| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id: ec.h,v 1.7 2004/03/23 15:19:32 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.h,v $ |
| 33 | * Revision 1.7 2004/03/23 15:19:32 mdw |
| 34 | * Test elliptic curves more thoroughly. |
| 35 | * |
| 36 | * Revision 1.6 2004/03/22 02:19:10 mdw |
| 37 | * Rationalise the sliding-window threshold. Drop guarantee that right |
| 38 | * arguments to EC @add@ are canonical, and fix up projective implementations |
| 39 | * to cope. |
| 40 | * |
| 41 | * Revision 1.5 2004/03/21 22:52:06 mdw |
| 42 | * Merge and close elliptic curve branch. |
| 43 | * |
| 44 | * Revision 1.4.4.3 2004/03/21 22:39:46 mdw |
| 45 | * Elliptic curves on binary fields work. |
| 46 | * |
| 47 | * Revision 1.4.4.2 2004/03/20 00:13:31 mdw |
| 48 | * Projective coordinates for prime curves |
| 49 | * |
| 50 | * Revision 1.4.4.1 2003/06/10 13:43:53 mdw |
| 51 | * Simple (non-projective) curves over prime fields now seem to work. |
| 52 | * |
| 53 | * Revision 1.4 2003/05/15 23:25:59 mdw |
| 54 | * Make elliptic curve stuff build. |
| 55 | * |
| 56 | * Revision 1.3 2002/01/13 13:48:44 mdw |
| 57 | * Further progress. |
| 58 | * |
| 59 | * Revision 1.2 2001/05/07 17:29:44 mdw |
| 60 | * Treat projective coordinates as an internal representation. Various |
| 61 | * minor interface changes. |
| 62 | * |
| 63 | * Revision 1.1 2001/04/29 18:12:33 mdw |
| 64 | * Prototype version. |
| 65 | * |
| 66 | */ |
| 67 | |
| 68 | #ifndef CATACOMB_EC_H |
| 69 | #define CATACOMB_EC_H |
| 70 | |
| 71 | #ifdef __cplusplus |
| 72 | extern "C" { |
| 73 | #endif |
| 74 | |
| 75 | /*----- Header files ------------------------------------------------------*/ |
| 76 | |
| 77 | #include "field.h" |
| 78 | #include "mp.h" |
| 79 | |
| 80 | /*----- Data structures ---------------------------------------------------*/ |
| 81 | |
| 82 | /* --- An elliptic curve representation --- */ |
| 83 | |
| 84 | typedef struct ec_curve { |
| 85 | const struct ec_ops *ops; /* Curve operations */ |
| 86 | field *f; /* Underlying field structure */ |
| 87 | } ec_curve; |
| 88 | |
| 89 | /* --- An elliptic curve point --- */ |
| 90 | |
| 91 | typedef struct ec { |
| 92 | mp *x, *y; /* Point coordinates */ |
| 93 | mp *z; /* Common denominator (or null) */ |
| 94 | } ec; |
| 95 | |
| 96 | /* --- A factor for simultaneous multiplication --- */ |
| 97 | |
| 98 | typedef struct ec_mulfactor { |
| 99 | ec base; /* The point */ |
| 100 | mp *exp; /* The exponent */ |
| 101 | } ec_mulfactor; |
| 102 | |
| 103 | /* --- Elliptic curve operations --- * |
| 104 | * |
| 105 | * All operations (apart from @destroy@ and @in@) are guaranteed to be |
| 106 | * performed on internal representations of points. |
| 107 | * |
| 108 | * (Historical note. We used to guarantee that the second to @add@ and @mul@ |
| 109 | * was the output of @in@ or @fix@, but this canonification turned out to |
| 110 | * make the precomputation in @ec_exp@ too slow. Projective implementations |
| 111 | * must therefore cope with a pair of arbitrary points.) |
| 112 | */ |
| 113 | |
| 114 | typedef struct ec_ops { |
| 115 | void (*destroy)(ec_curve */*c*/); |
| 116 | ec *(*in)(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 117 | ec *(*out)(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 118 | ec *(*fix)(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 119 | ec *(*find)(ec_curve */*c*/, ec */*d*/, mp */*x*/); |
