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Implement efficient reduction for pleasant-looking primes.
[u/mdw/catacomb] / ec-prime.c
CommitLineData
b0ab12e6 1/* -*-c-*-
2 *
f46efa79 3 * $Id: ec-prime.c,v 1.7 2004/03/27 00:04:46 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 $
f46efa79 33 * Revision 1.7 2004/03/27 00:04:46 mdw
34 * Implement efficient reduction for pleasant-looking primes.
35 *
bc985cef 36 * Revision 1.6 2004/03/23 15:19:32 mdw
37 * Test elliptic curves more thoroughly.
38 *
391faf42 39 * Revision 1.5 2004/03/22 02:19:10 mdw
40 * Rationalise the sliding-window threshold. Drop guarantee that right
41 * arguments to EC @add@ are canonical, and fix up projective implementations
42 * to cope.
43 *
c3caa2fa 44 * Revision 1.4 2004/03/21 22:52:06 mdw
45 * Merge and close elliptic curve branch.
46 *
ceb3f0c0 47 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
48 * Elliptic curves on binary fields work.
49 *
8823192f 50 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
51 * Projective coordinates for prime curves
52 *
dbfee00a 53 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
54 * Simple (non-projective) curves over prime fields now seem to work.
55 *
41cb1beb 56 * Revision 1.3 2003/05/15 23:25:59 mdw
57 * Make elliptic curve stuff build.
58 *
b085fd91 59 * Revision 1.2 2002/01/13 13:48:44 mdw
60 * Further progress.
61 *
b0ab12e6 62 * Revision 1.1 2001/04/29 18:12:33 mdw
63 * Prototype version.
64 *
65 */
66
67/*----- Header files ------------------------------------------------------*/
68
41cb1beb 69#include <mLib/sub.h>
70
b0ab12e6 71#include "ec.h"
72
73/*----- Data structures ---------------------------------------------------*/
74
75typedef struct ecctx {
76 ec_curve c;
77 mp *a, *b;
78} ecctx;
79
dbfee00a 80/*----- Simple prime curves -----------------------------------------------*/
b0ab12e6 81
8823192f 82static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
41cb1beb 83
84static ec *ecneg(ec_curve *c, ec *d, const ec *p)
b085fd91 85{
86 EC_COPY(d, p);
ceb3f0c0 87 if (d->y)
88 d->y = F_NEG(c->f, d->y, d->y);
b085fd91 89 return (d);
90}
91
8823192f 92static ec *ecfind(ec_curve *c, ec *d, mp *x)
93{
94 mp *p, *q;
95 ecctx *cc = (ecctx *)c;
96 field *f = c->f;
97
98 q = F_SQR(f, MP_NEW, x);
99 p = F_MUL(f, MP_NEW, x, q);
100 q = F_MUL(f, q, x, cc->a);
101 p = F_ADD(f, p, p, q);
102 p = F_ADD(f, p, p, cc->b);
103 MP_DROP(q);
104 p = F_SQRT(f, p, p);
105 if (!p)
106 return (0);
107 EC_DESTROY(d);
108 d->x = MP_COPY(x);
109 d->y = p;
110 d->z = MP_COPY(f->one);
b085fd91 111 return (d);
112}
113
114static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
b0ab12e6 115{
b085fd91 116 if (EC_ATINF(a))
117 EC_SETINF(d);
8823192f 118 else if (F_ZEROP(c->f, a->y))
b085fd91 119 EC_COPY(d, a);
120 else {
121 field *f = c->f;
122 ecctx *cc = (ecctx *)c;
123 mp *lambda;
124 mp *dy, *dx;
125
8823192f 126 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
127 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
128 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
129 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x^2 + A$% */
130 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
131 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
b085fd91 132
8823192f 133 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
134 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
135 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
136 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
137 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
138 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
b0ab12e6 139
b085fd91 140 EC_DESTROY(d);
141 d->x = dx;
142 d->y = dy;
143 d->z = 0;
144 MP_DROP(lambda);
145 }
146 return (d);
147}
148
8823192f 149static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
150{
151 if (EC_ATINF(a))
152 EC_SETINF(d);
153 else if (F_ZEROP(c->f, a->y))
154 EC_COPY(d, a);
155 else {
156 field *f = c->f;
157 ecctx *cc = (ecctx *)c;
158 mp *p, *q, *m, *s, *dx, *dy, *dz;
159
160 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
161 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
