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Test elliptic curves more thoroughly.
[u/mdw/catacomb] / ec-prime.c
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b0ab12e6 1/* -*-c-*-
2 *
bc985cef 3 * $Id: ec-prime.c,v 1.6 2004/03/23 15:19:32 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 $
bc985cef 33 * Revision 1.6 2004/03/23 15:19:32 mdw
34 * Test elliptic curves more thoroughly.
35 *
391faf42 36 * Revision 1.5 2004/03/22 02:19:10 mdw
37 * Rationalise the sliding-window threshold. Drop guarantee that right
38 * arguments to EC @add@ are canonical, and fix up projective implementations
39 * to cope.
40 *
c3caa2fa 41 * Revision 1.4 2004/03/21 22:52:06 mdw
42 * Merge and close elliptic curve branch.
43 *
ceb3f0c0 44 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
45 * Elliptic curves on binary fields work.
46 *
8823192f 47 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
48 * Projective coordinates for prime curves
49 *
dbfee00a 50 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
51 * Simple (non-projective) curves over prime fields now seem to work.
52 *
41cb1beb 53 * Revision 1.3 2003/05/15 23:25:59 mdw
54 * Make elliptic curve stuff build.
55 *
b085fd91 56 * Revision 1.2 2002/01/13 13:48:44 mdw
57 * Further progress.
58 *
b0ab12e6 59 * Revision 1.1 2001/04/29 18:12:33 mdw
60 * Prototype version.
61 *
62 */
63
64/*----- Header files ------------------------------------------------------*/
65
41cb1beb 66#include <mLib/sub.h>
67
b0ab12e6 68#include "ec.h"
69
70/*----- Data structures ---------------------------------------------------*/
71
72typedef struct ecctx {
73 ec_curve c;
74 mp *a, *b;
75} ecctx;
76
dbfee00a 77/*----- Simple prime curves -----------------------------------------------*/
b0ab12e6 78
8823192f 79static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
41cb1beb 80
81static ec *ecneg(ec_curve *c, ec *d, const ec *p)
b085fd91 82{
83 EC_COPY(d, p);
ceb3f0c0 84 if (d->y)
85 d->y = F_NEG(c->f, d->y, d->y);
b085fd91 86 return (d);
87}
88
8823192f 89static ec *ecfind(ec_curve *c, ec *d, mp *x)
90{
91 mp *p, *q;
92 ecctx *cc = (ecctx *)c;
93 field *f = c->f;
94
95 q = F_SQR(f, MP_NEW, x);
96 p = F_MUL(f, MP_NEW, x, q);
97 q = F_MUL(f, q, x, cc->a);
98 p = F_ADD(f, p, p, q);
99 p = F_ADD(f, p, p, cc->b);
100 MP_DROP(q);
101 p = F_SQRT(f, p, p);
102 if (!p)
103 return (0);
104 EC_DESTROY(d);
105 d->x = MP_COPY(x);
106 d->y = p;
107 d->z = MP_COPY(f->one);
b085fd91 108 return (d);
109}
110
111static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
b0ab12e6 112{
b085fd91 113 if (EC_ATINF(a))
114 EC_SETINF(d);
8823192f 115 else if (F_ZEROP(c->f, a->y))
b085fd91 116 EC_COPY(d, a);
117 else {
118 field *f = c->f;
119 ecctx *cc = (ecctx *)c;
120 mp *lambda;
121 mp *dy, *dx;
122
8823192f 123 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
124 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
125 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
126 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x^2 + A$% */
127 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
128 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
b085fd91 129
8823192f 130 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
131 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
132 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
133 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
134 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
135 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
b0ab12e6 136
b085fd91 137 EC_DESTROY(d);
138 d->x = dx;
139 d->y = dy;
140 d->z = 0;
141 MP_DROP(lambda);
142 }
143 return (d);
144}
145
8823192f 146static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
147{
148 if (EC_ATINF(a))
149 EC_SETINF(d);
150 else if (F_ZEROP(c->f, a->y))
151 EC_COPY(d, a);
152 else {
153 field *f = c->f;
154 ecctx *cc = (ecctx *)c;
155 mp *p, *q, *m, *s, *dx, *dy, *dz;
156
157 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
158 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
159 p = F_MUL(f, p, q, cc->a); /* %$A z^4$% */
160 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
161 m = F_TPL(f, m, m); /* %$3 x^2$% */
162 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
163
164 q = F_DBL(f, q, a->y); /* %$2 y$% */
165 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
166
167 p = F_SQR(f, p, q); /* %$4 y^2$% */
168 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
169 q = F_SQR(f, q, p); /* %$16 y^4$% */
170 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
171
172 p = F_DBL(f, p, s); /* %$2 s$% */
173 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
174 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
175
176 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
177 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
178 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
179
180 EC_DESTROY(d);
181 d->x = dx;
182 d->y = dy;
183 d->z = dz;
184 MP_DROP(m);
185 MP_DROP(q);
186 MP_DROP(s);
187 }
188 return (d);
189}
190
191static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
192{
193 if (EC_ATINF(a))
194 EC_SETINF(d);
195 else if (F_ZEROP(c->f, a->y))
196 EC_COPY(d, a);
197 else {
198 field *f = c->f;
199 mp *p, *q, *m, *s, *dx, *dy, *dz;
200
201 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
202 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
203 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
204 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
205 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
206
207 q = F_DBL(f, q, a->y); /* %$2 y$% */
208 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
209