| 120 | ec *(*neg)(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 121 | ec *(*add)(ec_curve */*c*/, ec */*d*/, const ec */*p*/, const ec */*q*/); |
| 122 | ec *(*sub)(ec_curve */*c*/, ec */*d*/, const ec */*p*/, const ec */*q*/); |
| 123 | ec *(*dbl)(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 124 | int (*check)(ec_curve */*c*/, const ec */*p*/); |
| 125 | } ec_ops; |
| 126 | |
| 127 | #define EC_IN(c, d, p) (c)->ops->in((c), (d), (p)) |
| 128 | #define EC_OUT(c, d, p) (c)->ops->out((c), (d), (p)) |
| 129 | #define EC_FIX(c, d, p) (c)->ops->fix((c), (d), (p)) |
| 130 | |
| 131 | #define EC_FIND(c, d, x) (c)->ops->find((c), (d), (x)) |
| 132 | #define EC_NEG(c, d, x) (c)->ops->neg((c), (d), (x)) |
| 133 | #define EC_ADD(c, d, p, q) (c)->ops->add((c), (d), (p), (q)) |
| 134 | #define EC_SUB(c, d, p, q) (c)->ops->sub((c), (d), (p), (q)) |
| 135 | #define EC_DBL(c, d, p) (c)->ops->dbl((c), (d), (p)) |
| 136 | #define EC_CHECK(c, p) (c)->ops->check((c), (p)) |
| 137 | |
| 138 | /*----- Simple memory management things -----------------------------------*/ |
| 139 | |
| 140 | /* --- @ec_create@ --- * |
| 141 | * |
| 142 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
| 143 | * |
| 144 | * Returns: The argument @p@. |
| 145 | * |
| 146 | * Use: Initializes a new point. The initial value is the additive |
| 147 | * identity (which is universal for all curves). |
| 148 | */ |
| 149 | |
| 150 | #define EC_INIT { MP_NEW, MP_NEW, MP_NEW } |
| 151 | |
| 152 | #define EC_CREATE(p) do { \ |
| 153 | ec *_p = (p); \ |
| 154 | _p->x = _p->y = _p->z = MP_NEW; \ |
| 155 | } while (0) |
| 156 | |
| 157 | extern ec *ec_create(ec */*p*/); |
| 158 | |
| 159 | /* --- @ec_destroy@ --- * |
| 160 | * |
| 161 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
| 162 | * |
| 163 | * Returns: --- |
| 164 | * |
| 165 | * Use: Destroys a point, making it invalid. |
| 166 | */ |
| 167 | |
| 168 | #define EC_DESTROY(p) do { \ |
| 169 | ec *_p = (p); \ |
| 170 | if (!EC_ATINF(_p)) { \ |
| 171 | MP_DROP(_p->x); \ |
| 172 | MP_DROP(_p->y); \ |
| 173 | if (_p->z) MP_DROP(_p->z); \ |
| 174 | } \ |
| 175 | } while (0) |
| 176 | |
| 177 | extern void ec_destroy(ec */*p*/); |
| 178 | |
| 179 | /* --- @ec_atinf@ --- * |
| 180 | * |
| 181 | * Arguments: @const ec *p@ = pointer to a point |
| 182 | * |
| 183 | * Returns: Nonzero if %$p = O$% is the point at infinity, zero |
| 184 | * otherwise. |
| 185 | */ |
| 186 | |
| 187 | #define EC_ATINF(p) ((p)->x == MP_NEW || (p)->x == MP_NEWSEC) |
| 188 | |
| 189 | extern int ec_atinf(const ec */*p*/); |
| 190 | |
| 191 | /* --- @ec_setinf@ --- * |
| 192 | * |
| 193 | * Arguments: @ec *p@ = pointer to a point |
| 194 | * |
| 195 | * Returns: The argument @p@. |
| 196 | * |
| 197 | * Use: Sets the given point to be the point %$O$% at infinity. |
| 198 | */ |
| 199 | |
| 200 | #define EC_SETINF(p) do { \ |
| 201 | ec *_p = (p); \ |
| 202 | if (!EC_ATINF(_p)) { \ |
| 203 | MP_DROP(_p->x); \ |
| 204 | MP_DROP(_p->y); \ |
| 205 | if (_p->z) MP_DROP(_p->z); \ |
| 206 | _p->x = _p->y = _p->z = MP_NEW; \ |
| 207 | _p->y = MP_NEW; \ |
| 208 | _p->z = MP_NEW; \ |
| 209 | } \ |
| 210 | } while (0) |
| 211 | |
| 212 | extern ec *ec_setinf(ec */*p*/); |
| 213 | |
| 214 | /* --- @ec_copy@ --- * |
| 215 | * |
