| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id$ |
| 4 | * |
| 5 | * Arithmetic for elliptic curves over binary fields |
| 6 | * |
| 7 | * (c) 2004 Straylight/Edgeware |
| 8 | */ |
| 9 | |
| 10 | /*----- Licensing notice --------------------------------------------------* |
| 11 | * |
| 12 | * This file is part of Catacomb. |
| 13 | * |
| 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. |
| 18 | * |
| 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. |
| 23 | * |
| 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, |
| 27 | * MA 02111-1307, USA. |
| 28 | */ |
| 29 | |
| 30 | /*----- Header files ------------------------------------------------------*/ |
| 31 | |
| 32 | #include <mLib/sub.h> |
| 33 | |
| 34 | #include "ec.h" |
| 35 | #include "ec-guts.h" |
| 36 | |
| 37 | /*----- Main code ---------------------------------------------------------*/ |
| 38 | |
| 39 | static const ec_ops ec_binops, ec_binprojops; |
| 40 | |
| 41 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) |
| 42 | { |
| 43 | EC_COPY(d, p); |
| 44 | if (d->x) |
| 45 | d->y = F_ADD(c->f, d->y, d->y, d->x); |
| 46 | return (d); |
| 47 | } |
| 48 | |
| 49 | static ec *ecprojneg(ec_curve *c, ec *d, const ec *p) |
| 50 | { |
| 51 | EC_COPY(d, p); |
| 52 | if (d->x) { |
| 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); |
| 55 | MP_DROP(t); |
| 56 | } |
| 57 | return (d); |
| 58 | } |
| 59 | |
| 60 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
| 61 | { |
| 62 | field *f = c->f; |
| 63 | mp *y, *u, *v; |
| 64 | |
| 65 | if (F_ZEROP(f, x)) |
| 66 | y = F_SQRT(f, MP_NEW, c->b); |
| 67 | else { |
| 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); /* %$A = x^3 + a x^2 + b$% */ |
| 73 | if (!F_ZEROP(f, y)) { |
| 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); /* %$y = z x$% */ |
| 78 | } |
| 79 | MP_DROP(u); |
| 80 | MP_DROP(v); |
| 81 | } |
| 82 | if (!y) return (0); |
| 83 | EC_DESTROY(d); |
| 84 | d->x = MP_COPY(x); |
| 85 | d->y = y; |
| 86 | d->z = MP_COPY(f->one); |
| 87 | return (d); |
| 88 | } |
| 89 | |
| 90 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) |
| 91 | { |
| 92 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
| 93 | EC_SETINF(d); |
| 94 | else { |
| 95 | field *f = c->f; |
| 96 | mp *lambda; |
| 97 | mp *dx, *dy; |
| 98 | |
| 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); /* %$\lambda = x + y/x$% */ |
| 102 | |
| 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); /* %$x' = a + \lambda^2 + \lambda$% */ |
| 106 | |
| 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'$% */ |
| 111 | |
| 112 | EC_DESTROY(d); |
| 113 | d->x = dx; |
| 114 | d->y = dy; |
| 115 | d->z = 0; |
| 116 | MP_DROP(lambda); |
| 117 | } |
| 118 | return (d); |
| 119 | } |
| 120 | |
| 121 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
| 122 | { |
| 123 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
| 124 | EC_SETINF(d); |
| 125 | else { |
| 126 | field *f = c->f; |
| 127 | ecctx_bin *cc = (ecctx_bin *)c; |
| 128 | mp *dx, *dy, *dz, *u, *v; |
| 129 | |
| 130 | dy = F_SQR(f, MP_NEW, a->z); /* %$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$% */ |
| 135 | |
| 136 | dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */ |
| 137 | |
| 138 | dy = F_SQR(f, dy, a->x); /* %$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$% */ |
| 142 | |
| 143 | v = F_SQR(f, MP_NEW, dy); /* %$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'$% */ |
| 147 | |
| 148 | EC_DESTROY(d); |
| 149 | d->x = dx; |
| 150 | d->y = dy; |
| 151 | d->z = dz; |
| 152 | MP_DROP(u); |
| 153 | MP_DROP(v); |
| 154 | } |
| 155 | return (d); |
| 156 | } |
| 157 | |
