| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id: ec.c,v 1.7 2004/03/27 17:54:11 mdw Exp $ |
| 4 | * |
| 5 | * Elliptic curve definitions |
| 6 | * |
| 7 | * (c) 2001 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 | /*----- Revision history --------------------------------------------------* |
| 31 | * |
| 32 | * $Log: ec.c,v $ |
| 33 | * Revision 1.7 2004/03/27 17:54:11 mdw |
| 34 | * Standard curves and curve checking. |
| 35 | * |
| 36 | * Revision 1.6 2004/03/23 15:19:32 mdw |
| 37 | * Test elliptic curves more thoroughly. |
| 38 | * |
| 39 | * Revision 1.5 2004/03/21 22:52:06 mdw |
| 40 | * Merge and close elliptic curve branch. |
| 41 | * |
| 42 | * Revision 1.4.4.2 2004/03/20 00:13:31 mdw |
| 43 | * Projective coordinates for prime curves |
| 44 | * |
| 45 | * Revision 1.4.4.1 2003/06/10 13:43:53 mdw |
| 46 | * Simple (non-projective) curves over prime fields now seem to work. |
| 47 | * |
| 48 | * Revision 1.4 2003/05/15 23:25:59 mdw |
| 49 | * Make elliptic curve stuff build. |
| 50 | * |
| 51 | * Revision 1.3 2002/01/13 13:48:44 mdw |
| 52 | * Further progress. |
| 53 | * |
| 54 | * Revision 1.2 2001/05/07 17:29:44 mdw |
| 55 | * Treat projective coordinates as an internal representation. Various |
| 56 | * minor interface changes. |
| 57 | * |
| 58 | * Revision 1.1 2001/04/29 18:12:33 mdw |
| 59 | * Prototype version. |
| 60 | * |
| 61 | */ |
| 62 | |
| 63 | /*----- Header files ------------------------------------------------------*/ |
| 64 | |
| 65 | #include "ec.h" |
| 66 | #include "ec-exp.h" |
| 67 | |
| 68 | /*----- Trivial wrappers --------------------------------------------------*/ |
| 69 | |
| 70 | /* --- @ec_create@ --- * |
| 71 | * |
| 72 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
| 73 | * |
| 74 | * Returns: The argument @p@. |
| 75 | * |
| 76 | * Use: Initializes a new point. The initial value is the additive |
| 77 | * identity (which is universal for all curves). |
| 78 | */ |
| 79 | |
| 80 | ec *ec_create(ec *p) { EC_CREATE(p); return (p); } |
| 81 | |
| 82 | /* --- @ec_destroy@ --- * |
| 83 | * |
| 84 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
| 85 | * |
| 86 | * Returns: --- |
| 87 | * |
| 88 | * Use: Destroys a point, making it invalid. |
| 89 | */ |
| 90 | |
| 91 | void ec_destroy(ec *p) { EC_DESTROY(p); } |
| 92 | |
| 93 | /* --- @ec_atinf@ --- * |
| 94 | * |
| 95 | * Arguments: @const ec *p@ = pointer to a point |
| 96 | * |
| 97 | * Returns: Nonzero if %$p = O$% is the point at infinity, zero |
| 98 | * otherwise. |
| 99 | */ |
| 100 | |
| 101 | int ec_atinf(const ec *p) { return (EC_ATINF(p)); } |
| 102 | |
| 103 | /* --- @ec_setinf@ --- * |
| 104 | * |
| 105 | * Arguments: @ec *p@ = pointer to a point |
| 106 | * |
| 107 | * Returns: The argument @p@. |
| 108 | * |
| 109 | * Use: Sets the given point to be the point %$O$% at infinity. |
| 110 | */ |
| 111 | |
| 112 | ec *ec_setinf(ec *p) { EC_SETINF(p); return (p); } |
| 113 | |
| 114 | /* --- @ec_copy@ --- * |
| 115 | * |
| 116 | * Arguments: @ec *d@ = pointer to destination point |
| 117 | * @const ec *p@ = pointer to source point |
| 118 | * |
| 119 | * Returns: The destination @d@. |
| 120 | * |
| 121 | * Use: Creates a copy of an elliptic curve point. |
| 122 | */ |
| 123 | |
| 124 | ec *ec_copy(ec *d, const ec *p) { EC_COPY(d, p); return (d); } |
| 125 | |
| 126 | /* --- @ec_eq@ --- * |
| 127 | * |
| 128 | * Arguments: @const ec *p, *q@ = two points |
| 129 | * |
| 130 | * Returns: Nonzero if the points are equal. Compares external-format |