| 216 | * Arguments: @ec *d@ = pointer to destination point |
| 217 | * @const ec *p@ = pointer to source point |
| 218 | * |
| 219 | * Returns: The destination @d@. |
| 220 | * |
| 221 | * Use: Creates a copy of an elliptic curve point. |
| 222 | */ |
| 223 | |
| 224 | #define EC_COPY(d, p) do { \ |
| 225 | ec *_d = (d); \ |
| 226 | const ec *_p = (p); \ |
| 227 | if (d != p) { \ |
| 228 | EC_DESTROY(d); \ |
| 229 | if (EC_ATINF(p)) \ |
| 230 | _d->x = _d->y = _d->z = MP_NEW; \ |
| 231 | else { \ |
| 232 | _d->x = MP_COPY(_p->x); \ |
| 233 | _d->y = MP_COPY(_p->y); \ |
| 234 | _d->z = _p->z ? MP_COPY(_p->z) : MP_NEW; \ |
| 235 | } \ |
| 236 | } \ |
| 237 | } while (0) |
| 238 | |
| 239 | extern ec *ec_copy(ec */*d*/, const ec */*p*/); |
| 240 | |
| 241 | /* --- @ec_eq@ --- * |
| 242 | * |
| 243 | * Arguments: @const ec *p, *q@ = two points |
| 244 | * |
| 245 | * Returns: Nonzero if the points are equal. Compares external-format |
| 246 | * points. |
| 247 | */ |
| 248 | |
| 249 | #define EC_EQ(p, q) \ |
| 250 | ((EC_ATINF(p) && EC_ATINF(q)) || \ |
| 251 | (!EC_ATINF(p) && !EC_ATINF(q) && \ |
| 252 | MP_EQ((p)->x, (q)->x) && \ |
| 253 | MP_EQ((p)->y, (q)->y))) |
| 254 | |
| 255 | extern int ec_eq(const ec *p, const ec *q); |
| 256 | |
| 257 | /*----- Interesting arithmetic --------------------------------------------*/ |
| 258 | |
| 259 | /* --- @ec_find@ --- * |
| 260 | * |
| 261 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 262 | * @ec *d@ = pointer to the destination point |
| 263 | * @mp *x@ = a possible x-coordinate |
| 264 | * |
| 265 | * Returns: The destination if OK, or null if no point was found. |
| 266 | * |
| 267 | * Use: Finds a point on an elliptic curve with a given |
| 268 | * x-coordinate. If there is no point with the given |
| 269 | * %$x$%-coordinate, a null pointer is returned and the |
| 270 | * destination is left invalid. |
| 271 | */ |
| 272 | |
| 273 | extern ec *ec_find(ec_curve */*c*/, ec */*d*/, mp */*x*/); |
| 274 | |
| 275 | /* --- @ec_rand@ --- * |
| 276 | * |
| 277 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 278 | * @ec *d@ = pointer to the destination point |
| 279 | * @grand *r@ = random number source |
| 280 | * |
| 281 | * Returns: The destination @d@. |
| 282 | * |
| 283 | * Use: Finds a random point on the given curve. |
| 284 | */ |
| 285 | |
| 286 | extern ec *ec_rand(ec_curve */*c*/, ec */*d*/, grand */*r*/); |
| 287 | |
| 288 | /* --- @ec_neg@ --- * |
| 289 | * |
| 290 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 291 | * @ec *d@ = pointer to the destination point |
| 292 | * @const ec *p@ = pointer to the operand point |
| 293 | * |
| 294 | * Returns: The destination point. |
| 295 | * |
| 296 | * Use: Computes the negation of the given point. |
| 297 | */ |
| 298 | |
| 299 | extern ec *ec_neg(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 300 | |
| 301 | /* --- @ec_add@ --- * |
| 302 | * |
| 303 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 304 | * @ec *d@ = pointer to the destination point |
| 305 | * @const ec *p, *q@ = pointers to the operand points |
| 306 | * |
| 307 | * Returns: The destination @d@. |
| 308 | * |
| 309 | * Use: Adds two points on an elliptic curve. |
| 310 | */ |
| 311 | |
| 312 | extern ec *ec_add(ec_curve */*c*/, ec */*d*/, |
| 313 | const ec */*p*/, const ec */*q*/); |
| 314 | |
| 315 | /* --- @ec_sub@ --- * |
