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 /*----- Header files ------------------------------------------------------*/
37 /*----- Main code ---------------------------------------------------------*/
39 static const ec_ops ec_binops, ec_binprojops;
41 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
45 d->y = F_ADD(c->f, d->y, d->y, d->x);
49 static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
53 mp *t = F_MUL(c->f, MP_NEW, d->x, d->z);
54 d->y = F_ADD(c->f, d->y, d->y, t);
60 static ec *ecfind(ec_curve *c, ec *d, mp *x)
66 y = F_SQRT(f, MP_NEW, c->b);
68 u = F_SQR(f, MP_NEW, x); /* %$x^2$% */
69 y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */
70 y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */
71 v = F_MUL(f, MP_NEW, u, x); /* %$x^3$% */
72 y = F_ADD(f, y, y, v); /* %$A = x^3 + a x^2 + b$% */
74 u = F_INV(f, u, u); /* %$x^{-2}$% */
75 v = F_MUL(f, v, u, y); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
76 y = F_QUADSOLVE(f, y, v); /* %$z^2 + z = B$% */
77 if (y) y = F_MUL(f, y, y, x); /* %$y = z x$% */
86 d->z = MP_COPY(f->one);
90 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
92 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
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, c->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'$% */
121 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
123 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
127 ecctx_bin *cc = (ecctx_bin *)c;
128 mp *dx, *dy, *dz, *u, *v;
130 dy = F_SQR(f, MP_NEW, a->z); /* %$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'$% */
158 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
162 else if (EC_ATINF(a))
164 else if (EC_ATINF(b))
171 if (!MP_EQ(a->x, b->x)) {
172 dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */
173 dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */
174 dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */
175 lambda = F_MUL(f, MP_NEW, dy, dx);
176 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
178 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
179 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
180 dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */
181 dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */
182 dx = F_ADD(f, dx, dx, b->x);
183 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
184 } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
188 dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
189 dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
190 lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
192 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
193 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
194 dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */
198 dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */
199 dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
200 dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
201 dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
212 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
215 c->ops->dbl(c, d, a);
216 else if (EC_ATINF(a))
218 else if (EC_ATINF(b))
222 mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
224 dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
225 u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */
226 t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */
227 s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */
229 dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */
230 uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */
231 t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */
232 ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */
234 w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */
235 r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */
244 return (c->ops->dbl(c, d, a));
252 l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */
254 dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */
256 ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */
257 t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */
258 v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */
260 t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */
262 uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */
263 dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */
264 uu = F_MUL(f, uu, t, r); /* %$t r$% */
265 dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */
266 r = F_SQR(f, r, w); /* %$w^2$% */
267 uu = F_MUL(f, uu, r, w); /* %$w^3$% */
268 dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */
270 r = F_SQR(f, r, l); /* %$l^2$% */
271 dy = F_MUL(f, uu, v, r); /* %$v l^2$% */
272 l = F_MUL(f, l, t, dx); /* %$t x'$% */
273 dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */
288 static int eccheck(ec_curve *c, const ec *p)
294 if (EC_ATINF(p)) return (0);
295 v = F_SQR(f, MP_NEW, p->x);
296 u = F_MUL(f, MP_NEW, v, p->x);
297 v = F_MUL(f, v, v, c->a);
298 u = F_ADD(f, u, u, v);
299 u = F_ADD(f, u, u, c->b);
300 v = F_MUL(f, v, p->x, p->y);
301 u = F_ADD(f, u, u, v);
302 v = F_SQR(f, v, p->y);
303 u = F_ADD(f, u, u, v);
304 rc = F_ZEROP(f, u) ? 0 : -1;
310 static int ecprojcheck(ec_curve *c, const ec *p)
315 c->ops->fix(c, &t, p);
321 static void ecdestroy(ec_curve *c)
323 ecctx_bin *cc = (ecctx_bin *)c;
326 if (cc->bb) MP_DROP(cc->bb);
330 /* --- @ec_bin@, @ec_binproj@ --- *
332 * Arguments: @field *f@ = the underlying field for this elliptic curve
333 * @mp *a, *b@ = the coefficients for this curve
335 * Returns: A pointer to the curve, or null.
337 * Use: Creates a curve structure for an elliptic curve defined over
338 * a binary field. The @binproj@ variant uses projective
339 * coordinates, which can be a win.
342 ec_curve *ec_bin(field *f, mp *a, mp *b)
344 ecctx_bin *cc = CREATE(ecctx_bin);
345 cc->c.ops = &ec_binops;
347 cc->c.a = F_IN(f, MP_NEW, a);
348 cc->c.b = F_IN(f, MP_NEW, b);
353 ec_curve *ec_binproj(field *f, mp *a, mp *b)
355 ecctx_bin *cc = CREATE(ecctx_bin);
356 cc->c.ops = &ec_binprojops;
358 cc->c.a = F_IN(f, MP_NEW, a);
359 cc->c.b = F_IN(f, MP_NEW, b);
360 cc->bb = F_SQRT(f, MP_NEW, cc->c.b);
362 cc->bb = F_SQRT(f, cc->bb, cc->bb);
372 static const ec_ops ec_binops = {
374 ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
375 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
378 static const ec_ops ec_binprojops = {
380 ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
381 ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
384 /*----- Test rig ----------------------------------------------------------*/
388 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
390 int main(int argc, char *argv[])
394 ec g = EC_INIT, d = EC_INIT;
395 mp *p, *a, *b, *r, *beta;
396 int i, n = argc == 1 ? 1 : atoi(argv[1]);
400 a = MP(0x7ffffffffffffffffffffffffffffffffffffffff);
401 b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
402 p = MP(0x800000000000000000000000000000000000000c9);
403 beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
404 r = MP(0x040000000000000000000292fe77e70c12a4234c32);
406 f = field_binnorm(p, beta);
407 c = ec_binproj(f, a, b);
408 g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
409 g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
411 for (i = 0; i < n; i++) {
412 ec_mul(c, &d, &g, r);
414 fprintf(stderr, "zero too early\n");
417 ec_add(c, &d, &d, &g);
419 fprintf(stderr, "didn't reach zero\n");
420 MP_EPRINTX("d.x", d.x);
421 MP_EPRINTX("d.y", d.y);
430 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta);
431 assert(!mparena_count(&mparena_global));
438 /*----- That's all, folks -------------------------------------------------*/