3 * $Id: ec-prime.c,v 1.5 2004/03/22 02:19:10 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 1.5 2004/03/22 02:19:10 mdw
34 * Rationalise the sliding-window threshold. Drop guarantee that right
35 * arguments to EC @add@ are canonical, and fix up projective implementations
38 * Revision 1.4 2004/03/21 22:52:06 mdw
39 * Merge and close elliptic curve branch.
41 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
42 * Elliptic curves on binary fields work.
44 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
45 * Projective coordinates for prime curves
47 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
48 * Simple (non-projective) curves over prime fields now seem to work.
50 * Revision 1.3 2003/05/15 23:25:59 mdw
51 * Make elliptic curve stuff build.
53 * Revision 1.2 2002/01/13 13:48:44 mdw
56 * Revision 1.1 2001/04/29 18:12:33 mdw
61 /*----- Header files ------------------------------------------------------*/
67 /*----- Data structures ---------------------------------------------------*/
69 typedef struct ecctx {
74 /*----- Simple prime curves -----------------------------------------------*/
76 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
78 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
82 d->y = F_NEG(c->f, d->y, d->y);
86 static ec *ecfind(ec_curve *c, ec *d, mp *x)
89 ecctx *cc = (ecctx *)c;
92 q = F_SQR(f, MP_NEW, x);
93 p = F_MUL(f, MP_NEW, x, q);
94 q = F_MUL(f, q, x, cc->a);
95 p = F_ADD(f, p, p, q);
96 p = F_ADD(f, p, p, cc->b);
104 d->z = MP_COPY(f->one);
108 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
112 else if (F_ZEROP(c->f, a->y))
116 ecctx *cc = (ecctx *)c;
120 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
121 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
122 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
123 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x^2 + A$% */
124 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
125 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
127 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
128 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
129 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
130 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
131 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
132 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
143 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
147 else if (F_ZEROP(c->f, a->y))
151 ecctx *cc = (ecctx *)c;
152 mp *p, *q, *m, *s, *dx, *dy, *dz;
154 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
155 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
156 p = F_MUL(f, p, q, cc->a); /* %$A z^4$% */
157 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
158 m = F_TPL(f, m, m); /* %$3 x^2$% */
159 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
161 q = F_DBL(f, q, a->y); /* %$2 y$% */
162 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
164 p = F_SQR(f, p, q); /* %$4 y^2$% */
165 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
166 q = F_SQR(f, q, p); /* %$16 y^4$% */
167 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
169 p = F_DBL(f, p, s); /* %$2 s$% */
170 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
171 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
173 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
174 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
175 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
188 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
192 else if (F_ZEROP(c->f, a->y))
196 mp *p, *q, *m, *s, *dx, *dy, *dz;
198 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
199 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
200 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
201 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
202 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
204 q = F_DBL(f, q, a->y); /* %$2 y$% */
205 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
207 p = F_SQR(f, p, q); /* %$4 y^2$% */
208 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
209 q = F_SQR(f, q, p); /* %$16 y^4$% */
210 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
212 p = F_DBL(f, p, s); /* %$2 s$% */
213 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
214 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
216 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
217 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
218 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
231 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
235 else if (EC_ATINF(a))
237 else if (EC_ATINF(b))
244 if (!MP_EQ(a->x, b->x)) {
245 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
246 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
247 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
248 lambda = F_MUL(f, MP_NEW, dy, dx);
249 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
250 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
254 ecctx *cc = (ecctx *)c;
255 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
256 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
257 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x_0^2 + A$% */
258 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
259 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
260 lambda = F_MUL(f, MP_NEW, dx, dy);
261 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
264 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
265 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
266 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
267 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
268 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
269 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
280 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
283 c->ops->dbl(c, d, a);
284 else if (EC_ATINF(a))
286 else if (EC_ATINF(b))
290 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
292 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
293 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
294 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
295 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
297 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
298 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
299 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
300 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
302 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
303 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
312 return (c->ops->dbl(c, d, a));
319 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
320 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
322 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
323 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
325 p = F_SQR(f, uu, w); /* %$w^2$% */
326 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
327 u = F_MUL(f, u, p, w); /* %$w^3$% */
328 p = F_MUL(f, p, u, s); /* %$m w^3$% */
330 dx = F_SQR(f, u, r); /* %$r^2$% */
331 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
333 s = F_DBL(f, s, dx); /* %$2 x'$% */
334 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
335 dy = F_MUL(f, s, q, r); /* %$v r$% */
336 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
337 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
351 static int eccheck(ec_curve *c, const ec *p)
353 ecctx *cc = (ecctx *)c;
356 mp *l = F_SQR(f, MP_NEW, p->y);
357 mp *x = F_SQR(f, MP_NEW, p->x);
358 mp *r = F_MUL(f, MP_NEW, x, p->x);
359 x = F_MUL(f, x, cc->a, p->x);
360 r = F_ADD(f, r, r, x);
361 r = F_ADD(f, r, r, cc->b);
362 rc = MP_EQ(l, r) ? 0 : -1;
369 static int ecprojcheck(ec_curve *c, const ec *p)
374 c->ops->fix(c, &t, p);
380 static void ecdestroy(ec_curve *c)
382 ecctx *cc = (ecctx *)c;
388 /* --- @ec_prime@, @ec_primeproj@ --- *
390 * Arguments: @field *f@ = the underlying field for this elliptic curve
391 * @mp *a, *b@ = the coefficients for this curve
393 * Returns: A pointer to the curve.
395 * Use: Creates a curve structure for an elliptic curve defined over
396 * a prime field. The @primeproj@ variant uses projective
397 * coordinates, which can be a win.
400 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
402 ecctx *cc = CREATE(ecctx);
403 cc->c.ops = &ec_primeops;
405 cc->a = F_IN(f, MP_NEW, a);
406 cc->b = F_IN(f, MP_NEW, b);
410 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
412 ecctx *cc = CREATE(ecctx);
415 ax = mp_add(MP_NEW, a, MP_THREE);
416 ax = F_IN(f, ax, ax);
418 cc->c.ops = &ec_primeprojxops;
420 cc->c.ops = &ec_primeprojops;
423 cc->a = F_IN(f, MP_NEW, a);
424 cc->b = F_IN(f, MP_NEW, b);
428 static const ec_ops ec_primeops = {
429 ecdestroy, ec_idin, ec_idout, ec_idfix,
430 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
433 static const ec_ops ec_primeprojops = {
434 ecdestroy, ec_projin, ec_projout, ec_projfix,
435 0, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
438 static const ec_ops ec_primeprojxops = {
439 ecdestroy, ec_projin, ec_projout, ec_projfix,
440 0, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
443 /*----- Test rig ----------------------------------------------------------*/
447 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
449 int main(int argc, char *argv[])
453 ec g = EC_INIT, d = EC_INIT;
455 int i, n = argc == 1 ? 1 : atoi(argv[1]);
457 printf("ec-prime: ");
460 b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
461 p = MP(6277101735386680763835789423207666416083908700390324961279);
462 r = MP(6277101735386680763835789423176059013767194773182842284080);
465 c = ec_primeproj(f, a, b);
467 g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
468 g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
470 for (i = 0; i < n; i++) {
471 ec_mul(c, &d, &g, r);
473 fprintf(stderr, "zero too early\n");
476 ec_add(c, &d, &d, &g);
478 fprintf(stderr, "didn't reach zero\n");
479 MP_EPRINT("d.x", d.x);
480 MP_EPRINT("d.y", d.y);
488 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
489 assert(!mparena_count(&mparena_global));
496 /*----- That's all, folks -------------------------------------------------*/