| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id: exp.h,v 1.1.4.1 2004/03/20 00:13:31 mdw Exp $ |
| 4 | * |
| 5 | * Generalized exponentiation |
| 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: exp.h,v $ |
| 33 | * Revision 1.1.4.1 2004/03/20 00:13:31 mdw |
| 34 | * Projective coordinates for prime curves |
| 35 | * |
| 36 | * Revision 1.1 2001/06/16 13:00:59 mdw |
| 37 | * New generic exponentation code. Includes sliding-window simultaneous |
| 38 | * exponentiation. |
| 39 | * |
| 40 | */ |
| 41 | |
| 42 | #ifdef CATACOMB_EXP_H |
| 43 | # error "Multiple inclusion of <catacomb/exp.h>" |
| 44 | #endif |
| 45 | |
| 46 | #define CATACOMB_EXP_H |
| 47 | |
| 48 | #ifdef __cplusplus |
| 49 | extern "C" { |
| 50 | #endif |
| 51 | |
| 52 | /*----- Header files ------------------------------------------------------*/ |
| 53 | |
| 54 | #include <stddef.h> |
| 55 | |
| 56 | #include <mLib/alloc.h> |
| 57 | |
| 58 | #ifndef CATACOMB_MP_H |
| 59 | # include "mp.h" |
| 60 | #endif |
| 61 | |
| 62 | /*----- Data structures ---------------------------------------------------*/ |
| 63 | |
| 64 | typedef struct exp_simulscan { |
| 65 | mpw w; |
| 66 | size_t len; |
| 67 | const mpw *v; |
| 68 | } exp_simulscan; |
| 69 | |
| 70 | typedef struct exp_simul { |
| 71 | unsigned b; |
| 72 | size_t o, n; |
| 73 | exp_simulscan *s; |
| 74 | } exp_simul; |
| 75 | |
| 76 | /*----- Macros provided ---------------------------------------------------*/ |
| 77 | |
| 78 | /* --- Parameters --- */ |
| 79 | |
| 80 | #ifndef EXP_WINSZ /* Sliding window size */ |
| 81 | # define EXP_WINSZ 4 /* Predefine if you need to */ |
| 82 | #endif |
| 83 | |
| 84 | /* --- These are determined from the window size --- */ |
| 85 | |
| 86 | #define EXP_TABSZ (1 << EXP_WINSZ) |
| 87 | #define EXP_THRESH (((MPW_BITS / EXP_WINSZ) << 2) + 1) |
| 88 | |
| 89 | /* --- Required operations --- * |
| 90 | * |
| 91 | * The macros here are independent of the underlying group elements. You |
| 92 | * must provide the necessary group operations and other definitions. The |
| 93 | * group operation is assumed to be written multiplicatively. |
| 94 | * |
| 95 | * @EXP_TYPE@ The type of a group element, e.g., @mp *@. |
| 96 | * |
| 97 | * @EXP_COPY(d, x)@ Makes @d@ be a copy of @x@. |
| 98 | * |
| 99 | * @EXP_DROP(x)@ Discards the element @x@, reclaiming any |
| 100 | * memory it used. |
| 101 | * |
| 102 | * @EXP_MUL(a, x)@ Multiplies @a@ by @x@ (writing the result |
| 103 | * back to @a@). |
| 104 | * |
| 105 | * @EXP_FIX(x)@ Makes @x@ be a canonical representation of |
| 106 | * its value. All multiplications have the |
| 107 | * right argument canonical. |
| 108 | * |
| 109 | * @EXP_SQR(a)@ Multiplies @a@ by itself. |
| 110 | * |
| 111 | * @EXP_SETMUL(d, x, y)@ Sets @d@ to be the product of @x@ and @y@. |
| 112 | * The value @d@ has not been initialized. |
| 113 | * |
| 114 | * @EXP_SETSQR(d, x)@ Sets @d@ to be the square of @x@. |
| 115 | * |
| 116 | * Only @EXP_TYPE@, @EXP_MUL@ and @EXP_SQR@ are required for simple |
| 117 | * exponentation. Sliding window and simultaneous exponentation require all |