| 316 | * |
| 317 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 318 | * @ec *d@ = pointer to the destination point |
| 319 | * @const ec *p, *q@ = pointers to the operand points |
| 320 | * |
| 321 | * Returns: The destination @d@. |
| 322 | * |
| 323 | * Use: Subtracts one point from another on an elliptic curve. |
| 324 | */ |
| 325 | |
| 326 | extern ec *ec_sub(ec_curve */*c*/, ec */*d*/, |
| 327 | const ec */*p*/, const ec */*q*/); |
| 328 | |
| 329 | /* --- @ec_dbl@ --- * |
| 330 | * |
| 331 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 332 | * @ec *d@ = pointer to the destination point |
| 333 | * @const ec *p@ = pointer to the operand point |
| 334 | * |
| 335 | * Returns: The destination @d@. |
| 336 | * |
| 337 | * Use: Doubles a point on an elliptic curve. |
| 338 | */ |
| 339 | |
| 340 | extern ec *ec_dbl(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 341 | |
| 342 | /* --- @ec_check@ --- * |
| 343 | * |
| 344 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 345 | * @const ec *p@ = pointer to the point |
| 346 | * |
| 347 | * Returns: Zero if OK, nonzero if this is an invalid point. |
| 348 | * |
| 349 | * Use: Checks that a point is actually on an elliptic curve. |
| 350 | */ |
| 351 | |
| 352 | extern int ec_check(ec_curve */*c*/, const ec */*p*/); |
| 353 | |
| 354 | /* --- @ec_mul@, @ec_imul@ --- * |
| 355 | * |
| 356 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 357 | * @ec *d@ = pointer to the destination point |
| 358 | * @const ec *p@ = pointer to the generator point |
| 359 | * @mp *n@ = integer multiplier |
| 360 | * |
| 361 | * Returns: The destination @d@. |
| 362 | * |
| 363 | * Use: Multiplies a point by a scalar, returning %$n p$%. The |
| 364 | * @imul@ variant uses internal representations for argument |
| 365 | * and result. |
| 366 | */ |
| 367 | |
| 368 | extern ec *ec_mul(ec_curve */*c*/, ec */*d*/, const ec */*p*/, mp */*n*/); |
| 369 | extern ec *ec_imul(ec_curve */*c*/, ec */*d*/, const ec */*p*/, mp */*n*/); |
| 370 | |
| 371 | /* --- @ec_mmul@, @ec_immul@ --- * |
| 372 | * |
| 373 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 374 | * @ec *d@ = pointer to the destination point |
| 375 | * @const ec_mulfactor *f@ = pointer to vector of factors |
| 376 | * @size_t n@ = number of factors |
| 377 | * |
| 378 | * Returns: The destination @d@. |
| 379 | * |
| 380 | * Use: Does simultaneous point multiplication. The @immul@ variant |
| 381 | * uses internal representations for arguments and result. |
| 382 | */ |
| 383 | |
| 384 | extern ec *ec_mmul(ec_curve */*c*/, ec */*d*/, |
| 385 | const ec_mulfactor */*f*/, size_t /*n*/); |
| 386 | extern ec *ec_immul(ec_curve */*c*/, ec */*d*/, |
| 387 | const ec_mulfactor */*f*/, size_t /*n*/); |
| 388 | |
| 389 | /*----- Standard curve operations -----------------------------------------*/ |
| 390 | |
| 391 | /* --- @ec_idin@, @ec_idout@, @ec_idfix@ --- * |
| 392 | * |
| 393 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 394 | * @ec *d@ = pointer to the destination |
| 395 | * @const ec *p@ = pointer to a source point |
| 396 | * |
| 397 | * Returns: The destination @d@. |
| 398 | * |
| 399 | * Use: An identity operation if your curve has no internal |
| 400 | * representation. (The field internal representation is still |
| 401 | * used.) |
| 402 | */ |
| 403 | |
