3 * $Id: ec-prime.c,v 1.4 2004/03/21 22:52:06 mdw Exp $
5 * Elliptic curves over prime fields
7 * (c) 2001 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
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.
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.
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,
30 /*----- Revision history --------------------------------------------------*
32 * $Log: ec-prime.c,v $
33 * Revision 1.4 2004/03/21 22:52:06 mdw
34 * Merge and close elliptic curve branch.
36 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
37 * Elliptic curves on binary fields work.
39 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
40 * Projective coordinates for prime curves
42 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
43 * Simple (non-projective) curves over prime fields now seem to work.
45 * Revision 1.3 2003/05/15 23:25:59 mdw
46 * Make elliptic curve stuff build.
48 * Revision 1.2 2002/01/13 13:48:44 mdw
51 * Revision 1.1 2001/04/29 18:12:33 mdw
56 /*----- Header files ------------------------------------------------------*/
62 /*----- Data structures ---------------------------------------------------*/
64 typedef struct ecctx {
69 /*----- Simple prime curves -----------------------------------------------*/
71 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
73 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
77 d->y = F_NEG(c->f, d->y, d->y);
81 static ec *ecfind(ec_curve *c, ec *d, mp *x)
84 ecctx *cc = (ecctx *)c;
87 q = F_SQR(f, MP_NEW, x);
88 p = F_MUL(f, MP_NEW, x, q);
89 q = F_MUL(f, q, x, cc->a);
90 p = F_ADD(f, p, p, q);
91 p = F_ADD(f, p, p, cc->b);
99 d->z = MP_COPY(f->one);
103 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
107 else if (F_ZEROP(c->f, a->y))
111 ecctx *cc = (ecctx *)c;
115 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
116 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
117 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
118 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x^2 + A$% */
119 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
120 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
122 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
123 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
124 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
125 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
126 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
127 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
138 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
142 else if (F_ZEROP(c->f, a->y))
146 ecctx *cc = (ecctx *)c;
147 mp *p, *q, *m, *s, *dx, *dy, *dz;
149 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
150 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
151 p = F_MUL(f, p, q, cc->a); /* %$A z^4$% */
152 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
153 m = F_TPL(f, m, m); /* %$3 x^2$% */
154 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
156 q = F_DBL(f, q, a->y); /* %$2 y$% */
157 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
159 p = F_SQR(f, p, q); /* %$4 y^2$% */
160 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
161 q = F_SQR(f, q, p); /* %$16 y^4$% */
162 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
164 p = F_DBL(f, p, s); /* %$2 s$% */
165 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
166 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
168 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
169 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
170 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
183 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
187 else if (F_ZEROP(c->f, a->y))
191 mp *p, *q, *m, *s, *dx, *dy, *dz;
193 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
194 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
195 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
196 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
197 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
199 q = F_DBL(f, q, a->y); /* %$2 y$% */
200 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
202 p = F_SQR(f, p, q); /* %$4 y^2$% */
203 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
204 q = F_SQR(f, q, p); /* %$16 y^4$% */
205 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
207 p = F_DBL(f, p, s); /* %$2 s$% */
208 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
209 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
211 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
212 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
213 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
226 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
230 else if (EC_ATINF(a))
232 else if (EC_ATINF(b))
239 if (!MP_EQ(a->x, b->x)) {
240 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
241 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
242 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