162 p = F_MUL(f, p, q, cc->a); /* %$A z^4$% */
163 m = F_SQR(f, MP_NEW, a->x); /* %$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 {
260 ecctx *cc = (ecctx *)c;
8823192f 261 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
262 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
263 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x_0^2 + A$% */
264 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
265 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
41cb1beb 266 lambda = F_MUL(f, MP_NEW, dx, dy);
8823192f 267 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
b085fd91 268 }
269
8823192f 270 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
271 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
272 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
273 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
274 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
ceb3f0c0 275 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
b0ab12e6 276
b085fd91 277 EC_DESTROY(d);
278 d->x = dx;
279 d->y = dy;
280 d->z = 0;
281 MP_DROP(lambda);
b0ab12e6 282 }
b085fd91 283 return (d);
b0ab12e6 284}
285
8823192f 286static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
287{
288 if (a == b)
289 c->ops->dbl(c, d, a);
290 else if (EC_ATINF(a))
291 EC_COPY(d, b);
292 else if (EC_ATINF(b))
293 EC_COPY(d, a);
294 else {
295 field *f = c->f;
391faf42 296 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
8823192f 297
298 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
299 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
300 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
301 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
302
391faf42 303 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
304 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
305 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
306 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
307
308 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
309 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
8823192f 310 if (F_ZEROP(f, w)) {
ceb3f0c0 311 MP_DROP(w);
312 MP_DROP(u);
313 MP_DROP(s);
391faf42 314 MP_DROP(uu);
315 MP_DROP(ss);
8823192f 316 if (F_ZEROP(f, r)) {
8823192f 317 MP_DROP(r);
8823192f 318 return (c->ops->dbl(c, d, a));
319 } else {
8823192f 320 MP_DROP(r);
8823192f 321 EC_SETINF(d);
322 return (d);
323 }
324 }
391faf42 325 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
326 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
8823192f 327
391faf42 328 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
329 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
8823192f 330
391faf42 331 p = F_SQR(f, uu, w); /* %$w^2$% */
8823192f 332 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
333 u = F_MUL(f, u, p, w); /* %$w^3$% */
334 p = F_MUL(f, p, u, s); /* %$m w^3$% */
335
336 dx = F_SQR(f, u, r); /* %$r^2$% */
337 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
338
339 s = F_DBL(f, s, dx); /* %$2 x'$% */
340 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
341 dy = F_MUL(f, s, q, r); /* %$v r$% */
342 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
343 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
344
345 EC_DESTROY(d);
346 d->x = dx;
347 d->y = dy;
348 d->z = dz;
349 MP_DROP(p);
350 MP_DROP(q);
351 MP_DROP(r);
352 MP_DROP(w);
353 }
354 return (d);
355}
356
357static int eccheck(ec_curve *c, const ec *p)
358{
359 ecctx *cc = (ecctx *)c;
360 field *f = c->f;
361 int rc;
362 mp *l = F_SQR(f, MP_NEW, p->y);
363 mp *x = F_SQR(f, MP_NEW, p->x);
364 mp *r = F_MUL(f, MP_NEW, x, p->x);
365 x = F_MUL(f, x, cc->a, p->x);
366 r = F_ADD(f, r, r, x);
367 r = F_ADD(f, r, r, cc->b);
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{
388 ecctx *cc = (ecctx *)c;
389 MP_DROP(cc->a);
390 MP_DROP(cc->b);
391 DESTROY(cc);
392}
393
394/* --- @ec_prime@, @ec_primeproj@ --- *
395 *
dbfee00a 396 * Arguments: @field *f@ = the underlying field for this elliptic curve
41cb1beb 397 * @mp *a, *b@ = the coefficients for this curve
398 *
399 * Returns: A pointer to the curve.