| 131 | * points. |
| 132 | */ |
| 133 | |
| 134 | int ec_eq(const ec *p, const ec *q) { return (EC_EQ(p, q)); } |
| 135 | |
| 136 | /*----- Standard curve operations -----------------------------------------*/ |
| 137 | |
| 138 | /* --- @ec_idin@, @ec_idout@, @ec_idfix@ --- * |
| 139 | * |
| 140 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 141 | * @ec *d@ = pointer to the destination |
| 142 | * @const ec *p@ = pointer to a source point |
| 143 | * |
| 144 | * Returns: The destination @d@. |
| 145 | * |
| 146 | * Use: An identity operation if your curve has no internal |
| 147 | * representation. (The field internal representation is still |
| 148 | * used.) |
| 149 | */ |
| 150 | |
| 151 | ec *ec_idin(ec_curve *c, ec *d, const ec *p) |
| 152 | { |
| 153 | if (EC_ATINF(p)) |
| 154 | EC_SETINF(d); |
| 155 | else { |
| 156 | field *f = c->f; |
| 157 | d->x = F_IN(f, d->x, p->x); |
| 158 | d->y = F_IN(f, d->y, p->y); |
| 159 | mp_drop(d->z); d->z = 0; |
| 160 | } |
| 161 | return (d); |
| 162 | } |
| 163 | |
| 164 | ec *ec_idout(ec_curve *c, ec *d, const ec *p) |
| 165 | { |
| 166 | if (EC_ATINF(p)) |
| 167 | EC_SETINF(d); |
| 168 | else { |
| 169 | field *f = c->f; |
| 170 | d->x = F_OUT(f, d->x, p->x); |
| 171 | d->y = F_OUT(f, d->y, p->y); |
| 172 | mp_drop(d->z); d->z = 0; |
| 173 | } |
| 174 | return (d); |
| 175 | } |
| 176 | |
| 177 | ec *ec_idfix(ec_curve *c, ec *d, const ec *p) |
| 178 | { |
| 179 | EC_COPY(d, p); |
| 180 | return (d); |
| 181 | } |
| 182 | |
| 183 | /* --- @ec_projin@, @ec_projout@ --- * |
| 184 | * |
| 185 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 186 | * @ec *d@ = pointer to the destination |
| 187 | * @const ec *p@ = pointer to a source point |
| 188 | * |
| 189 | * Returns: The destination @d@. |
| 190 | * |
| 191 | * Use: Conversion functions if your curve operations use a |
| 192 | * projective representation. |
| 193 | */ |
| 194 | |
| 195 | ec *ec_projin(ec_curve *c, ec *d, const ec *p) |
| 196 | { |
| 197 | if (EC_ATINF(p)) |
| 198 | EC_SETINF(d); |
| 199 | else { |
| 200 | field *f = c->f; |
| 201 | d->x = F_IN(f, d->x, p->x); |
| 202 | d->y = F_IN(f, d->y, p->y); |
| 203 | mp_drop(d->z); d->z = MP_COPY(f->one); |
| 204 | } |
| 205 | return (d); |
| 206 | } |
| 207 | |
| 208 | ec *ec_projout(ec_curve *c, ec *d, const ec *p) |
| 209 | { |
| 210 | if (EC_ATINF(p)) |
| 211 | EC_SETINF(d); |
| 212 | else { |
| 213 | mp *x, *y, *z, *zz; |
| 214 | field *f = c->f; |
| 215 | z = F_INV(f, MP_NEW, p->z); |
| 216 | zz = F_SQR(f, MP_NEW, z); |
| 217 | z = F_MUL(f, z, zz, z); |
| 218 | x = F_MUL(f, d->x, p->x, zz); |
| 219 | y = F_MUL(f, d->y, p->y, z); |
| 220 | mp_drop(z); |
| 221 | mp_drop(zz); |
| 222 | mp_drop(d->z); |
| 223 | d->x = F_OUT(f, x, x); |
| 224 | d->y = F_OUT(f, y, y); |
| 225 | d->z = 0; |
| 226 | } |
| 227 | return (d); |
| 228 | } |
| 229 | |
| 230 | ec *ec_projfix(ec_curve *c, ec *d, const ec *p) |
| 231 | { |
| 232 | if (EC_ATINF(p)) |
| 233 | EC_SETINF(d); |
| 234 | else if (d->z == c->f->one) |
| 235 | EC_COPY(d, p); |
| 236 | else { |
| 237 | mp *z, *zz; |
| 238 | field *f = c->f; |
| 239 | z = F_INV(f, MP_NEW, p->z); |
| 240 | zz = F_SQR(f, MP_NEW, z); |
| 241 | z = F_MUL(f, z, zz, z); |
| 242 | d->x = F_MUL(f, d->x, p->x, zz); |
| 243 | d->y = F_MUL(f, d->y, p->y, z); |
| 244 | mp_drop(z); |
| 245 | mp_drop(zz); |
| 246 | mp_drop(d->z); |
| 247 | d->z = MP_COPY(f->one); |
| 248 | } |
| 249 | return (d); |
| 250 | } |
| 251 | |
