| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id: mpmont.c,v 1.7 1999/12/11 01:51:14 mdw Exp $ |
| 4 | * |
| 5 | * Montgomery reduction |
| 6 | * |
| 7 | * (c) 1999 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: mpmont.c,v $ |
| 33 | * Revision 1.7 1999/12/11 01:51:14 mdw |
| 34 | * Use a Karatsuba-based reduction for large moduli. |
| 35 | * |
| 36 | * Revision 1.6 1999/12/10 23:18:39 mdw |
| 37 | * Change interface for suggested destinations. |
| 38 | * |
| 39 | * Revision 1.5 1999/11/22 13:58:40 mdw |
| 40 | * Add an option to disable Montgomery reduction, so that performance |
| 41 | * comparisons can be done. |
| 42 | * |
| 43 | * Revision 1.4 1999/11/21 12:27:06 mdw |
| 44 | * Remove a division from the Montgomery setup by calculating |
| 45 | * %$R^2 \bmod m$% first and then %$R \bmod m$% by Montgomery reduction of |
| 46 | * %$R^2$%. |
| 47 | * |
| 48 | * Revision 1.3 1999/11/21 11:35:10 mdw |
| 49 | * Performance improvement: use @mp_sqr@ and @mpmont_reduce@ instead of |
| 50 | * @mpmont_mul@ for squaring in exponentiation. |
| 51 | * |
| 52 | * Revision 1.2 1999/11/19 13:17:26 mdw |
| 53 | * Add extra interface to exponentiation which returns a Montgomerized |
| 54 | * result. |
| 55 | * |
| 56 | * Revision 1.1 1999/11/17 18:02:16 mdw |
| 57 | * New multiprecision integer arithmetic suite. |
| 58 | * |
| 59 | */ |
| 60 | |
| 61 | /*----- Header files ------------------------------------------------------*/ |
| 62 | |
| 63 | #include "mp.h" |
| 64 | #include "mpmont.h" |
| 65 | |
| 66 | /*----- Tweakables --------------------------------------------------------*/ |
| 67 | |
| 68 | /* --- @MPMONT_DISABLE@ --- * |
| 69 | * |
| 70 | * Replace all the clever Montgomery reduction with good old-fashioned long |
| 71 | * division. |
| 72 | */ |
| 73 | |
| 74 | /* #define MPMONT_DISABLE */ |
| 75 | |
| 76 | /*----- Main code ---------------------------------------------------------*/ |
| 77 | |
| 78 | /* --- @mpmont_create@ --- * |
| 79 | * |
| 80 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 81 | * @mp *m@ = modulus to use |
| 82 | * |
| 83 | * Returns: --- |
| 84 | * |
| 85 | * Use: Initializes a Montgomery reduction context ready for use. |
| 86 | * The argument @m@ must be a positive odd integer. |
| 87 | */ |
| 88 | |
| 89 | #ifdef MPMONT_DISABLE |
| 90 | |
| 91 | void mpmont_create(mpmont *mm, mp *m) |
| 92 | { |
| 93 | mp_shrink(m); |
| 94 | mm->m = MP_COPY(m); |
| 95 | mm->r = MP_ONE; |
| 96 | mm->r2 = MP_ONE; |
| 97 | mm->mi = MP_ONE; |
| 98 | } |
| 99 | |
| 100 | #else |
| 101 | |
| 102 | void mpmont_create(mpmont *mm, mp *m) |
| 103 | { |
