| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * Elliptic curves over prime fields |
| 4 | * |
| 5 | * (c) 2001 Straylight/Edgeware |
| 6 | */ |
| 7 | |
| 8 | /*----- Licensing notice --------------------------------------------------* |
| 9 | * |
| 10 | * This file is part of Catacomb. |
| 11 | * |
| 12 | * Catacomb is free software; you can redistribute it and/or modify |
| 13 | * it under the terms of the GNU Library General Public License as |
| 14 | * published by the Free Software Foundation; either version 2 of the |
| 15 | * License, or (at your option) any later version. |
| 16 | * |
| 17 | * Catacomb is distributed in the hope that it will be useful, |
| 18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| 20 | * GNU Library General Public License for more details. |
| 21 | * |
| 22 | * You should have received a copy of the GNU Library General Public |
| 23 | * License along with Catacomb; if not, write to the Free |
| 24 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
| 25 | * MA 02111-1307, USA. |
| 26 | */ |
| 27 | |
| 28 | /*----- Header files ------------------------------------------------------*/ |
| 29 | |
| 30 | #include <mLib/sub.h> |
| 31 | |
| 32 | #include "ec.h" |
| 33 | |
| 34 | /*----- Simple prime curves -----------------------------------------------*/ |
| 35 | |
| 36 | static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops; |
| 37 | |
| 38 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) |
| 39 | { |
| 40 | EC_COPY(d, p); |
| 41 | if (d->y) |
| 42 | d->y = F_NEG(c->f, d->y, d->y); |
| 43 | return (d); |
| 44 | } |
| 45 | |
| 46 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
| 47 | { |
| 48 | mp *p, *q; |
| 49 | field *f = c->f; |
| 50 | |
| 51 | q = F_SQR(f, MP_NEW, x); |
| 52 | p = F_MUL(f, MP_NEW, x, q); |
| 53 | q = F_MUL(f, q, x, c->a); |
| 54 | p = F_ADD(f, p, p, q); |
| 55 | p = F_ADD(f, p, p, c->b); |
| 56 | MP_DROP(q); |
| 57 | p = F_SQRT(f, p, p); |
| 58 | if (!p) |
| 59 | return (0); |
| 60 | EC_DESTROY(d); |
| 61 | d->x = MP_COPY(x); |
| 62 | d->y = p; |
| 63 | d->z = MP_COPY(f->one); |
| 64 | return (d); |
| 65 | } |
| 66 | |
| 67 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) |
| 68 | { |
| 69 | if (EC_ATINF(a) || F_ZEROP(c->f, a->y)) |
| 70 | EC_SETINF(d); |
| 71 | else { |
| 72 | field *f = c->f; |
| 73 | mp *lambda; |
| 74 | mp *dy, *dx; |
| 75 | |
| 76 | dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */ |
| 77 | dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */ |
| 78 | dx = F_TPL(f, dx, dx); /* %$3 x^2$% */ |
| 79 | dx = F_ADD(f, dx, dx, c->a); /* %$3 x^2 + A$% */ |
| 80 | dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */ |
| 81 | lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */ |
| 82 | |
| 83 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
| 84 | dy = F_DBL(f, dy, a->x); /* %$2 x$% */ |
| 85 | dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */ |
| 86 | dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */ |
| 87 | dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */ |
| 88 | dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */ |
| 89 | |
| 90 | EC_DESTROY(d); |
| 91 | d->x = dx; |
| 92 | d->y = dy; |
| 93 | d->z = 0; |
| 94 | MP_DROP(lambda); |
| 95 | } |
| 96 | return (d); |
| 97 | } |
| 98 | |
| 99 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
| 100 | { |
| 101 | if (EC_ATINF(a) || F_ZEROP(c->f, a->y)) |
| 102 | EC_SETINF(d); |
| 103 | else { |
| 104 | field *f = c->f; |
| 105 | mp *p, *q, *m, *s, *dx, *dy, *dz; |
| 106 | |
| 107 | p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
| 108 | q = F_SQR(f, MP_NEW, p); /* %$z^4$% */ |
| 109 | p = F_MUL(f, p, q, c->a); /* %$A z^4$% */ |
| 110 | m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */ |
| 111 | m = F_TPL(f, m, m); /* %$3 x^2$% */ |
| 112 | m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */ |
| 113 | |
| 114 | q = F_DBL(f, q, a->y); /* %$2 y$% */ |