| 252 | /* --- @ec_stdsub@ --- * |
| 253 | * |
| 254 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 255 | * @ec *d@ = pointer to the destination |
| 256 | * @const ec *p, *q@ = the operand points |
| 257 | * |
| 258 | * Returns: The destination @d@. |
| 259 | * |
| 260 | * Use: Standard point subtraction operation, in terms of negation |
| 261 | * and addition. This isn't as efficient as a ready-made |
| 262 | * subtraction operator. |
| 263 | */ |
| 264 | |
| 265 | ec *ec_stdsub(ec_curve *c, ec *d, const ec *p, const ec *q) |
| 266 | { |
| 267 | ec t = EC_INIT; |
| 268 | EC_NEG(c, &t, q); |
| 269 | EC_FIX(c, &t, &t); |
| 270 | EC_ADD(c, d, p, &t); |
| 271 | EC_DESTROY(&t); |
| 272 | return (d); |
| 273 | } |
| 274 | |
| 275 | /*----- Creating curves ---------------------------------------------------*/ |
| 276 | |
| 277 | /* --- @ec_destroycurve@ --- * |
| 278 | * |
| 279 | * Arguments: @ec_curve *c@ = pointer to an ellptic curve |
| 280 | * |
| 281 | * Returns: --- |
| 282 | * |
| 283 | * Use: Destroys a description of an elliptic curve. |
| 284 | */ |
| 285 | |
| 286 | void ec_destroycurve(ec_curve *c) { c->ops->destroy(c); } |
| 287 | |
| 288 | /*----- Real arithmetic ---------------------------------------------------*/ |
| 289 | |
| 290 | /* --- @ec_find@ --- * |
| 291 | * |
| 292 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 293 | * @ec *d@ = pointer to the destination point |
| 294 | * @mp *x@ = a possible x-coordinate |
| 295 | * |
| 296 | * Returns: Zero if OK, nonzero if there isn't a point there. |
| 297 | * |
| 298 | * Use: Finds a point on an elliptic curve with a given x-coordinate. |
| 299 | */ |
| 300 | |
| 301 | ec *ec_find(ec_curve *c, ec *d, mp *x) |
| 302 | { |
| 303 | x = F_IN(c->f, MP_NEW, x); |
| 304 | if ((d = EC_FIND(c, d, x)) != 0) |
| 305 | EC_OUT(c, d, d); |
| 306 | MP_DROP(x); |
| 307 | return (d); |
| 308 | } |
| 309 | |
| 310 | /* --- @ec_neg@ --- * |
| 311 | * |
| 312 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 313 | * @ec *d@ = pointer to the destination point |
| 314 | * @const ec *p@ = pointer to the operand point |
| 315 | * |
| 316 | * Returns: The destination point. |
| 317 | * |
| 318 | * Use: Computes the negation of the given point. |
| 319 | */ |
| 320 | |
| 321 | ec *ec_neg(ec_curve *c, ec *d, const ec *p) |
| 322 | { |
| 323 | EC_IN(c, d, p); |
| 324 | EC_NEG(c, d, d); |
| 325 | return (EC_OUT(c, d, d)); |
| 326 | } |
| 327 | |
| 328 | /* --- @ec_add@ --- * |
| 329 | * |
| 330 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 331 | * @ec *d@ = pointer to the destination point |
| 332 | * @const ec *p, *q@ = pointers to the operand points |
| 333 | * |
| 334 | * Returns: --- |
| 335 | * |
| 336 | * Use: Adds two points on an elliptic curve. |
| 337 | */ |
| 338 | |
| 339 | ec *ec_add(ec_curve *c, ec *d, const ec *p, const ec *q) |
| 340 | { |
| 341 | ec pp = EC_INIT, qq = EC_INIT; |
| 342 | EC_IN(c, &pp, p); |
| 343 | EC_IN(c, &qq, q); |
| 344 | EC_ADD(c, d, &pp, &qq); |
| 345 | EC_OUT(c, d, d); |
| 346 | EC_DESTROY(&pp); |
| 347 | EC_DESTROY(&qq); |
| 348 | return (d); |
| 349 | } |
| 350 | |
| 351 | /* --- @ec_sub@ --- * |
| 352 | * |
| 353 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 354 | * @ec *d@ = pointer to the destination point |
| 355 | * @const ec *p, *q@ = pointers to the operand points |
| 356 | * |
| 357 | * Returns: The destination @d@. |
| 358 | * |
| 359 | * Use: Subtracts one point from another on an elliptic curve. |
| 360 | */ |