| 104 | size_t n = MP_LEN(m); |
| 105 | mp *r2 = mp_create(2 * n + 1); |
| 106 | mp r; |
| 107 | |
| 108 | /* --- Validate the arguments --- */ |
| 109 | |
| 110 | assert(((void)"Montgomery modulus must be positive", |
| 111 | (m->f & MP_NEG) == 0)); |
| 112 | assert(((void)"Montgomery modulus must be odd", m->v[0] & 1)); |
| 113 | |
| 114 | /* --- Take a copy of the modulus --- */ |
| 115 | |
| 116 | mp_shrink(m); |
| 117 | mm->m = MP_COPY(m); |
| 118 | |
| 119 | /* --- Determine %$R^2$% --- */ |
| 120 | |
| 121 | mm->n = n; |
| 122 | MPX_ZERO(r2->v, r2->vl - 1); |
| 123 | r2->vl[-1] = 1; |
| 124 | |
| 125 | /* --- Find the magic value @mi@ --- */ |
| 126 | |
| 127 | mp_build(&r, r2->v + n, r2->vl); |
| 128 | mm->mi = MP_NEW; |
| 129 | mp_gcd(0, 0, &mm->mi, &r, m); |
| 130 | mm->mi = mp_sub(mm->mi, &r, mm->mi); |
| 131 | |
| 132 | /* --- Discover the values %$R \bmod m$% and %$R^2 \bmod m$% --- */ |
| 133 | |
| 134 | mm->r2 = MP_NEW; |
| 135 | mp_div(0, &mm->r2, r2, m); |
| 136 | mm->r = mpmont_reduce(mm, MP_NEW, mm->r2); |
| 137 | MP_DROP(r2); |
| 138 | } |
| 139 | |
| 140 | #endif |
| 141 | |
| 142 | /* --- @mpmont_destroy@ --- * |
| 143 | * |
| 144 | * Arguments: @mpmont *mm@ = pointer to a Montgomery reduction context |
| 145 | * |
| 146 | * Returns: --- |
| 147 | * |
| 148 | * Use: Disposes of a context when it's no longer of any use to |
| 149 | * anyone. |
| 150 | */ |
| 151 | |
| 152 | void mpmont_destroy(mpmont *mm) |
| 153 | { |
| 154 | MP_DROP(mm->m); |
| 155 | MP_DROP(mm->r); |
| 156 | MP_DROP(mm->r2); |
| 157 | MP_DROP(mm->mi); |
| 158 | } |
| 159 | |
| 160 | /* --- @mpmont_reduce@ --- * |
| 161 | * |
| 162 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 163 | * @mp *d@ = destination |
| 164 | * @mp *a@ = source, assumed positive |
| 165 | * |
| 166 | * Returns: Result, %$a R^{-1} \bmod m$%. |
| 167 | */ |
| 168 | |
| 169 | #ifdef MPMONT_DISABLE |
| 170 | |
| 171 | mp *mpmont_reduce(mpmont *mm, mp *d, mp *a) |
| 172 | { |
| 173 | mp_div(0, &d, a, mm->m); |
| 174 | return (d); |
| 175 | } |
| 176 | |
| 177 | #else |
| 178 | |
| 179 | mp *mpmont_reduce(mpmont *mm, mp *d, mp *a) |
| 180 | { |
| 181 | size_t n = mm->n; |
| 182 | |
| 183 | /* --- Check for serious Karatsuba reduction --- */ |
| 184 | |
| 185 | if (n > KARATSUBA_CUTOFF * 2) { |
| 186 | mp al; |
| 187 | mpw *vl; |
| 188 | mp *u; |
| 189 | |
| 190 | if (MP_LEN(a) >= n) |
| 191 | vl = a->v + n; |
| 192 | else |
| 193 | vl = a->vl; |
| 194 | mp_build(&al, a->v, vl); |
| 195 | u = mp_mul(MP_NEW, &al, mm->mi); |
| 196 | if (MP_LEN(u) > n) |
| 197 | u->vl = u->v + n; |
| 198 | u = mp_mul(u, u, mm->m); |