| 115 | dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */ |
| 116 | |
| 117 | p = F_SQR(f, p, q); /* %$4 y^2$% */ |
| 118 | s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */ |
| 119 | q = F_SQR(f, q, p); /* %$16 y^4$% */ |
| 120 | q = F_HLV(f, q, q); /* %$t = 8 y^4$% */ |
| 121 | |
| 122 | p = F_DBL(f, p, s); /* %$2 s$% */ |
| 123 | dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */ |
| 124 | dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */ |
| 125 | |
| 126 | s = F_SUB(f, s, s, dx); /* %$s - x'$% */ |
| 127 | dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */ |
| 128 | dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */ |
| 129 | |
| 130 | EC_DESTROY(d); |
| 131 | d->x = dx; |
| 132 | d->y = dy; |
| 133 | d->z = dz; |
| 134 | MP_DROP(m); |
| 135 | MP_DROP(q); |
| 136 | MP_DROP(s); |
| 137 | } |
| 138 | return (d); |
| 139 | } |
| 140 | |
| 141 | static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a) |
| 142 | { |
| 143 | if (EC_ATINF(a) || F_ZEROP(c->f, a->y)) |
| 144 | EC_SETINF(d); |
| 145 | else { |
| 146 | field *f = c->f; |
| 147 | mp *p, *q, *m, *s, *dx, *dy, *dz; |
| 148 | |
| 149 | m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
| 150 | p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */ |
| 151 | q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */ |
| 152 | m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */ |
| 153 | m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */ |
| 154 | |
| 155 | q = F_DBL(f, q, a->y); /* %$2 y$% */ |
| 156 | dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */ |
| 157 | |
| 158 | p = F_SQR(f, p, q); /* %$4 y^2$% */ |
| 159 | s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */ |
| 160 | q = F_SQR(f, q, p); /* %$16 y^4$% */ |
| 161 | q = F_HLV(f, q, q); /* %$t = 8 y^4$% */ |
| 162 | |
| 163 | p = F_DBL(f, p, s); /* %$2 s$% */ |
| 164 | dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */ |
| 165 | dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */ |
| 166 | |
| 167 | s = F_SUB(f, s, s, dx); /* %$s - x'$% */ |
| 168 | dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */ |
| 169 | dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */ |
| 170 | |
| 171 | EC_DESTROY(d); |
| 172 | d->x = dx; |
| 173 | d->y = dy; |
| 174 | d->z = dz; |
| 175 | MP_DROP(m); |
| 176 | MP_DROP(q); |
| 177 | MP_DROP(s); |
| 178 | } |
| 179 | return (d); |
| 180 | } |
| 181 | |
| 182 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
| 183 | { |
| 184 | if (a == b) |
| 185 | ecdbl(c, d, a); |
| 186 | else if (EC_ATINF(a)) |
| 187 | EC_COPY(d, b); |
| 188 | else if (EC_ATINF(b)) |
| 189 | EC_COPY(d, a); |
| 190 | else { |
| 191 | field *f = c->f; |
| 192 | mp *lambda; |
| 193 | mp *dy, *dx; |
| 194 | |
| 195 | if (!MP_EQ(a->x, b->x)) { |
| 196 | dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */ |
| 197 | dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */ |
| 198 | dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */ |
| 199 | lambda = F_MUL(f, MP_NEW, dy, dx); |
| 200 | /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */ |
| 201 | } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) { |
| 202 | EC_SETINF(d); |
| 203 | return (d); |
| 204 | } else { |
| 205 | dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */ |
| 206 | dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */ |
| 207 | dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */ |
| 208 | dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */ |
| 209 | dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */ |
| 210 | lambda = F_MUL(f, MP_NEW, dx, dy); |
| 211 | /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */ |
| 212 | } |
| 213 | |
| 214 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
| 215 | dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */ |
| 216 | dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */ |
| 217 | dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */ |
