3 * $Id: ec-prime.c,v 1.10 2004/04/03 03:32:05 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 1.10 2004/04/03 03:32:05 mdw
34 * General robustification.
36 * Revision 1.9 2004/04/01 12:50:09 mdw
37 * Add cyclic group abstraction, with test code. Separate off exponentation
38 * functions for better static linking. Fix a buttload of bugs on the way.
39 * Generally ensure that negative exponents do inversion correctly. Add
40 * table of standard prime-field subgroups. (Binary field subgroups are
41 * currently unimplemented but easy to add if anyone ever finds a good one.)
43 * Revision 1.8 2004/03/27 17:54:11 mdw
44 * Standard curves and curve checking.
46 * Revision 1.7 2004/03/27 00:04:46 mdw
47 * Implement efficient reduction for pleasant-looking primes.
49 * Revision 1.6 2004/03/23 15:19:32 mdw
50 * Test elliptic curves more thoroughly.
52 * Revision 1.5 2004/03/22 02:19:10 mdw
53 * Rationalise the sliding-window threshold. Drop guarantee that right
54 * arguments to EC @add@ are canonical, and fix up projective implementations
57 * Revision 1.4 2004/03/21 22:52:06 mdw
58 * Merge and close elliptic curve branch.
60 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
61 * Elliptic curves on binary fields work.
63 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
64 * Projective coordinates for prime curves
66 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
67 * Simple (non-projective) curves over prime fields now seem to work.
69 * Revision 1.3 2003/05/15 23:25:59 mdw
70 * Make elliptic curve stuff build.
72 * Revision 1.2 2002/01/13 13:48:44 mdw
75 * Revision 1.1 2001/04/29 18:12:33 mdw
80 /*----- Header files ------------------------------------------------------*/
86 /*----- Simple prime curves -----------------------------------------------*/
88 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
90 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
94 d->y = F_NEG(c->f, d->y, d->y);
98 static ec *ecfind(ec_curve *c, ec *d, mp *x)
103 q = F_SQR(f, MP_NEW, x);
104 p = F_MUL(f, MP_NEW, x, q);
105 q = F_MUL(f, q, x, c->a);
106 p = F_ADD(f, p, p, q);
107 p = F_ADD(f, p, p, c->b);
115 d->z = MP_COPY(f->one);
119 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
123 else if (F_ZEROP(c->f, a->y))
130 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
131 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
132 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
133 dx = F_ADD(f, dx, dx, c->a); /* %$3 x^2 + A$% */
134 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
135 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
137 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
138 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
139 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
140 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
141 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
142 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
153 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
157 else if (F_ZEROP(c->f, a->y))
161 mp *p, *q, *m, *s, *dx, *dy, *dz;
163 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
164 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
165 p = F_MUL(f, p, q, c->a); /* %$A z^4$% */
166 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
167 m = F_TPL(f, m, m); /* %$3 x^2$% */
168 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
170 q = F_DBL(f, q, a->y); /* %$2 y$% */
171 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
173 p = F_SQR(f, p, q); /* %$4 y^2$% */
174 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
175 q = F_SQR(f, q, p); /* %$16 y^4$% */
176 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
178 p = F_DBL(f, p, s); /* %$2 s$% */
179 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
180 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
182 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
183 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
184 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
197 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
201 else if (F_ZEROP(c->f, a->y))
205 mp *p, *q, *m, *s, *dx, *dy, *dz;
207 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
208 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
209 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
210 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
211 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
213 q = F_DBL(f, q, a->y); /* %$2 y$% */
214 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
216 p = F_SQR(f, p, q); /* %$4 y^2$% */
217 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
218 q = F_SQR(f, q, p); /* %$16 y^4$% */
219 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
221 p = F_DBL(f, p, s); /* %$2 s$% */
222 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
223 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
225 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
226 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
227 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
240 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
244 else if (EC_ATINF(a))
246 else if (EC_ATINF(b))
253 if (!MP_EQ(a->x, b->x)) {
254 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
255 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