| 199 | d = mp_add(d, a, u); |
| 200 | mp_drop(u); |
| 201 | } |
| 202 | |
| 203 | /* --- Otherwise do it the hard way --- */ |
| 204 | |
| 205 | else { |
| 206 | mpw *dv, *dvl; |
| 207 | mpw *mv, *mvl; |
| 208 | mpw mi; |
| 209 | size_t k = n; |
| 210 | |
| 211 | /* --- Initial conditioning of the arguments --- */ |
| 212 | |
| 213 | if (d == a) |
| 214 | MP_MODIFY(d, 2 * n + 1); |
| 215 | else { |
| 216 | MP_MODIFY(d, 2 * n + 1); |
| 217 | MPX_COPY(d->v, d->vl, a->v, a->vl); |
| 218 | } |
| 219 | |
| 220 | dv = d->v; dvl = d->vl; |
| 221 | mv = mm->m->v; mvl = mm->m->vl; |
| 222 | |
| 223 | /* --- Let's go to work --- */ |
| 224 | |
| 225 | mi = mm->mi->v[0]; |
| 226 | while (k--) { |
| 227 | mpw u = MPW(*dv * mi); |
| 228 | MPX_UMLAN(dv, dvl, mv, mvl, u); |
| 229 | dv++; |
| 230 | } |
| 231 | } |
| 232 | |
| 233 | /* --- Wrap everything up --- */ |
| 234 | |
| 235 | d->f = a->f & MP_BURN; |
| 236 | memmove(d->v, d->v + n, MPWS(MP_LEN(d) - n)); |
| 237 | d->vl -= n; |
| 238 | if (MP_CMP(d, >=, mm->m)) |
| 239 | d = mp_sub(d, d, mm->m); |
| 240 | MP_SHRINK(d); |
| 241 | return (d); |
| 242 | } |
| 243 | |
| 244 | #endif |
| 245 | |
| 246 | /* --- @mpmont_mul@ --- * |
| 247 | * |
| 248 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 249 | * @mp *d@ = destination |
| 250 | * @mp *a, *b@ = sources, assumed positive |
| 251 | * |
| 252 | * Returns: Result, %$a b R^{-1} \bmod m$%. |
| 253 | */ |
| 254 | |
| 255 | #ifdef MPMONT_DISABLE |
| 256 | |
| 257 | mp *mpmont_mul(mpmont *mm, mp *d, mp *a, mp *b) |
| 258 | { |
| 259 | d = mp_mul(d, a, b); |
| 260 | mp_div(0, &d, d, mm->m); |
| 261 | return (d); |
| 262 | } |
| 263 | |
| 264 | #else |
| 265 | |
| 266 | mp *mpmont_mul(mpmont *mm, mp *d, mp *a, mp *b) |
| 267 | { |
| 268 | if (mm->n > KARATSUBA_CUTOFF * 2) { |
| 269 | d = mp_mul(d, a, b); |
| 270 | d = mpmont_reduce(mm, d, d); |
| 271 | } else { |
| 272 | mpw *dv, *dvl; |
| 273 | mpw *av, *avl; |
| 274 | mpw *bv, *bvl; |
| 275 | mpw *mv, *mvl; |
| 276 | mpw y; |
| 277 | size_t n, i; |
| 278 | mpw mi; |
| 279 | |
| 280 | /* --- Initial conditioning of the arguments --- */ |
| 281 | |
| 282 | if (MP_LEN(a) > MP_LEN(b)) { |
| 283 | mp *t = a; a = b; b = t; |
| 284 | } |
| 285 | n = MP_LEN(mm->m); |
| 286 | |
| 287 | a = MP_COPY(a); |
| 288 | b = MP_COPY(b); |
| 289 | MP_MODIFY(d, 2 * n + 1); |
| 290 | dv = d->v; dvl = d->vl; |
| 291 | MPX_ZERO(dv, dvl); |
| 292 | av = a->v; avl = a->vl; |
| 293 | bv = b->v; bvl = b->vl; |
| 294 | mv = mm->m->v; mvl = mm->m->vl; |
| 295 | y = *bv; |
| 296 | |
| 297 | /* --- Montgomery multiplication phase --- */ |
| 298 | |
| 299 | i = 0; |