210 p = F_SQR(f, p, q); /* %$4 y^2$% */
211 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
212 q = F_SQR(f, q, p); /* %$16 y^4$% */
213 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
214
215 p = F_DBL(f, p, s); /* %$2 s$% */
216 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
217 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
218
219 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
220 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
221 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
222
223 EC_DESTROY(d);
224 d->x = dx;
225 d->y = dy;
226 d->z = dz;
227 MP_DROP(m);
228 MP_DROP(q);
229 MP_DROP(s);
230 }
231 return (d);
232}
233
b085fd91 234static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
235{
b0ab12e6 236 if (a == b)
237 ecdbl(c, d, a);
238 else if (EC_ATINF(a))
239 EC_COPY(d, b);
240 else if (EC_ATINF(b))
241 EC_COPY(d, a);
b085fd91 242 else {
243 field *f = c->f;
244 mp *lambda;
245 mp *dy, *dx;
246
247 if (!MP_EQ(a->x, b->x)) {
8823192f 248 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
249 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
250 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
b085fd91 251 lambda = F_MUL(f, MP_NEW, dy, dx);
8823192f 252 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
253 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
b0ab12e6 254 EC_SETINF(d);
b085fd91 255 return (d);
256 } else {
257 ecctx *cc = (ecctx *)c;
8823192f 258 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
259 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
260 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x_0^2 + A$% */
261 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
262 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
41cb1beb 263 lambda = F_MUL(f, MP_NEW, dx, dy);
8823192f 264 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
b085fd91 265 }
266
8823192f 267 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
268 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
269 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
270 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
271 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
ceb3f0c0 272 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
b0ab12e6 273
b085fd91 274 EC_DESTROY(d);
275 d->x = dx;
276 d->y = dy;
277 d->z = 0;
278 MP_DROP(lambda);
b0ab12e6 279 }
b085fd91 280 return (d);
b0ab12e6 281}
282
8823192f 283static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
284{
285 if (a == b)
286 c->ops->dbl(c, d, a);
287 else if (EC_ATINF(a))
288 EC_COPY(d, b);
289 else if (EC_ATINF(b))
290 EC_COPY(d, a);
291 else {
292 field *f = c->f;
391faf42 293 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
8823192f 294
295 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
296 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
297 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
298 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
299
391faf42 300 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
301 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
302 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
303 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
304
305 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
306 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
8823192f 307 if (F_ZEROP(f, w)) {
ceb3f0c0 308 MP_DROP(w);
309 MP_DROP(u);
310 MP_DROP(s);
391faf42 311 MP_DROP(uu);
312 MP_DROP(ss);
8823192f 313 if (F_ZEROP(f, r)) {
8823192f 314 MP_DROP(r);
8823192f 315 return (c->ops->dbl(c, d, a));
316 } else {
8823192f 317 MP_DROP(r);
8823192f 318 EC_SETINF(d);
319 return (d);
320 }
321 }
391faf42 322 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
323 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
8823192f 324
391faf42 325 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
326 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
8823192f 327
391faf42 328 p = F_SQR(f, uu, w); /* %$w^2$% */
8823192f 329 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
330 u = F_MUL(f, u, p, w); /* %$w^3$% */
331 p = F_MUL(f, p, u, s); /* %$m w^3$% */
332
333 dx = F_SQR(f, u, r); /* %$r^2$% */
334 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
335
336 s = F_DBL(f, s, dx); /* %$2 x'$% */
337 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
338 dy = F_MUL(f, s, q, r); /* %$v r$% */
339 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
340 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
341
342 EC_DESTROY(d);
343 d->x = dx;
344 d->y = dy;
345 d->z = dz;
346 MP_DROP(p);
347 MP_DROP(q);
348 MP_DROP(r);
349 MP_DROP(w);
350 }
351 return (d);
352}
353
354static int eccheck(ec_curve *c, const ec *p)
355{
356 ecctx *cc = (ecctx *)c;
357 field *f = c->f;
358 int rc;
359 mp *l = F_SQR(f, MP_NEW, p->y);
360 mp *x = F_SQR(f, MP_NEW, p->x);
361 mp *r = F_MUL(f, MP_NEW, x, p->x);
362 x = F_MUL(f, x, cc->a, p->x);
363 r = F_ADD(f, r, r, x);
364 r = F_ADD(f, r, r, cc->b);
365 rc = MP_EQ(l, r) ? 0 : -1;
366 mp_drop(l);
367 mp_drop(x);
368 mp_drop(r);
369 return (rc);
370}
371
372static int ecprojcheck(ec_curve *c, const ec *p)
373{
374 ec t = EC_INIT;
375 int rc;
376
377 c->ops->fix(c, &t, p);
378 rc = eccheck(c, &t);
379 EC_DESTROY(&t);
380 return (rc);
381}
382
41cb1beb 383static void ecdestroy(ec_curve *c)
384{
385 ecctx *cc = (ecctx *)c;
386 MP_DROP(cc->a);
387 MP_DROP(cc->b);
388 DESTROY(cc);
389}
390
391/* --- @ec_prime@, @ec_primeproj@ --- *
392 *
dbfee00a 393 * Arguments: @field *f@ = the underlying field for this elliptic curve
41cb1beb 394 * @mp *a, *b@ = the coefficients for this curve
395 *
396 * Returns: A pointer to the curve.
