);
41cb1beb 461
dbfee00a 462 printf("ec-prime: ");
463 fflush(stdout);
41cb1beb 464 a = MP(-3);
432c4e18 465 b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
466 p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
467 r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
41cb1beb 468
f46efa79 469 f = field_niceprime(p);
ceb3f0c0 470 c = ec_primeproj(f, a, b);
41cb1beb 471
432c4e18 472 g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
473 g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
41cb1beb 474
ceb3f0c0 475 for (i = 0; i < n; i++) {
476 ec_mul(c, &d, &g, r);
477 if (EC_ATINF(&d)) {
478 fprintf(stderr, "zero too early\n");
479 return (1);
480 }
481 ec_add(c, &d, &d, &g);
482 if (!EC_ATINF(&d)) {
483 fprintf(stderr, "didn't reach zero\n");
484 MP_EPRINT("d.x", d.x);
485 MP_EPRINT("d.y", d.y);
486 return (1);
487 }
488 ec_destroy(&d);
dbfee00a 489 }
41cb1beb 490 ec_destroy(&g);
491 ec_destroycurve(c);
492 F_DESTROY(f);
dbfee00a 493 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
494 assert(!mparena_count(&mparena_global));
495 printf("ok\n");
41cb1beb 496 return (0);
497}
498
499#endif
500
b0ab12e6 501/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)