| 300 | mi = mm->mi->v[0]; |
| 301 | while (i < n && av < avl) { |
| 302 | mpw x = *av++; |
| 303 | mpw u = MPW((*dv + x * y) * mi); |
| 304 | MPX_UMLAN(dv, dvl, bv, bvl, x); |
| 305 | MPX_UMLAN(dv, dvl, mv, mvl, u); |
| 306 | dv++; |
| 307 | i++; |
| 308 | } |
| 309 | |
| 310 | /* --- Simpler Montgomery reduction phase --- */ |
| 311 | |
| 312 | while (i < n) { |
| 313 | mpw u = MPW(*dv * mi); |
| 314 | MPX_UMLAN(dv, dvl, mv, mvl, u); |
| 315 | dv++; |
| 316 | i++; |
| 317 | } |
| 318 | |
| 319 | /* --- Done --- */ |
| 320 | |
| 321 | memmove(d->v, dv, MPWS(dvl - dv)); |
| 322 | d->vl -= dv - d->v; |
| 323 | MP_SHRINK(d); |
| 324 | d->f = (a->f | b->f) & MP_BURN; |
| 325 | if (MP_CMP(d, >=, mm->m)) |
| 326 | d = mp_sub(d, d, mm->m); |
| 327 | MP_DROP(a); |
| 328 | MP_DROP(b); |
| 329 | } |
| 330 | |
| 331 | return (d); |
| 332 | } |
| 333 | |
| 334 | #endif |
| 335 | |
| 336 | /* --- @mpmont_expr@ --- * |
| 337 | * |
| 338 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 339 | * @mp *d@ = fake destination |
| 340 | * @mp *a@ = base |
| 341 | * @mp *e@ = exponent |
| 342 | * |
| 343 | * Returns: Result, %$a^e R \bmod m$%. |
| 344 | */ |
| 345 | |
| 346 | mp *mpmont_expr(mpmont *mm, mp *d, mp *a, mp *e) |
| 347 | { |
| 348 | mpscan sc; |
| 349 | mp *ar = mpmont_mul(mm, MP_NEW, a, mm->r2); |
| 350 | mp *x = MP_COPY(mm->r); |
| 351 | mp *spare = MP_NEW; |
| 352 | |
| 353 | mp_scan(&sc, e); |
| 354 | |
| 355 | if (MP_STEP(&sc)) { |
| 356 | size_t sq = 0; |
| 357 | for (;;) { |
| 358 | mp *dd; |
| 359 | if (MP_BIT(&sc)) { |
| 360 | while (sq) { |
| 361 | dd = mp_sqr(spare, ar); |
| 362 | dd = mpmont_reduce(mm, dd, dd); |
| 363 | spare = ar; ar = dd; |
| 364 | sq--; |
| 365 | } |
| 366 | dd = mpmont_mul(mm, spare, x, ar); |
| 367 | spare = x; x = dd; |
| 368 | } |
| 369 | sq++; |
| 370 | if (!MP_STEP(&sc)) |
| 371 | break; |
| 372 | } |
| 373 | } |
| 374 | MP_DROP(ar); |
| 375 | if (spare != MP_NEW) |
| 376 | MP_DROP(spare); |
| 377 | if (d != MP_NEW) |
| 378 | MP_DROP(d); |
| 379 | return (x); |
| 380 | } |
| 381 | |
| 382 | /* --- @mpmont_exp@ --- * |
| 383 | * |
| 384 | * Arguments: @mpmont *mm@ = pointer to Montgomery reduction context |
| 385 | * @mp *d@ = fake destination |
| 386 | * @mp *a@ = base |
| 387 | * @mp *e@ = exponent |
| 388 | * |
| 389 | * Returns: Result, %$a^e \bmod m$%. |
| 390 | */ |
| 391 | |
| 392 | mp *mpmont_exp(mpmont *mm, mp *d, mp *a, mp *e) |
| 393 | { |
| 394 | d = mpmont_expr(mm, d, a, e); |
| 395 | d = mpmont_reduce(mm, d, d); |
| 396 | return (d); |
| 397 | } |
| 398 | |
| 399 | /*----- Test rig ----------------------------------------------------------*/ |