397 *
398 * Use: Creates a curve structure for an elliptic curve defined over
399 * a prime field. The @primeproj@ variant uses projective
400 * coordinates, which can be a win.
401 */
402
403extern ec_curve *ec_prime(field *f, mp *a, mp *b)
404{
405 ecctx *cc = CREATE(ecctx);
406 cc->c.ops = &ec_primeops;
407 cc->c.f = f;
dbfee00a 408 cc->a = F_IN(f, MP_NEW, a);
409 cc->b = F_IN(f, MP_NEW, b);
41cb1beb 410 return (&cc->c);
411}
412
8823192f 413extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
414{
415 ecctx *cc = CREATE(ecctx);
416 mp *ax;
417
418 ax = mp_add(MP_NEW, a, MP_THREE);
419 ax = F_IN(f, ax, ax);
420 if (F_ZEROP(f, ax))
421 cc->c.ops = &ec_primeprojxops;
422 else
423 cc->c.ops = &ec_primeprojops;
424 MP_DROP(ax);
425 cc->c.f = f;
426 cc->a = F_IN(f, MP_NEW, a);
427 cc->b = F_IN(f, MP_NEW, b);
41cb1beb 428 return (&cc->c);
429}
430
431static const ec_ops ec_primeops = {
8823192f 432 ecdestroy, ec_idin, ec_idout, ec_idfix,
bc985cef 433 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
8823192f 434};
435
436static const ec_ops ec_primeprojops = {
437 ecdestroy, ec_projin, ec_projout, ec_projfix,
bc985cef 438 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
8823192f 439};
440
441static const ec_ops ec_primeprojxops = {
442 ecdestroy, ec_projin, ec_projout, ec_projfix,
bc985cef 443 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
41cb1beb 444};
445
446/*----- Test rig ----------------------------------------------------------*/
447
448#ifdef TEST_RIG
449
450#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
451
ceb3f0c0 452int main(int argc, char *argv[])
41cb1beb 453{
454 field *f;
455 ec_curve *c;
456 ec g = EC_INIT, d = EC_INIT;
457 mp *p, *a, *b, *r;
ceb3f0c0 458 int i, n = argc == 1 ? 1 : atoi(argv[1]);
41cb1beb 459
dbfee00a 460 printf("ec-prime: ");
461 fflush(stdout);
41cb1beb 462 a = MP(-3);
463 b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
464 p = MP(6277101735386680763835789423207666416083908700390324961279);
dbfee00a 465 r = MP(6277101735386680763835789423176059013767194773182842284080);
41cb1beb 466
467 f = field_prime(p);
ceb3f0c0 468 c = ec_primeproj(f, a, b);
41cb1beb 469
470 g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
471 g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
472
ceb3f0c0 473 for (i = 0; i < n; i++) {
474 ec_mul(c, &d, &g, r);
475 if (EC_ATINF(&d)) {
476 fprintf(stderr, "zero too early\n");
477 return (1);
478 }
479 ec_add(c, &d, &d, &g);
480 if (!EC_ATINF(&d)) {
481 fprintf(stderr, "didn't reach zero\n");
482 MP_EPRINT("d.x", d.x);
483 MP_EPRINT("d.y", d.y);
484 return (1);
485 }
486 ec_destroy(&d);
dbfee00a 487 }
41cb1beb 488 ec_destroy(&g);
489 ec_destroycurve(c);
490 F_DESTROY(f);
dbfee00a 491 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
492 assert(!mparena_count(&mparena_global));
493 printf("ok\n");
41cb1beb 494 return (0);
495}
496
497#endif
498
b0ab12e6 499/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)