243 lambda = F_MUL(f, MP_NEW, dy, dx);
244 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
245 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
249 ecctx *cc = (ecctx *)c;
250 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
251 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
252 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x_0^2 + A$% */
253 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
254 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
255 lambda = F_MUL(f, MP_NEW, dx, dy);
256 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
259 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
260 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
261 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
262 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
263 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
264 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
275 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
278 c->ops->dbl(c, d, a);
279 else if (EC_ATINF(a))
281 else if (EC_ATINF(b))
285 mp *p, *q, *r, *w, *u, *s, *dx, *dy, *dz;
287 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
288 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
289 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
290 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
292 w = F_SUB(f, p, a->x, u); /* %$w = x_0 - u$% */
293 r = F_SUB(f, MP_NEW, a->y, s); /* %$r = y_0 - s$% */
300 return (c->ops->dbl(c, d, a));
307 u = F_ADD(f, u, u, a->x); /* %$t = x_0 + u$% */
308 s = F_ADD(f, s, s, a->y); /* %$m = y_0 + r$% */
310 dz = F_MUL(f, MP_NEW, a->z, w); /* %$z' = z_0 w$% */
312 p = F_SQR(f, MP_NEW, w); /* %$w^2$% */
313 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
314 u = F_MUL(f, u, p, w); /* %$w^3$% */
315 p = F_MUL(f, p, u, s); /* %$m w^3$% */
317 dx = F_SQR(f, u, r); /* %$r^2$% */
318 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
320 s = F_DBL(f, s, dx); /* %$2 x'$% */
321 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
322 dy = F_MUL(f, s, q, r); /* %$v r$% */
323 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
324 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
338 static int eccheck(ec_curve *c, const ec *p)
340 ecctx *cc = (ecctx *)c;
343 mp *l = F_SQR(f, MP_NEW, p->y);
344 mp *x = F_SQR(f, MP_NEW, p->x);
345 mp *r = F_MUL(f, MP_NEW, x, p->x);
346 x = F_MUL(f, x, cc->a, p->x);
347 r = F_ADD(f, r, r, x);
348 r = F_ADD(f, r, r, cc->b);
349 rc = MP_EQ(l, r) ? 0 : -1;
356 static int ecprojcheck(ec_curve *c, const ec *p)
361 c->ops->fix(c, &t, p);
367 static void ecdestroy(ec_curve *c)
369 ecctx *cc = (ecctx *)c;
375 /* --- @ec_prime@, @ec_primeproj@ --- *
377 * Arguments: @field *f@ = the underlying field for this elliptic curve
378 * @mp *a, *b@ = the coefficients for this curve
380 * Returns: A pointer to the curve.
382 * Use: Creates a curve structure for an elliptic curve defined over
383 * a prime field. The @primeproj@ variant uses projective
384 * coordinates, which can be a win.
387 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
389 ecctx *cc = CREATE(ecctx);
390 cc->c.ops = &ec_primeops;
392 cc->a = F_IN(f, MP_NEW, a);
393 cc->b = F_IN(f, MP_NEW, b);
397 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
399 ecctx *cc = CREATE(ecctx);
402 ax = mp_add(MP_NEW, a, MP_THREE);
403 ax = F_IN(f, ax, ax);
405 cc->c.ops = &ec_primeprojxops;
407 cc->c.ops = &ec_primeprojops;
410 cc->a = F_IN(f, MP_NEW, a);
411 cc->b = F_IN(f, MP_NEW, b);
415 static const ec_ops ec_primeops = {
416 ecdestroy, ec_idin, ec_idout, ec_idfix,
417 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
420 static const ec_ops ec_primeprojops = {
421 ecdestroy, ec_projin, ec_projout, ec_projfix,
422 0, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
425 static const ec_ops ec_primeprojxops = {
426 ecdestroy, ec_projin, ec_projout, ec_projfix,
427 0, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
430 /*----- Test rig ----------------------------------------------------------*/
434 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
436 int main(int argc, char *argv[])
440 ec g = EC_INIT, d = EC_INIT;
442 int i, n = argc == 1 ? 1 : atoi(argv[1]);
444 printf("ec-prime: ");
447 b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
448 p = MP(6277101735386680763835789423207666416083908700390324961279);
449 r = MP(6277101735386680763835789423176059013767194773182842284080);
452 c = ec_primeproj(f, a, b);
454 g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
455 g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
457 for (i = 0; i < n; i++) {
458 ec_mul(c, &d, &g, r);
460 fprintf(stderr, "zero too early\n");
463 ec_add(c, &d, &d, &g);
465 fprintf(stderr, "didn't reach zero\n");
466 MP_EPRINT("d.x", d.x);
467 MP_EPRINT("d.y", d.y);
475 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
476 assert(!mparena_count(&mparena_global));
483 /*----- That's all, folks -------------------------------------------------*/