| 400 | |
| 401 | #ifdef TEST_RIG |
| 402 | |
| 403 | static int tcreate(dstr *v) |
| 404 | { |
| 405 | mp *m = *(mp **)v[0].buf; |
| 406 | mp *mi = *(mp **)v[1].buf; |
| 407 | mp *r = *(mp **)v[2].buf; |
| 408 | mp *r2 = *(mp **)v[3].buf; |
| 409 | |
| 410 | mpmont mm; |
| 411 | int ok = 1; |
| 412 | |
| 413 | mpmont_create(&mm, m); |
| 414 | |
| 415 | if (mm.mi->v[0] != mi->v[0]) { |
| 416 | fprintf(stderr, "\n*** bad mi: found %lu, expected %lu", |
| 417 | (unsigned long)mm.mi->v[0], (unsigned long)mi->v[0]); |
| 418 | fputs("\nm = ", stderr); mp_writefile(m, stderr, 10); |
| 419 | fputc('\n', stderr); |
| 420 | ok = 0; |
| 421 | } |
| 422 | |
| 423 | if (MP_CMP(mm.r, !=, r)) { |
| 424 | fputs("\n*** bad r", stderr); |
| 425 | fputs("\nm = ", stderr); mp_writefile(m, stderr, 10); |
| 426 | fputs("\nexpected ", stderr); mp_writefile(r, stderr, 10); |
| 427 | fputs("\n found ", stderr); mp_writefile(mm.r, stderr, 10); |
| 428 | fputc('\n', stderr); |
| 429 | ok = 0; |
| 430 | } |
| 431 | |
| 432 | if (MP_CMP(mm.r2, !=, r2)) { |
| 433 | fputs("\n*** bad r2", stderr); |
| 434 | fputs("\nm = ", stderr); mp_writefile(m, stderr, 10); |
| 435 | fputs("\nexpected ", stderr); mp_writefile(r2, stderr, 10); |
| 436 | fputs("\n found ", stderr); mp_writefile(mm.r2, stderr, 10); |
| 437 | fputc('\n', stderr); |
| 438 | ok = 0; |
| 439 | } |
| 440 | |
| 441 | MP_DROP(m); |
| 442 | MP_DROP(mi); |
| 443 | MP_DROP(r); |
| 444 | MP_DROP(r2); |
| 445 | mpmont_destroy(&mm); |
| 446 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
| 447 | return (ok); |
| 448 | } |
| 449 | |
| 450 | static int tmul(dstr *v) |
| 451 | { |
| 452 | mp *m = *(mp **)v[0].buf; |
| 453 | mp *a = *(mp **)v[1].buf; |
| 454 | mp *b = *(mp **)v[2].buf; |
| 455 | mp *r = *(mp **)v[3].buf; |
| 456 | int ok = 1; |
| 457 | |
| 458 | mpmont mm; |
| 459 | mpmont_create(&mm, m); |
| 460 | |
| 461 | { |
| 462 | mp *qr = mp_mul(MP_NEW, a, b); |
| 463 | mp_div(0, &qr, qr, m); |
| 464 | |
| 465 | if (MP_CMP(qr, !=, r)) { |
| 466 | fputs("\n*** classical modmul failed", stderr); |
| 467 | fputs("\n m = ", stderr); mp_writefile(m, stderr, 10); |
| 468 | fputs("\n a = ", stderr); mp_writefile(a, stderr, 10); |
| 469 | fputs("\n b = ", stderr); mp_writefile(b, stderr, 10); |
| 470 | fputs("\n r = ", stderr); mp_writefile(r, stderr, 10); |
| 471 | fputs("\nqr = ", stderr); mp_writefile(qr, stderr, 10); |
| 472 | fputc('\n', stderr); |
| 473 | ok = 0; |
| 474 | } |
| 475 | |
| 476 | mp_drop(qr); |
| 477 | } |
| 478 | |
| 479 | { |
| 480 | mp *ar = mpmont_mul(&mm, MP_NEW, a, mm.r2); |
| 481 | mp *br = mpmont_mul(&mm, MP_NEW, b, mm.r2); |