| 218 | dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */ |
| 219 | dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */ |
| 220 | |
| 221 | EC_DESTROY(d); |
| 222 | d->x = dx; |
| 223 | d->y = dy; |
| 224 | d->z = 0; |
| 225 | MP_DROP(lambda); |
| 226 | } |
| 227 | return (d); |
| 228 | } |
| 229 | |
| 230 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
| 231 | { |
| 232 | if (a == b) |
| 233 | c->ops->dbl(c, d, a); |
| 234 | else if (EC_ATINF(a)) |
| 235 | EC_COPY(d, b); |
| 236 | else if (EC_ATINF(b)) |
| 237 | EC_COPY(d, a); |
| 238 | else { |
| 239 | field *f = c->f; |
| 240 | mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz; |
| 241 | |
| 242 | q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */ |
| 243 | u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */ |
| 244 | p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */ |
| 245 | s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */ |
| 246 | |
| 247 | q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ |
| 248 | uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/ |
| 249 | p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */ |
| 250 | ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */ |
| 251 | |
| 252 | w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */ |
| 253 | r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */ |
| 254 | if (F_ZEROP(f, w)) { |
| 255 | MP_DROP(w); |
| 256 | MP_DROP(u); |
| 257 | MP_DROP(s); |
| 258 | MP_DROP(uu); |
| 259 | MP_DROP(ss); |
| 260 | if (F_ZEROP(f, r)) { |
| 261 | MP_DROP(r); |
| 262 | return (c->ops->dbl(c, d, a)); |
| 263 | } else { |
| 264 | MP_DROP(r); |
| 265 | EC_SETINF(d); |
| 266 | return (d); |
| 267 | } |
| 268 | } |
| 269 | u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */ |
| 270 | s = F_ADD(f, s, s, ss); /* %$m = ss + s$% */ |
| 271 | |
| 272 | uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */ |
| 273 | dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */ |
| 274 | |
| 275 | p = F_SQR(f, uu, w); /* %$w^2$% */ |
| 276 | q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */ |
| 277 | u = F_MUL(f, u, p, w); /* %$w^3$% */ |
| 278 | p = F_MUL(f, p, u, s); /* %$m w^3$% */ |
| 279 | |
| 280 | dx = F_SQR(f, u, r); /* %$r^2$% */ |
| 281 | dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */ |
| 282 | |
| 283 | s = F_DBL(f, s, dx); /* %$2 x'$% */ |
| 284 | q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */ |
| 285 | dy = F_MUL(f, s, q, r); /* %$v r$% */ |
| 286 | dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */ |
| 287 | dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */ |
| 288 | |
| 289 | EC_DESTROY(d); |
| 290 | d->x = dx; |
| 291 | d->y = dy; |
| 292 | d->z = dz; |
| 293 | MP_DROP(p); |
| 294 | MP_DROP(q); |
| 295 | MP_DROP(r); |
| 296 | MP_DROP(w); |
| 297 | } |
| 298 | return (d); |
| 299 | } |
| 300 | |
| 301 | static int eccheck(ec_curve *c, const ec *p) |
| 302 | { |
| 303 | field *f = c->f; |
| 304 | mp *l, *x, *r; |
| 305 | int rc; |
| 306 | if (EC_ATINF(p)) return (0); |
| 307 | l = F_SQR(f, MP_NEW, p->y); |
| 308 | x = F_SQR(f, MP_NEW, p->x); |
| 309 | r = F_MUL(f, MP_NEW, x, p->x); |
| 310 | x = F_MUL(f, x, c->a, p->x); |
| 311 | r = F_ADD(f, r, r, x); |
| 312 | r = F_ADD(f, r, r, c->b); |
| 313 | rc = MP_EQ(l, r) ? 0 : -1; |
| 314 | mp_drop(l); |
| 315 | mp_drop(x); |
| 316 | mp_drop(r); |
| 317 | return (rc); |
| 318 | } |
| 319 | |
| 320 | static int ecprojcheck(ec_curve *c, const ec *p) |
| 321 | { |
| 322 | ec t = EC_INIT; |
| 323 | int rc; |
| 324 | |
| 325 | c->ops->fix(c, &t, p); |
| 326 | rc = eccheck(c, &t); |
| 327 | EC_DESTROY(&t); |
| 328 | return (rc); |
| 329 | } |
| 330 | |
| 331 | static int eccompr(ec_curve *c, const ec *p) |
| 332 | { |
| 333 | /* --- Just take the LSB of %$y$% --- |
| 334 | * |
| 335 | * Since @p@ is odd, either %$y$% or %$-y = p - y$% must be odd, so this |
| 336 | * disambiguates. |
