3 * $Id: ec-bin.c,v 1.3 2004/03/22 02:19:09 mdw Exp $
5 * Arithmetic for elliptic curves over binary fields
7 * (c) 2004 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 --------------------------------------------------*
33 * Revision 1.3 2004/03/22 02:19:09 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 1.2 2004/03/21 22:52:06 mdw
39 * Merge and close elliptic curve branch.
41 * Revision 1.1.2.1 2004/03/21 22:39:46 mdw
42 * Elliptic curves on binary fields work.
46 /*----- Header files ------------------------------------------------------*/
52 /*----- Data structures ---------------------------------------------------*/
54 typedef struct ecctx {
60 /*----- Main code ---------------------------------------------------------*/
62 static const ec_ops ec_binops, ec_binprojops;
64 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
68 d->y = F_ADD(c->f, d->y, d->y, d->x);
72 static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
76 mp *t = F_MUL(c->f, MP_NEW, d->x, d->z);
77 d->y = F_ADD(c->f, d->y, d->y, t);
83 static ec *ecfind(ec_curve *c, ec *d, mp *x)
89 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
91 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
95 ecctx *cc = (ecctx *)c;
99 dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
100 dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
101 lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
103 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
104 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
105 dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */
107 dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */
108 dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
109 dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
110 dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
121 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
123 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
127 ecctx *cc = (ecctx *)c;
128 mp *dx, *dy, *dz, *u, *v;
130 dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
131 dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */
132 dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */
133 dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */
134 dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */
136 dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */
138 dy = F_SQR(f, dy, a->x); /* %$x^2$% */
139 u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */
140 u = F_ADD(f, u, u, dz); /* %$z' + y z$% */
141 u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */
143 v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */
144 dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */
145 v = F_MUL(f, v, u, dx); /* %$u x'$% */
146 dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */
154 assert(!(d->x->f & MP_DESTROYED));
155 assert(!(d->y->f & MP_DESTROYED));
156 assert(!(d->z->f & MP_DESTROYED));
161 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
165 else if (EC_ATINF(a))
167 else if (EC_ATINF(b))
171 ecctx *cc = (ecctx *)c;
175 if (!MP_EQ(a->x, b->x)) {
176 dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */
177 dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */
178 dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */
179 lambda = F_MUL(f, MP_NEW, dy, dx);
180 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
182 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
183 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
184 dx = F_ADD(f, dx, dx, cc->a); /* %$a + \lambda^2 + \lambda$% */
185 dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */
186 dx = F_ADD(f, dx, dx, b->x);
187 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
188 } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
192 dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
193 dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
194 lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
196 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
197 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
198 dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */
202 dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */
203 dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
204 dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
205 dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
216 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
219 c->ops->dbl(c, d, a);
220 else if (EC_ATINF(a))
222 else if (EC_ATINF(b))
226 ecctx *cc = (ecctx *)c;
227 mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
229 dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
230 u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */
231 t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */
232 s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */
234 dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */
235 uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */
236 t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */
237 ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */
239 w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */
240 r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */
249 return (c->ops->dbl(c, d, a));
257 l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */
259 dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */
261 ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */
262 t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */
263 v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */
265 t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */
267 uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */
268 dx = F_MUL(f, MP_NEW, uu, cc->a); /* %$a z'^2$% */
269 uu = F_MUL(f, uu, t, r); /* %$t r$% */
270 dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */
271 r = F_SQR(f, r, w); /* %$w^2$% */
272 uu = F_MUL(f, uu, r, w); /* %$w^3$% */
273 dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */
275 r = F_SQR(f, r, l); /* %$l^2$% */
276 dy = F_MUL(f, uu, v, r); /* %$v l^2$% */
277 l = F_MUL(f, l, t, dx); /* %$t x'$% */
278 dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */
293 static int eccheck(ec_curve *c, const ec *p)
295 ecctx *cc = (ecctx *)c;
300 v = F_SQR(f, MP_NEW, p->x);
301 u = F_MUL(f, MP_NEW, v, p->x);
302 v = F_MUL(f, v, v, cc->a);
303 u = F_ADD(f, u, u, v);
304 u = F_ADD(f, u, u, cc->b);
305 v = F_MUL(f, v, p->x, p->y);
306 u = F_ADD(f, u, u, v);
307 v = F_SQR(f, v, p->y);
308 u = F_ADD(f, u, u, v);
315 static int ecprojcheck(ec_curve *c, const ec *p)
320 c->ops->fix(c, &t, p);
326 static void ecdestroy(ec_curve *c)
328 ecctx *cc = (ecctx *)c;
331 if (cc->bb) MP_DROP(cc->bb);
335 /* --- @ec_bin@, @ec_binproj@ --- *
337 * Arguments: @field *f@ = the underlying field for this elliptic curve
338 * @mp *a, *b@ = the coefficients for this curve
340 * Returns: A pointer to the curve.
342 * Use: Creates a curve structure for an elliptic curve defined over
343 * a binary field. The @binproj@ variant uses projective
344 * coordinates, which can be a win.
347 ec_curve *ec_bin(field *f, mp *a, mp *b)
349 ecctx *cc = CREATE(ecctx);
350 cc->c.ops = &ec_binops;
352 cc->a = F_IN(f, MP_NEW, a);
353 cc->b = F_IN(f, MP_NEW, b);
358 ec_curve *ec_binproj(field *f, mp *a, mp *b)
360 ecctx *cc = CREATE(ecctx);
361 cc->c.ops = &ec_binprojops;
363 cc->a = F_IN(f, MP_NEW, a);
364 cc->b = F_IN(f, MP_NEW, b);
365 cc->bb = F_SQRT(f, MP_NEW, b);
366 cc->bb = F_SQRT(f, cc->bb, cc->bb);
370 static const ec_ops ec_binops = {
371 ecdestroy, ec_idin, ec_idout, ec_idfix,
372 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
375 static const ec_ops ec_binprojops = {
376 ecdestroy, ec_projin, ec_projout, ec_projfix,
377 0, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
380 /*----- Test rig ----------------------------------------------------------*/
384 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
386 int main(int argc, char *argv[])
390 ec g = EC_INIT, d = EC_INIT;
392 int i, n = argc == 1 ? 1 : atoi(argv[1]);
397 b = MP(0x066647ede6c332c7f8c0923bb58213b333b20e9ce4281fe115f7d8f90ad);
398 p = MP(0x20000000000000000000000000000000000000004000000000000000001);
400 MP(6901746346790563787434755862277025555839812737345013555379383634485462);
402 f = field_binpoly(p);
403 c = ec_binproj(f, a, b);
405 g.x = MP(0x0fac9dfcbac8313bb2139f1bb755fef65bc391f8b36f8f8eb7371fd558b);
406 g.y = MP(0x1006a08a41903350678e58528bebf8a0beff867a7ca36716f7e01f81052);
408 for (i = 0; i < n; i++) {
409 ec_mul(c, &d, &g, r);
411 fprintf(stderr, "zero too early\n");
414 ec_add(c, &d, &d, &g);
416 fprintf(stderr, "didn't reach zero\n");
417 MP_EPRINTX("d.x", d.x);
418 MP_EPRINTX("d.y", d.y);
427 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
428 assert(!mparena_count(&mparena_global));
435 /*----- That's all, folks -------------------------------------------------*/