| 482 | mp *mr = mpmont_mul(&mm, MP_NEW, ar, br); |
| 483 | mr = mpmont_reduce(&mm, mr, mr); |
| 484 | if (MP_CMP(mr, !=, r)) { |
| 485 | fputs("\n*** montgomery modmul failed", stderr); |
| 486 | fputs("\n m = ", stderr); mp_writefile(m, stderr, 10); |
| 487 | fputs("\n a = ", stderr); mp_writefile(a, stderr, 10); |
| 488 | fputs("\n b = ", stderr); mp_writefile(b, stderr, 10); |
| 489 | fputs("\n r = ", stderr); mp_writefile(r, stderr, 10); |
| 490 | fputs("\nmr = ", stderr); mp_writefile(mr, stderr, 10); |
| 491 | fputc('\n', stderr); |
| 492 | ok = 0; |
| 493 | } |
| 494 | MP_DROP(ar); MP_DROP(br); |
| 495 | mp_drop(mr); |
| 496 | } |
| 497 | |
| 498 | |
| 499 | MP_DROP(m); |
| 500 | MP_DROP(a); |
| 501 | MP_DROP(b); |
| 502 | MP_DROP(r); |
| 503 | mpmont_destroy(&mm); |
| 504 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
| 505 | return ok; |
| 506 | } |
| 507 | |
| 508 | static int texp(dstr *v) |
| 509 | { |
| 510 | mp *m = *(mp **)v[0].buf; |
| 511 | mp *a = *(mp **)v[1].buf; |
| 512 | mp *b = *(mp **)v[2].buf; |
| 513 | mp *r = *(mp **)v[3].buf; |
| 514 | mp *mr; |
| 515 | int ok = 1; |
| 516 | |
| 517 | mpmont mm; |
| 518 | mpmont_create(&mm, m); |
| 519 | |
| 520 | mr = mpmont_exp(&mm, MP_NEW, a, b); |
| 521 | |
| 522 | if (MP_CMP(mr, !=, r)) { |
| 523 | fputs("\n*** montgomery modexp failed", stderr); |
| 524 | fputs("\n m = ", stderr); mp_writefile(m, stderr, 10); |
| 525 | fputs("\n a = ", stderr); mp_writefile(a, stderr, 10); |
| 526 | fputs("\n e = ", stderr); mp_writefile(b, stderr, 10); |
| 527 | fputs("\n r = ", stderr); mp_writefile(r, stderr, 10); |
| 528 | fputs("\nmr = ", stderr); mp_writefile(mr, stderr, 10); |
| 529 | fputc('\n', stderr); |
| 530 | ok = 0; |
| 531 | } |
| 532 | |
| 533 | MP_DROP(m); |
| 534 | MP_DROP(a); |
| 535 | MP_DROP(b); |
| 536 | MP_DROP(r); |
| 537 | MP_DROP(mr); |
| 538 | mpmont_destroy(&mm); |
| 539 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
| 540 | return ok; |
| 541 | } |
| 542 | |
| 543 | |
| 544 | static test_chunk tests[] = { |
| 545 | { "create", tcreate, { &type_mp, &type_mp, &type_mp, &type_mp, 0 } }, |
| 546 | { "mul", tmul, { &type_mp, &type_mp, &type_mp, &type_mp, 0 } }, |
| 547 | { "exp", texp, { &type_mp, &type_mp, &type_mp, &type_mp, 0 } }, |
| 548 | { 0, 0, { 0 } }, |
| 549 | }; |
| 550 | |
| 551 | int main(int argc, char *argv[]) |
| 552 | { |
| 553 | sub_init(); |
| 554 | test_run(argc, argv, tests, SRCDIR "/tests/mpmont"); |
| 555 | return (0); |
| 556 | } |
| 557 | |
| 558 | #endif |
| 559 | |
| 560 | /*----- That's all, folks -------------------------------------------------*/ |