256 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
257 lambda = F_MUL(f, MP_NEW, dy, dx);
258 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
259 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
263 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
264 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
265 dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */
266 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
267 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
268 lambda = F_MUL(f, MP_NEW, dx, dy);
269 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
272 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
273 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
274 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
275 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
276 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
277 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
288 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
291 c->ops->dbl(c, d, a);
292 else if (EC_ATINF(a))
294 else if (EC_ATINF(b))
298 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
300 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
301 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
302 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
303 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
305 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
306 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
307 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
308 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
310 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
311 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
320 return (c->ops->dbl(c, d, a));
327 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
328 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
330 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
331 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
333 p = F_SQR(f, uu, w); /* %$w^2$% */
334 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
335 u = F_MUL(f, u, p, w); /* %$w^3$% */
336 p = F_MUL(f, p, u, s); /* %$m w^3$% */
338 dx = F_SQR(f, u, r); /* %$r^2$% */
339 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
341 s = F_DBL(f, s, dx); /* %$2 x'$% */
342 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
343 dy = F_MUL(f, s, q, r); /* %$v r$% */
344 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
345 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
359 static int eccheck(ec_curve *c, const ec *p)
364 if (EC_ATINF(p)) return (0);
365 l = F_SQR(f, MP_NEW, p->y);
366 x = F_SQR(f, MP_NEW, p->x);
367 r = F_MUL(f, MP_NEW, x, p->x);
368 x = F_MUL(f, x, c->a, p->x);
369 r = F_ADD(f, r, r, x);
370 r = F_ADD(f, r, r, c->b);
371 rc = MP_EQ(l, r) ? 0 : -1;
378 static int ecprojcheck(ec_curve *c, const ec *p)
383 c->ops->fix(c, &t, p);
389 static void ecdestroy(ec_curve *c)
396 /* --- @ec_prime@, @ec_primeproj@ --- *
398 * Arguments: @field *f@ = the underlying field for this elliptic curve
399 * @mp *a, *b@ = the coefficients for this curve
401 * Returns: A pointer to the curve, or null.
403 * Use: Creates a curve structure for an elliptic curve defined over
404 * a prime field. The @primeproj@ variant uses projective
405 * coordinates, which can be a win.
408 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
410 ec_curve *c = CREATE(ec_curve);
411 c->ops = &ec_primeops;
413 c->a = F_IN(f, MP_NEW, a);
414 c->b = F_IN(f, MP_NEW, b);
418 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
420 ec_curve *c = CREATE(ec_curve);
423 ax = mp_add(MP_NEW, a, MP_THREE);
424 ax = F_IN(f, ax, ax);
426 c->ops = &ec_primeprojxops;
428 c->ops = &ec_primeprojops;
431 c->a = F_IN(f, MP_NEW, a);
432 c->b = F_IN(f, MP_NEW, b);
436 static const ec_ops ec_primeops = {
437 ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
438 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
441 static const ec_ops ec_primeprojops = {
442 ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
443 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
446 static const ec_ops ec_primeprojxops = {
447 ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
448 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
451 /*----- Test rig ----------------------------------------------------------*/
455 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
457 int main(int argc, char *argv[])
461 ec g = EC_INIT, d = EC_INIT;
463 int i, n = argc == 1 ? 1 : atoi(argv[1]);
465 printf("ec-prime: ");
468 b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
469 p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
470 r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
472 f = field_niceprime(p);
473 c = ec_primeproj(f, a, b);
475 g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
476 g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
478 for (i = 0; i < n; i++) {
479 ec_mul(c, &d, &g, r);
481 fprintf(stderr, "zero too early\n");
484 ec_add(c, &d, &d, &g);
486 fprintf(stderr, "didn't reach zero\n");
487 MP_EPRINT("d.x", d.x);
488 MP_EPRINT("d.y", d.y);
496 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
497 assert(!mparena_count(&mparena_global));
504 /*----- That's all, folks -------------------------------------------------*/