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 /*----- Header files ------------------------------------------------------*/
36 /*----- Simple prime curves -----------------------------------------------*/
38 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
40 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
44 d->y = F_NEG(c->f, d->y, d->y);
48 static ec *ecfind(ec_curve *c, ec *d, mp *x)
53 q = F_SQR(f, MP_NEW, x);
54 p = F_MUL(f, MP_NEW, x, q);
55 q = F_MUL(f, q, x, c->a);
56 p = F_ADD(f, p, p, q);
57 p = F_ADD(f, p, p, c->b);
65 d->z = MP_COPY(f->one);
69 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
71 if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
78 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
79 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
80 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
81 dx = F_ADD(f, dx, dx, c->a); /* %$3 x^2 + A$% */
82 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
83 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
85 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
86 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
87 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
88 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
89 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
90 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
101 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
103 if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
107 mp *p, *q, *m, *s, *dx, *dy, *dz;
109 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
110 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
111 p = F_MUL(f, p, q, c->a); /* %$A z^4$% */
112 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
113 m = F_TPL(f, m, m); /* %$3 x^2$% */
114 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
116 q = F_DBL(f, q, a->y); /* %$2 y$% */
117 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
119 p = F_SQR(f, p, q); /* %$4 y^2$% */
120 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
121 q = F_SQR(f, q, p); /* %$16 y^4$% */
122 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
124 p = F_DBL(f, p, s); /* %$2 s$% */
125 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
126 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
128 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
129 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
130 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
143 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
145 if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
149 mp *p, *q, *m, *s, *dx, *dy, *dz;
151 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
152 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
153 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
154 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
155 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
157 q = F_DBL(f, q, a->y); /* %$2 y$% */
158 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
160 p = F_SQR(f, p, q); /* %$4 y^2$% */
161 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
162 q = F_SQR(f, q, p); /* %$16 y^4$% */
163 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
165 p = F_DBL(f, p, s); /* %$2 s$% */
166 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
167 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
169 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
170 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
171 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
184 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
188 else if (EC_ATINF(a))
190 else if (EC_ATINF(b))
197 if (!MP_EQ(a->x, b->x)) {
198 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
199 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
200 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
201 lambda = F_MUL(f, MP_NEW, dy, dx);
202 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
203 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
207 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
208 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
209 dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */
210 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
211 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
212 lambda = F_MUL(f, MP_NEW, dx, dy);
213 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
216 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
217 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
218 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
219 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
220 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
221 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
232 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
235 c->ops->dbl(c, d, a);
236 else if (EC_ATINF(a))
238 else if (EC_ATINF(b))
242 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
244 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
245 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
246 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
247 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
249 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
250 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
251 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
252 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
254 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
255 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
264 return (c->ops->dbl(c, d, a));
271 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
272 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
274 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
275 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
277 p = F_SQR(f, uu, w); /* %$w^2$% */
278 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
279 u = F_MUL(f, u, p, w); /* %$w^3$% */
280 p = F_MUL(f, p, u, s); /* %$m w^3$% */
282 dx = F_SQR(f, u, r); /* %$r^2$% */
283 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
285 s = F_DBL(f, s, dx); /* %$2 x'$% */
286 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
287 dy = F_MUL(f, s, q, r); /* %$v r$% */
288 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
289 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
303 static int eccheck(ec_curve *c, const ec *p)
308 if (EC_ATINF(p)) return (0);
309 l = F_SQR(f, MP_NEW, p->y);
310 x = F_SQR(f, MP_NEW, p->x);
311 r = F_MUL(f, MP_NEW, x, p->x);
312 x = F_MUL(f, x, c->a, p->x);
313 r = F_ADD(f, r, r, x);
314 r = F_ADD(f, r, r, c->b);
315 rc = MP_EQ(l, r) ? 0 : -1;
322 static int ecprojcheck(ec_curve *c, const ec *p)
327 c->ops->fix(c, &t, p);
333 static void ecdestroy(ec_curve *c)
340 /* --- @ec_prime@, @ec_primeproj@ --- *
342 * Arguments: @field *f@ = the underlying field for this elliptic curve
343 * @mp *a, *b@ = the coefficients for this curve
345 * Returns: A pointer to the curve, or null.
347 * Use: Creates a curve structure for an elliptic curve defined over
348 * a prime field. The @primeproj@ variant uses projective
349 * coordinates, which can be a win.
352 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
354 ec_curve *c = CREATE(ec_curve);
355 c->ops = &ec_primeops;
357 c->a = F_IN(f, MP_NEW, a);
358 c->b = F_IN(f, MP_NEW, b);
362 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
364 ec_curve *c = CREATE(ec_curve);
367 ax = mp_add(MP_NEW, a, MP_THREE);
368 ax = F_IN(f, ax, ax);
370 c->ops = &ec_primeprojxops;
372 c->ops = &ec_primeprojops;
375 c->a = F_IN(f, MP_NEW, a);
376 c->b = F_IN(f, MP_NEW, b);
380 static const ec_ops ec_primeops = {
382 ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
383 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
386 static const ec_ops ec_primeprojops = {
388 ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
389 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
392 static const ec_ops ec_primeprojxops = {
394 ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
395 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
398 /*----- Test rig ----------------------------------------------------------*/
402 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
404 int main(int argc, char *argv[])
408 ec g = EC_INIT, d = EC_INIT;
410 int i, n = argc == 1 ? 1 : atoi(argv[1]);
412 printf("ec-prime: ");
415 b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
416 p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
417 r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
419 f = field_niceprime(p);
420 c = ec_primeproj(f, a, b);
422 g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
423 g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
425 for (i = 0; i < n; i++) {
426 ec_mul(c, &d, &g, r);
428 fprintf(stderr, "zero too early\n");
431 ec_add(c, &d, &d, &g);
433 fprintf(stderr, "didn't reach zero\n");
434 MP_EPRINT("d.x", d.x);
435 MP_EPRINT("d.y", d.y);
443 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
444 assert(!mparena_count(&mparena_global));
451 /*----- That's all, folks -------------------------------------------------*/