| 118 | * of the operations. |
| 119 | */ |
| 120 | |
| 121 | #ifndef EXP_TYPE |
| 122 | # error "EXP_TYPE not defined for <catacomb/exp.h>" |
| 123 | #endif |
| 124 | |
| 125 | /* --- @EXP_SIMPLE@ --- * |
| 126 | * |
| 127 | * Arguments: @a@ = the result object, initially a multiplicative identity |
| 128 | * @g@ = the object to exponentiate |
| 129 | * @x@ = the exponent, as a multiprecision integer |
| 130 | * |
| 131 | * Use: Performs a simple left-to-right exponentiation. At the end |
| 132 | * of the code, the answer is left in @a@; @g@ and @x@ are |
| 133 | * unchanged. |
| 134 | */ |
| 135 | |
| 136 | #define EXP_SIMPLE(a, g, x) do { \ |
| 137 | mpscan sc; \ |
| 138 | unsigned sq = 0; \ |
| 139 | \ |
| 140 | /* --- Begin scanning --- */ \ |
| 141 | \ |
| 142 | mp_rscan(&sc, x); \ |
| 143 | if (!MP_RSTEP(&sc)) \ |
| 144 | goto exp_simple_exit; \ |
| 145 | while (!MP_RBIT(&sc)) \ |
| 146 | MP_RSTEP(&sc); \ |
| 147 | \ |
| 148 | /* --- Do the main body of the work --- */ \ |
| 149 | \ |
| 150 | EXP_FIX(g); \ |
| 151 | for (;;) { \ |
| 152 | EXP_MUL(a, g); \ |
| 153 | sq = 0; \ |
| 154 | for (;;) { \ |
| 155 | if (!MP_RSTEP(&sc)) \ |
| 156 | goto exp_simple_done; \ |
| 157 | sq++; \ |
| 158 | if (MP_RBIT(&sc)) \ |
| 159 | break; \ |
| 160 | } \ |
| 161 | while (sq--) EXP_SQR(a); \ |
| 162 | } \ |
| 163 | \ |
| 164 | /* --- Do a final round of squaring --- */ \ |
| 165 | \ |
| 166 | exp_simple_done: \ |
| 167 | while (sq--) EXP_SQR(a); \ |
| 168 | exp_simple_exit:; \ |
| 169 | } while (0) |
| 170 | |
| 171 | /* --- @EXP_WINDOW@ --- * |
| 172 | * |
| 173 | * Arguments: @a@ = the result object, initially a multiplicative identity |
| 174 | * @g@ = the object to exponentiate |
| 175 | * @x@ = the exponent, as a multiprecision integer |
| 176 | * |
| 177 | * Use: Performs a sliding-window exponentiation. At the end of the |
| 178 | * code, the answer is left in @a@; @g@ and @x@ are unchanged. |
| 179 | */ |
| 180 | |
| 181 | #define EXP_WINDOW(a, g, x) do { \ |
| 182 | EXP_TYPE *v; \ |
| 183 | EXP_TYPE g2; \ |
| 184 | unsigned i, sq = 0; \ |
| 185 | mpscan sc; \ |
| 186 | \ |
| 187 | /* --- Get going --- */ \ |
| 188 | \ |
| 189 | mp_rscan(&sc, x); \ |
| 190 | if (!MP_RSTEP(&sc)) \ |
| 191 | goto exp_window_exit; \ |
| 192 | \ |
| 193 | /* --- Do the precomputation --- */ \ |
| 194 | \ |
| 195 | EXP_FIX(g); \ |
| 196 | EXP_SETSQR(g2, g); \ |
| 197 | EXP_FIX(g2); \ |
| 198 | v = xmalloc(EXP_TABSZ * sizeof(EXP_TYPE)); \ |
| 199 | EXP_COPY(v[0], g); \ |
| 200 | for (i = 1; i < EXP_TABSZ; i++) { \ |
| 201 | EXP_SETMUL(v[i], v[i - 1], g2); \ |
| 202 | EXP_FIX(v[i]); \ |
| 203 | } \ |
| 204 | EXP_DROP(g2); \ |
| 205 | \ |
| 206 | /* --- Skip top-end zero bits --- * \ |
| 207 | * \ |
| 208 | * If the initial step worked, there must be a set bit somewhere, so \ |
| 209 | * keep stepping until I find it. \ |
| 210 | */ \ |
| 211 | \ |
| 212 | while (!MP_RBIT(&sc)) \ |
| 213 | MP_RSTEP(&sc); \ |
| 214 | \ |
| 215 | /* --- Now for the main work --- */ \ |
| 216 | \ |
| 217 | for (;;) { \ |
| 218 | unsigned l = 1; \ |