| 404 | extern ec *ec_idin(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 405 | extern ec *ec_idout(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 406 | extern ec *ec_idfix(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 407 | |
| 408 | /* --- @ec_projin@, @ec_projout@, @ec_projfix@ --- * |
| 409 | * |
| 410 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 411 | * @ec *d@ = pointer to the destination |
| 412 | * @const ec *p@ = pointer to a source point |
| 413 | * |
| 414 | * Returns: The destination @d@. |
| 415 | * |
| 416 | * Use: Conversion functions if your curve operations use a |
| 417 | * projective representation. |
| 418 | */ |
| 419 | |
| 420 | extern ec *ec_projin(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 421 | extern ec *ec_projout(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 422 | extern ec *ec_projfix(ec_curve */*c*/, ec */*d*/, const ec */*p*/); |
| 423 | |
| 424 | /* --- @ec_stdsub@ --- * |
| 425 | * |
| 426 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
| 427 | * @ec *d@ = pointer to the destination |
| 428 | * @const ec *p, *q@ = the operand points |
| 429 | * |
| 430 | * Returns: The destination @d@. |
| 431 | * |
| 432 | * Use: Standard point subtraction operation, in terms of negation |
| 433 | * and addition. This isn't as efficient as a ready-made |
| 434 | * subtraction operator. |
| 435 | */ |
| 436 | |
| 437 | extern ec *ec_stdsub(ec_curve */*c*/, ec */*d*/, |
| 438 | const ec */*p*/, const ec */*q*/); |
| 439 | |
| 440 | /*----- Creating curves ---------------------------------------------------*/ |
| 441 | |
| 442 | /* --- @ec_destroycurve@ --- * |
| 443 | * |
| 444 | * Arguments: @ec_curve *c@ = pointer to an ellptic curve |
| 445 | * |
| 446 | * Returns: --- |
| 447 | * |
| 448 | * Use: Destroys a description of an elliptic curve. |
| 449 | */ |
| 450 | |
| 451 | extern void ec_destroycurve(ec_curve */*c*/); |
| 452 | |
| 453 | /* --- @ec_prime@, @ec_primeproj@ --- * |
| 454 | * |
| 455 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
| 456 | * @mp *a, *b@ = the coefficients for this curve |
| 457 | * |
| 458 | * Returns: A pointer to the curve. |
| 459 | * |
| 460 | * Use: Creates a curve structure for an elliptic curve defined over |
| 461 | * a prime field. The @primeproj@ variant uses projective |
| 462 | * coordinates, which can be a win. |
| 463 | */ |
| 464 | |
| 465 | extern ec_curve *ec_prime(field */*f*/, mp */*a*/, mp */*b*/); |
| 466 | extern ec_curve *ec_primeproj(field */*f*/, mp */*a*/, mp */*b*/); |
| 467 | |
| 468 | /* --- @ec_bin@, @ec_binproj@ --- * |
| 469 | * |
| 470 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
| 471 | * @mp *a, *b@ = the coefficients for this curve |
| 472 | * |
| 473 | * Returns: A pointer to the curve. |
| 474 | * |
| 475 | * Use: Creates a curve structure for an elliptic curve defined over |
| 476 | * a binary field. The @binproj@ variant uses projective |
| 477 | * coordinates, which can be a win. |
| 478 | */ |
| 479 | |
| 480 | extern ec_curve *ec_bin(field */*f*/, mp */*a*/, mp */*b*/); |
| 481 | extern ec_curve *ec_binproj(field */*f*/, mp */*a*/, mp */*b*/); |
| 482 | |
| 483 | /*----- That's all, folks -------------------------------------------------*/ |
| 484 | |
| 485 | #ifdef __cplusplus |
| 486 | } |
| 487 | #endif |
| 488 | |
| 489 | #endif |