| 337 | */ |
| 338 | |
| 339 | return (MP_ODDP(p->y)); |
| 340 | } |
| 341 | |
| 342 | static void ecdestroy(ec_curve *c) |
| 343 | { |
| 344 | MP_DROP(c->a); |
| 345 | MP_DROP(c->b); |
| 346 | DESTROY(c); |
| 347 | } |
| 348 | |
| 349 | /* --- @ec_prime@, @ec_primeproj@ --- * |
| 350 | * |
| 351 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
| 352 | * @mp *a, *b@ = the coefficients for this curve |
| 353 | * |
| 354 | * Returns: A pointer to the curve, or null. |
| 355 | * |
| 356 | * Use: Creates a curve structure for an elliptic curve defined over |
| 357 | * a prime field. The @primeproj@ variant uses projective |
| 358 | * coordinates, which can be a win. |
| 359 | */ |
| 360 | |
| 361 | extern ec_curve *ec_prime(field *f, mp *a, mp *b) |
| 362 | { |
| 363 | ec_curve *c = CREATE(ec_curve); |
| 364 | c->ops = &ec_primeops; |
| 365 | c->f = f; |
| 366 | c->a = F_IN(f, MP_NEW, a); |
| 367 | c->b = F_IN(f, MP_NEW, b); |
| 368 | return (c); |
| 369 | } |
| 370 | |
| 371 | extern ec_curve *ec_primeproj(field *f, mp *a, mp *b) |
| 372 | { |
| 373 | ec_curve *c = CREATE(ec_curve); |
| 374 | mp *ax; |
| 375 | |
| 376 | ax = mp_add(MP_NEW, a, MP_THREE); |
| 377 | ax = F_IN(f, ax, ax); |
| 378 | if (F_ZEROP(f, ax)) |
| 379 | c->ops = &ec_primeprojxops; |
| 380 | else |
| 381 | c->ops = &ec_primeprojops; |
| 382 | MP_DROP(ax); |
| 383 | c->f = f; |
| 384 | c->a = F_IN(f, MP_NEW, a); |
| 385 | c->b = F_IN(f, MP_NEW, b); |
| 386 | return (c); |
| 387 | } |
| 388 | |
| 389 | static const ec_ops ec_primeops = { |
| 390 | "prime", |
| 391 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
| 392 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck, eccompr |
| 393 | }; |
| 394 | |
| 395 | static const ec_ops ec_primeprojops = { |
| 396 | "primeproj", |
| 397 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
| 398 | ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck, eccompr |
| 399 | }; |
| 400 | |
| 401 | static const ec_ops ec_primeprojxops = { |
| 402 | "primeproj", |
| 403 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
| 404 | ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck, eccompr |
| 405 | }; |
| 406 | |
| 407 | /*----- Test rig ----------------------------------------------------------*/ |
| 408 | |
| 409 | #ifdef TEST_RIG |
| 410 | |
| 411 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
| 412 | |
| 413 | int main(int argc, char *argv[]) |
| 414 | { |
| 415 | field *f; |
| 416 | ec_curve *c; |
| 417 | ec g = EC_INIT, d = EC_INIT; |
| 418 | mp *p, *a, *b, *r; |
| 419 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
| 420 | |
| 421 | printf("ec-prime: "); |
| 422 | fflush(stdout); |
| 423 | a = MP(-3); |
| 424 | b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef); |
| 425 | p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319); |
| 426 | r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642); |
| 427 | |
| 428 | f = field_niceprime(p); |
| 429 | c = ec_primeproj(f, a, b); |
| 430 | |
| 431 | g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7); |
| 432 | g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f); |
| 433 | |
| 434 | for (i = 0; i < n; i++) { |
| 435 | ec_mul(c, &d, &g, r); |
| 436 | if (EC_ATINF(&d)) { |
| 437 | fprintf(stderr, "zero too early\n"); |
| 438 | return (1); |
| 439 | } |
| 440 | ec_add(c, &d, &d, &g); |
| 441 | if (!EC_ATINF(&d)) { |
| 442 | fprintf(stderr, "didn't reach zero\n"); |
| 443 | MP_EPRINT("d.x", d.x); |
| 444 | MP_EPRINT("d.y", d.y); |
| 445 | return (1); |
| 446 | } |
| 447 | ec_destroy(&d); |
| 448 | } |
| 449 | ec_destroy(&g); |
| 450 | ec_destroycurve(c); |
| 451 | F_DESTROY(f); |
| 452 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); |
| 453 | assert(!mparena_count(&mparena_global)); |
| 454 | printf("ok\n"); |
| 455 | return (0); |
| 456 | } |
| 457 | |
| 458 | #endif |
| 459 | |
| 460 | /*----- That's all, folks -------------------------------------------------*/ |