| 219 | unsigned z = 0; \ |
| 220 | \ |
| 221 | /* --- The next bit is set, so read a window index --- * \ |
| 222 | * \ |
| 223 | * Reset @i@ to zero and increment @sq@. Then, until either I read \ |
| 224 | * @WINSZ@ bits or I run out of bits, scan in a bit: if it's clear, \ |
| 225 | * bump the @z@ counter; if it's set, push a set bit into @i@, \ |
| 226 | * shift it over by @z@ bits, bump @sq@ by @z + 1@ and clear @z@. \ |
| 227 | * By the end of this palaver, @i@ is an index to the precomputed \ |
| 228 | * value in @v@. \ |
| 229 | */ \ |
| 230 | \ |
| 231 | i = 0; \ |
| 232 | sq++; \ |
| 233 | while (l < EXP_WINSZ && MP_RSTEP(&sc)) { \ |
| 234 | l++; \ |
| 235 | if (!MP_RBIT(&sc)) \ |
| 236 | z++; \ |
| 237 | else { \ |
| 238 | i = ((i << 1) | 1) << z; \ |
| 239 | sq += z + 1; \ |
| 240 | z = 0; \ |
| 241 | } \ |
| 242 | } \ |
| 243 | \ |
| 244 | /* --- Do the squaring --- * \ |
| 245 | * \ |
| 246 | * Remember that @sq@ carries over from the zero-skipping stuff \ |
| 247 | * below. \ |
| 248 | */ \ |
| 249 | \ |
| 250 | while (sq--) EXP_SQR(a); \ |
| 251 | \ |
| 252 | /* --- Do the multiply --- */ \ |
| 253 | \ |
| 254 | EXP_MUL(a, v[i]); \ |
| 255 | \ |
| 256 | /* --- Now grind along through the rest of the bits --- */ \ |
| 257 | \ |
| 258 | sq = z; \ |
| 259 | for (;;) { \ |
| 260 | if (!MP_RSTEP(&sc)) \ |
| 261 | goto exp_window_done; \ |
| 262 | if (MP_RBIT(&sc)) \ |
| 263 | break; \ |
| 264 | sq++; \ |
| 265 | } \ |
| 266 | } \ |
| 267 | \ |
| 268 | /* --- Do a final round of squaring --- */ \ |
| 269 | \ |
| 270 | exp_window_done: \ |
| 271 | while (sq--) EXP_SQR(a); \ |
| 272 | for (i = 0; i < EXP_TABSZ; i++) \ |
| 273 | EXP_DROP(v[i]); \ |
| 274 | xfree(v); \ |
| 275 | exp_window_exit:; \ |
| 276 | } while (0) |
| 277 | |
| 278 | /* --- @EXP_SIMUL@ --- * |
| 279 | * |
| 280 | * Arguments: @a@ = the result object, initially a multiplicative identity |
| 281 | * @f@ = pointer to a vector of base/exp pairs |
| 282 | * @n@ = the number of base/exp pairs |
| 283 | * |
| 284 | * Use: Performs a simultaneous sliding-window exponentiation. The |
| 285 | * @f@ table is an array of structures containing members @base@ |
| 286 | * of type @EXP_TYPE@, and @exp@ of type @mp *@. |
| 287 | */ |
| 288 | |
| 289 | #define EXP_SIMUL(a, f, n) do { \ |
| 290 | size_t i, j, jj, k; \ |
| 291 | size_t vn = 1 << (EXP_WINSZ * n), m = (1 << n) - 1; \ |
| 292 | EXP_TYPE *v = xmalloc(vn * sizeof(EXP_TYPE)); \ |
| 293 | exp_simul e; \ |
| 294 | unsigned sq = 0; \ |
| 295 | \ |
| 296 | /* --- Fill in the precomputed table --- */ \ |
| 297 | \ |
| 298 | j = 1; \ |
| 299 | for (i = 0; i < n; i++) { \ |
| 300 | EXP_COPY(v[j], f[n - 1 - i].base); \ |
| 301 | EXP_FIX(v[j]); \ |
| 302 | j <<= 1; \ |
| 303 | } \ |
| 304 | k = n * EXP_WINSZ; \ |
| 305 | jj = 1; \ |
| 306 | for (; i < k; i++) { \ |
| 307 | EXP_SETSQR(v[j], v[jj]); \ |
| 308 | EXP_FIX(v[j]); \ |
| 309 | j <<= 1; jj <<= 1; \ |
| 310 | } \ |
| 311 | for (i = 1; i < vn; i <<= 1) { \ |
| 312 | for (j = 1; j < i; j++) { \ |
| 313 | EXP_SETMUL(v[j + i], v[j], v[i]); \ |
| 314 | EXP_FIX(v[j + i]); \ |