400 *
401 * Use: Creates a curve structure for an elliptic curve defined over
402 * a prime field. The @primeproj@ variant uses projective
403 * coordinates, which can be a win.
404 */
405
406extern ec_curve *ec_prime(field *f, mp *a, mp *b)
407{
408 ecctx *cc = CREATE(ecctx);
409 cc->c.ops = &ec_primeops;
410 cc->c.f = f;
dbfee00a 411 cc->a = F_IN(f, MP_NEW, a);
412 cc->b = F_IN(f, MP_NEW, b);
41cb1beb 413 return (&cc->c);
414}
415
8823192f 416extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
417{
418 ecctx *cc = CREATE(ecctx);
419 mp *ax;
420
421 ax = mp_add(MP_NEW, a, MP_THREE);
422 ax = F_IN(f, ax, ax);
423 if (F_ZEROP(f, ax))
424 cc->c.ops = &ec_primeprojxops;
425 else
426 cc->c.ops = &ec_primeprojops;
427 MP_DROP(ax);
428 cc->c.f = f;
429 cc->a = F_IN(f, MP_NEW, a);
430 cc->b = F_IN(f, MP_NEW, b);
41cb1beb 431 return (&cc->c);
432}
433
434static const ec_ops ec_primeops = {
8823192f 435 ecdestroy, ec_idin, ec_idout, ec_idfix,
bc985cef 436 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
8823192f 437};
438
439static const ec_ops ec_primeprojops = {
440 ecdestroy, ec_projin, ec_projout, ec_projfix,
bc985cef 441 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
8823192f 442};
443
444static const ec_ops ec_primeprojxops = {
445 ecdestroy, ec_projin, ec_projout, ec_projfix,
bc985cef 446 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
41cb1beb 447};
448
449/*----- Test rig ----------------------------------------------------------*/
450
451#ifdef TEST_RIG
452
453#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
454
ceb3f0c0 455int main(int argc, char *argv[])
41cb1beb 456{
457 field *f;
458 ec_curve *c;
459 ec g = EC_INIT, d = EC_INIT;
460 mp *p, *a, *b, *r;
ceb3f0c0 461 int i, n = argc == 1 ? 1 : atoi(argv[1]);
41cb1beb 462
dbfee00a 463 printf("ec-prime: ");
464 fflush(stdout);
41cb1beb 465 a = MP(-3);
466 b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
467 p = MP(6277101735386680763835789423207666416083908700390324961279);
dbfee00a 468 r = MP(6277101735386680763835789423176059013767194773182842284080);
41cb1beb 469
f46efa79 470 f = field_niceprime(p);
ceb3f0c0 471 c = ec_primeproj(f, a, b);
41cb1beb 472
473 g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
474 g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
475
ceb3f0c0 476 for (i = 0; i < n; i++) {
477 ec_mul(c, &d, &g, r);
478 if (EC_ATINF(&d)) {
479 fprintf(stderr, "zero too early\n");
480 return (1);
481 }
482 ec_add(c, &d, &d, &g);
483 if (!EC_ATINF(&d)) {
484 fprintf(stderr, "didn't reach zero\n");
485 MP_EPRINT("d.x", d.x);
486 MP_EPRINT("d.y", d.y);
487 return (1);
488 }
489 ec_destroy(&d);
dbfee00a 490 }
41cb1beb 491 ec_destroy(&g);
492 ec_destroycurve(c);
493 F_DESTROY(f);
dbfee00a 494 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
495 assert(!mparena_count(&mparena_global));
496 printf("ok\n");
41cb1beb 497 return (0);
498}
499
500#endif
501
b0ab12e6 502/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)