| 361 | |
| 362 | ec *ec_sub(ec_curve *c, ec *d, const ec *p, const ec *q) |
| 363 | { |
| 364 | ec pp = EC_INIT, qq = EC_INIT; |
| 365 | EC_IN(c, &pp, p); |
| 366 | EC_IN(c, &qq, q); |
| 367 | EC_SUB(c, d, &pp, &qq); |
| 368 | EC_OUT(c, d, d); |
| 369 | EC_DESTROY(&pp); |
| 370 | EC_DESTROY(&qq); |
| 371 | return (d); |
| 372 | } |
| 373 | |
| 374 | /* --- @ec_dbl@ --- * |
| 375 | * |
| 376 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 377 | * @ec *d@ = pointer to the destination point |
| 378 | * @const ec *p@ = pointer to the operand point |
| 379 | * |
| 380 | * Returns: --- |
| 381 | * |
| 382 | * Use: Doubles a point on an elliptic curve. |
| 383 | */ |
| 384 | |
| 385 | ec *ec_dbl(ec_curve *c, ec *d, const ec *p) |
| 386 | { |
| 387 | EC_IN(c, d, p); |
| 388 | EC_DBL(c, d, d); |
| 389 | return (EC_OUT(c, d, d)); |
| 390 | } |
| 391 | |
| 392 | /* --- @ec_check@ --- * |
| 393 | * |
| 394 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 395 | * @const ec *p@ = pointer to the point |
| 396 | * |
| 397 | * Returns: Zero if OK, nonzero if this is an invalid point. |
| 398 | * |
| 399 | * Use: Checks that a point is actually on an elliptic curve. |
| 400 | */ |
| 401 | |
| 402 | int ec_check(ec_curve *c, const ec *p) |
| 403 | { |
| 404 | ec t = EC_INIT; |
| 405 | int rc; |
| 406 | |
| 407 | if (EC_ATINF(p)) |
| 408 | return (0); |
| 409 | EC_IN(c, &t, p); |
| 410 | rc = EC_CHECK(c, &t); |
| 411 | EC_DESTROY(&t); |
| 412 | return (rc); |
| 413 | } |
| 414 | |
| 415 | /* --- @ec_rand@ --- * |
| 416 | * |
| 417 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 418 | * @ec *d@ = pointer to the destination point |
| 419 | * @grand *r@ = random number source |
| 420 | * |
| 421 | * Returns: The destination @d@. |
| 422 | * |
| 423 | * Use: Finds a random point on the given curve. |
| 424 | */ |
| 425 | |
| 426 | ec *ec_rand(ec_curve *c, ec *d, grand *r) |
| 427 | { |
| 428 | mp *x = MP_NEW; |
| 429 | do x = F_RAND(c->f, x, r); while (!EC_FIND(c, d, x)); |
| 430 | mp_drop(x); |
| 431 | if (grand_range(r, 2)) EC_NEG(c, d, d); |
| 432 | return (EC_OUT(c, d, d)); |
| 433 | } |
| 434 | |
| 435 | /* --- @ec_imul@, @ec_mul@ --- * |
| 436 | * |
| 437 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 438 | * @ec *d@ = pointer to the destination point |
| 439 | * @const ec *p@ = pointer to the generator point |
| 440 | * @mp *n@ = integer multiplier |
| 441 | * |
| 442 | * Returns: The destination @d@. |
| 443 | * |
| 444 | * Use: Multiplies a point by a scalar, returning %$n p$%. The |
| 445 | * @imul@ variant uses internal representations for argument |
| 446 | * and result. |
| 447 | */ |
| 448 | |
| 449 | ec *ec_imul(ec_curve *c, ec *d, const ec *p, mp *n) |
| 450 | { |
| 451 | ec t = EC_INIT; |
| 452 | |
| 453 | EC_COPY(&t, p); |
| 454 | if (t.x && (n->f & MP_BURN)) |
| 455 | t.x->f |= MP_BURN; |
| 456 | MP_SHRINK(n); |
| 457 | EC_SETINF(d); |
| 458 | if (MP_LEN(n) == 0) |
| 459 | ; |
| 460 | else { |
| 461 | if (n->f & MP_NEG) |
| 462 | EC_NEG(c, &t, &t); |
| 463 | if (MP_LEN(n) < EXP_THRESH) |
| 464 | EXP_SIMPLE(*d, t, n); |
| 465 | else |
| 466 | EXP_WINDOW(*d, t, n); |
| 467 | } |
| 468 | EC_DESTROY(&t); |
| 469 | return (d); |
| 470 | } |
| 471 | |
| 472 | ec *ec_mul(ec_curve *c, ec *d, const ec *p, mp *n) |
| 473 | { |
| 474 | EC_IN(c, d, p); |
| 475 | ec_imul(c, d, d, n); |
| 476 | return (EC_OUT(c, d, d)); |
| 477 | } |
| 478 | |
| 479 | /*----- That's all, folks -------------------------------------------------*/ |