3 * $Id: ec-prime.c,v 1.8 2004/03/27 17:54:11 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 1.8 2004/03/27 17:54:11 mdw
34 * Standard curves and curve checking.
36 * Revision 1.7 2004/03/27 00:04:46 mdw
37 * Implement efficient reduction for pleasant-looking primes.
39 * Revision 1.6 2004/03/23 15:19:32 mdw
40 * Test elliptic curves more thoroughly.
42 * Revision 1.5 2004/03/22 02:19:10 mdw
43 * Rationalise the sliding-window threshold. Drop guarantee that right
44 * arguments to EC @add@ are canonical, and fix up projective implementations
47 * Revision 1.4 2004/03/21 22:52:06 mdw
48 * Merge and close elliptic curve branch.
50 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
51 * Elliptic curves on binary fields work.
53 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
54 * Projective coordinates for prime curves
56 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
57 * Simple (non-projective) curves over prime fields now seem to work.
59 * Revision 1.3 2003/05/15 23:25:59 mdw
60 * Make elliptic curve stuff build.
62 * Revision 1.2 2002/01/13 13:48:44 mdw
65 * Revision 1.1 2001/04/29 18:12:33 mdw
70 /*----- Header files ------------------------------------------------------*/
76 /*----- Simple prime curves -----------------------------------------------*/
78 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
80 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
84 d->y = F_NEG(c->f, d->y, d->y);
88 static ec *ecfind(ec_curve *c, ec *d, mp *x)
93 q = F_SQR(f, MP_NEW, x);
94 p = F_MUL(f, MP_NEW, x, q);
95 q = F_MUL(f, q, x, c->a);
96 p = F_ADD(f, p, p, q);
97 p = F_ADD(f, p, p, c->b);
105 d->z = MP_COPY(f->one);
109 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
113 else if (F_ZEROP(c->f, a->y))
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, c->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$% */
143 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
147 else if (F_ZEROP(c->f, a->y))
151 mp *p, *q, *m, *s, *dx, *dy, *dz;
153 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
154 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
155 p = F_MUL(f, p, q, c->a); /* %$A z^4$% */
156 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
157 m = F_TPL(f, m, m); /* %$3 x^2$% */
158 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
160 q = F_DBL(f, q, a->y); /* %$2 y$% */
161 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
163 p = F_SQR(f, p, q); /* %$4 y^2$% */
164 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
165 q = F_SQR(f, q, p); /* %$16 y^4$% */
166 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
168 p = F_DBL(f, p, s); /* %$2 s$% */
169 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
170 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
172 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
173 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
174 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
187 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
191 else if (F_ZEROP(c->f, a->y))
195 mp *p, *q, *m, *s, *dx, *dy, *dz;
197 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
198 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
199 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
200 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
201 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
203 q = F_DBL(f, q, a->y); /* %$2 y$% */
204 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
206 p = F_SQR(f, p, q); /* %$4 y^2$% */
207 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
208 q = F_SQR(f, q, p); /* %$16 y^4$% */
209 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
211 p = F_DBL(f, p, s); /* %$2 s$% */
212 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
213 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
215 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
216 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
217 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
230 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
234 else if (EC_ATINF(a))
236 else if (EC_ATINF(b))
243 if (!MP_EQ(a->x, b->x)) {
244 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
245 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
246 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
247 lambda = F_MUL(f, MP_NEW, dy, dx);
248 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
249 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
253 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
254 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
255 dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */
256 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
257 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
258 lambda = F_MUL(f, MP_NEW, dx, dy);
259 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
262 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
263 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
264 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
265 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
266 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
267 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
278 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
281 c->ops->dbl(c, d, a);
282 else if (EC_ATINF(a))
284 else if (EC_ATINF(b))
288 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
290 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
291 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
292 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
293 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
295 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
296 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
297 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
298 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
300 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
301 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
310 return (c->ops->dbl(c, d, a));
317 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
318 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
320 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
321 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
323 p = F_SQR(f, uu, w); /* %$w^2$% */
324 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
325 u = F_MUL(f, u, p, w); /* %$w^3$% */
326 p = F_MUL(f, p, u, s); /* %$m w^3$% */
328 dx = F_SQR(f, u, r); /* %$r^2$% */
329 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
331 s = F_DBL(f, s, dx); /* %$2 x'$% */
332 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
333 dy = F_MUL(f, s, q, r); /* %$v r$% */
334 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
335 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
349 static int eccheck(ec_curve *c, const ec *p)
353 mp *l = F_SQR(f, MP_NEW, p->y);
354 mp *x = F_SQR(f, MP_NEW, p->x);
355 mp *r = F_MUL(f, MP_NEW, x, p->x);
356 x = F_MUL(f, x, c->a, p->x);
357 r = F_ADD(f, r, r, x);
358 r = F_ADD(f, r, r, c->b);
359 rc = MP_EQ(l, r) ? 0 : -1;
366 static int ecprojcheck(ec_curve *c, const ec *p)
371 c->ops->fix(c, &t, p);
377 static void ecdestroy(ec_curve *c)
384 /* --- @ec_prime@, @ec_primeproj@ --- *
386 * Arguments: @field *f@ = the underlying field for this elliptic curve
387 * @mp *a, *b@ = the coefficients for this curve
389 * Returns: A pointer to the curve.
391 * Use: Creates a curve structure for an elliptic curve defined over
392 * a prime field. The @primeproj@ variant uses projective
393 * coordinates, which can be a win.
396 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
398 ec_curve *c = CREATE(ec_curve);
399 c->ops = &ec_primeops;
401 c->a = F_IN(f, MP_NEW, a);
402 c->b = F_IN(f, MP_NEW, b);
406 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
408 ec_curve *c = CREATE(ec_curve);
411 ax = mp_add(MP_NEW, a, MP_THREE);
412 ax = F_IN(f, ax, ax);
414 c->ops = &ec_primeprojxops;
416 c->ops = &ec_primeprojops;
419 c->a = F_IN(f, MP_NEW, a);
420 c->b = F_IN(f, MP_NEW, b);
424 static const ec_ops ec_primeops = {
425 ecdestroy, ec_idin, ec_idout, ec_idfix,
426 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
429 static const ec_ops ec_primeprojops = {
430 ecdestroy, ec_projin, ec_projout, ec_projfix,
431 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
434 static const ec_ops ec_primeprojxops = {
435 ecdestroy, ec_projin, ec_projout, ec_projfix,
436 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
439 /*----- Test rig ----------------------------------------------------------*/
443 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
445 int main(int argc, char *argv[])
449 ec g = EC_INIT, d = EC_INIT;
451 int i, n = argc == 1 ? 1 : atoi(argv[1]);
453 printf("ec-prime: ");
456 b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
457 p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
458 r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
460 f = field_niceprime(p);
461 c = ec_primeproj(f, a, b);
463 g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
464 g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
466 for (i = 0; i < n; i++) {
467 ec_mul(c, &d, &g, r);
469 fprintf(stderr, "zero too early\n");
472 ec_add(c, &d, &d, &g);
474 fprintf(stderr, "didn't reach zero\n");
475 MP_EPRINT("d.x", d.x);
476 MP_EPRINT("d.y", d.y);
484 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
485 assert(!mparena_count(&mparena_global));
492 /*----- That's all, folks -------------------------------------------------*/