| 158 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
| 159 | { |
| 160 | if (a == b) |
| 161 | ecdbl(c, d, a); |
| 162 | else if (EC_ATINF(a)) |
| 163 | EC_COPY(d, b); |
| 164 | else if (EC_ATINF(b)) |
| 165 | EC_COPY(d, a); |
| 166 | else { |
| 167 | field *f = c->f; |
| 168 | mp *lambda; |
| 169 | mp *dx, *dy; |
| 170 | |
| 171 | if (!MP_EQ(a->x, b->x)) { |
| 172 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$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)$% */ |
| 177 | |
| 178 | dx = F_SQR(f, dx, lambda); /* %$\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$% */ |
| 184 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
| 185 | EC_SETINF(d); |
| 186 | return (d); |
| 187 | } else { |
| 188 | dx = F_INV(f, MP_NEW, a->x); /* %$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$% */ |
| 191 | |
| 192 | dx = F_SQR(f, dx, lambda); /* %$\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$% */ |
| 195 | dy = MP_NEW; |
| 196 | } |
| 197 | |
| 198 | dy = F_ADD(f, dy, a->x, dx); /* %$ 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'$% */ |
| 202 | |
| 203 | EC_DESTROY(d); |
| 204 | d->x = dx; |
| 205 | d->y = dy; |
| 206 | d->z = 0; |
| 207 | MP_DROP(lambda); |
| 208 | } |
| 209 | return (d); |
| 210 | } |
| 211 | |
| 212 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
| 213 | { |
| 214 | if (a == b) |
| 215 | c->ops->dbl(c, d, a); |
| 216 | else if (EC_ATINF(a)) |
| 217 | EC_COPY(d, b); |
| 218 | else if (EC_ATINF(b)) |
| 219 | EC_COPY(d, a); |
| 220 | else { |
| 221 | field *f = c->f; |
| 222 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
| 223 | |
| 224 | dz = F_SQR(f, MP_NEW, b->z); /* %$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$% */ |
| 228 | |
| 229 | dz = F_SQR(f, dz, a->z); /* %$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$% */ |
| 233 | |
| 234 | w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */ |
| 235 | r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */ |
| 236 | if (F_ZEROP(f, w)) { |
| 237 | MP_DROP(w); |
| 238 | MP_DROP(uu); |
| 239 | MP_DROP(ss); |
| 240 | MP_DROP(t); |
| 241 | MP_DROP(dz); |
| 242 | if (F_ZEROP(f, r)) { |
| 243 | MP_DROP(r); |
| 244 | return (c->ops->dbl(c, d, a)); |
| 245 | } else { |
| 246 | MP_DROP(r); |
| 247 | EC_SETINF(d); |
| 248 | return (d); |
| 249 | } |
| 250 | } |
| 251 | |
| 252 | l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */ |
| 253 | |
| 254 | dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */ |
| 255 | |
| 256 | ss = F_MUL(f, ss, r, b->x); /* %$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$% */ |
| 259 | |
| 260 | t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */ |
| 261 | |
| 262 | uu = F_SQR(f, MP_NEW, dz); /* %$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$% */ |
| 269 | |
| 270 | r = F_SQR(f, r, l); /* %$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$% */ |
| 274 | |
| 275 | EC_DESTROY(d); |
| 276 | d->x = dx; |
| 277 | d->y = dy; |
| 278 | d->z = dz; |
| 279 | MP_DROP(l); |
| 280 | MP_DROP(r); |
| 281 | MP_DROP(w); |
| 282 | MP_DROP(t); |
| 283 | MP_DROP(v); |
| 284 | } |
| 285 | return (d); |
| 286 | } |
| 287 | |
| 288 | static int eccheck(ec_curve *c, const ec *p) |
| 289 | { |
| 290 | field *f = c->f; |
| 291 | int rc; |
| 292 | mp *u, *v; |
| 293 | |
| 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; |
| 305 | mp_drop(u); |
| 306 | mp_drop(v); |
| 307 | return (rc); |
| 308 | } |
| 309 | |
| 310 | static int ecprojcheck(ec_curve *c, const ec *p) |
| 311 | { |
| 312 | ec t = EC_INIT; |
| 313 | int rc; |
| 314 | |
| 315 | c->ops->fix(c, &t, p); |
| 316 | rc = eccheck(c, &t); |
| 317 | EC_DESTROY(&t); |
| 318 | return (rc); |
| 319 | } |
| 320 | |
| 321 | static void ecdestroy(ec_curve *c) |
| 322 | { |
| 323 | ecctx_bin *cc = (ecctx_bin *)c; |
| 324 | MP_DROP(cc->c.a); |
| 325 | MP_DROP(cc->c.b); |
| 326 | if (cc->bb) MP_DROP(cc->bb); |
| 327 | DESTROY(cc); |
| 328 | } |
| 329 | |
| 330 | /* --- @ec_bin@, @ec_binproj@ --- * |
| 331 | * |
| 332 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
| 333 | * @mp *a, *b@ = the coefficients for this curve |
| 334 | * |
| 335 | * Returns: A pointer to the curve, or null. |
| 336 | * |
| 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. |
| 340 | */ |
| 341 | |
| 342 | ec_curve *ec_bin(field *f, mp *a, mp *b) |
| 343 | { |
| 344 | ecctx_bin *cc = CREATE(ecctx_bin); |
| 345 | cc->c.ops = &ec_binops; |
| 346 | cc->c.f = f; |
| 347 | cc->c.a = F_IN(f, MP_NEW, a); |
| 348 | cc->c.b = F_IN(f, MP_NEW, b); |
| 349 | cc->bb = 0; |
| 350 | return (&cc->c); |
| 351 | } |
| 352 | |
| 353 | ec_curve *ec_binproj(field *f, mp *a, mp *b) |
| 354 | { |
| 355 | ecctx_bin *cc = CREATE(ecctx_bin); |
| 356 | cc->c.ops = &ec_binprojops; |
| 357 | cc->c.f = f; |
| 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); |
| 361 | if (cc->bb) |
| 362 | cc->bb = F_SQRT(f, cc->bb, cc->bb); |
| 363 | if (!cc->bb) { |
| 364 | MP_DROP(cc->c.a); |
| 365 | MP_DROP(cc->c.b); |
| 366 | DESTROY(cc); |
| 367 | return (0); |
| 368 | } |
| 369 | return (&cc->c); |
| 370 | } |
| 371 | |
| 372 | static const ec_ops ec_binops = { |
| 373 | "bin", |
| 374 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
| 375 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
| 376 | }; |
| 377 | |
| 378 | static const ec_ops ec_binprojops = { |
| 379 | "binproj", |
| 380 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
| 381 | ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
| 382 | }; |
| 383 | |
| 384 | /*----- Test rig ----------------------------------------------------------*/ |
| 385 | |
| 386 | #ifdef TEST_RIG |
| 387 | |
| 388 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
| 389 | |
| 390 | int main(int argc, char *argv[]) |
| 391 | { |
| 392 | field *f; |
| 393 | ec_curve *c; |
| 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]); |
| 397 | |
| 398 | printf("ec-bin: "); |
| 399 | fflush(stdout); |
| 400 | a = MP(0x7ffffffffffffffffffffffffffffffffffffffff); |
| 401 | b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb); |
| 402 | p = MP(0x800000000000000000000000000000000000000c9); |
| 403 | beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef); |
| 404 | r = MP(0x040000000000000000000292fe77e70c12a4234c32); |
| 405 | |
| 406 | f = field_binnorm(p, beta); |
| 407 | c = ec_binproj(f, a, b); |
| 408 | g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8); |
| 409 | g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f); |
| 410 | |
| 411 | for (i = 0; i < n; i++) { |
| 412 | ec_mul(c, &d, &g, r); |
| 413 | if (EC_ATINF(&d)) { |
| 414 | fprintf(stderr, "zero too early\n"); |
| 415 | return (1); |
| 416 | } |
| 417 | ec_add(c, &d, &d, &g); |
| 418 | if (!EC_ATINF(&d)) { |
| 419 | fprintf(stderr, "didn't reach zero\n"); |
| 420 | MP_EPRINTX("d.x", d.x); |
| 421 | MP_EPRINTX("d.y", d.y); |
| 422 | return (1); |
| 423 | } |
| 424 | ec_destroy(&d); |
| 425 | } |
| 426 | |
| 427 | ec_destroy(&g); |
| 428 | ec_destroycurve(c); |
| 429 | F_DESTROY(f); |
| 430 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta); |
| 431 | assert(!mparena_count(&mparena_global)); |
| 432 | printf("ok\n"); |
| 433 | return (0); |
| 434 | } |
| 435 | |
| 436 | #endif |
| 437 | |
| 438 | /*----- That's all, folks -------------------------------------------------*/ |