| 315 | } \ |
| 316 | } \ |
| 317 | \ |
| 318 | /* --- Set up the bitscanners --- * \ |
| 319 | * \ |
| 320 | * Got to use custom scanners, to keep them all in sync. \ |
| 321 | */ \ |
| 322 | \ |
| 323 | e.n = n; \ |
| 324 | e.b = 0; \ |
| 325 | e.s = xmalloc(n * sizeof(*e.s)); \ |
| 326 | e.o = 0; \ |
| 327 | for (i = 0; i < n; i++) { \ |
| 328 | MP_SHRINK(f[i].exp); \ |
| 329 | e.s[i].len = MP_LEN(f[i].exp); \ |
| 330 | e.s[i].v = f[i].exp->v; \ |
| 331 | if (e.s[i].len > e.o) \ |
| 332 | e.o = e.s[i].len; \ |
| 333 | } \ |
| 334 | \ |
| 335 | /* --- Skip as far as a nonzero column in the exponent matrix --- */ \ |
| 336 | \ |
| 337 | do { \ |
| 338 | if (!e.o && !e.b) \ |
| 339 | goto exp_simul_done; \ |
| 340 | i = exp_simulnext(&e, 0); \ |
| 341 | } while (!(i & m)); \ |
| 342 | \ |
| 343 | /* --- Now for the main work --- */ \ |
| 344 | \ |
| 345 | for (;;) { \ |
| 346 | unsigned l = 1; \ |
| 347 | unsigned z = 0; \ |
| 348 | \ |
| 349 | /* --- Just read a nonzero column, so read a window index --- * \ |
| 350 | * \ |
| 351 | * Clear high bits of @i@ and increment @sq@. Then, until either I \ |
| 352 | * read @WINSZ@ columns or I run out, scan in a column and append \ |
| 353 | * it to @i@. If it's zero, bump the @z@ counter; if it's nonzero, \ |
| 354 | * bump @sq@ by @z + 1@ and clear @z@. By the end of this palaver, \ |
| 355 | * @i@ is an index to the precomputed value in @v@, followed by \ |
| 356 | * @n * z@ zero bits. \ |
| 357 | */ \ |
| 358 | \ |
| 359 | sq++; \ |
| 360 | while (l < EXP_WINSZ && (e.o || e.b)) { \ |
| 361 | l++; \ |
| 362 | i = exp_simulnext(&e, i); \ |
| 363 | if (!(i & m)) \ |
| 364 | z++; \ |
| 365 | else { \ |
| 366 | sq += z + 1; \ |
| 367 | z = 0; \ |
| 368 | } \ |
| 369 | } \ |
| 370 | \ |
| 371 | /* --- Do the squaring --- * \ |
| 372 | * \ |
| 373 | * Remember that @sq@ carries over from the zero-skipping stuff \ |
| 374 | * below. \ |
| 375 | */ \ |
| 376 | \ |
| 377 | while (sq--) EXP_SQR(a); \ |
| 378 | \ |
| 379 | /* --- Do the multiply --- */ \ |
| 380 | \ |
| 381 | i >>= (z * n); \ |
| 382 | EXP_MUL(a, v[i]); \ |
| 383 | \ |
| 384 | /* --- Now grind along through the rest of the bits --- */ \ |
| 385 | \ |
| 386 | sq = z; \ |
| 387 | for (;;) { \ |
| 388 | if (!e.o && !e.b) \ |
| 389 | goto exp_simul_done; \ |
| 390 | if ((i = exp_simulnext(&e, 0)) != 0) \ |
| 391 | break; \ |
| 392 | sq++; \ |
| 393 | } \ |
| 394 | } \ |
| 395 | \ |
| 396 | /* --- Do a final round of squaring --- */ \ |
| 397 | \ |
| 398 | exp_simul_done: \ |
| 399 | while (sq--) EXP_SQR(a); \ |
| 400 | for (i = 1; i < vn; i++) \ |
| 401 | EXP_DROP(v[i]); \ |
| 402 | xfree(v); \ |
| 403 | } while (0) |
| 404 | |
| 405 | /*----- Functions provided ------------------------------------------------*/ |
| 406 | |
| 407 | /* --- @exp_simulnext@ --- * |
| 408 | * |
| 409 | * Arguments: @exp_simul *e@ = pointer to state structure |
| 410 | * @size_t x@ = a current accumulator |
| 411 | * |
| 412 | * Returns: The next column of bits. |
| 413 | * |
| 414 | * Use: Scans the next column of bits for a simultaneous |
| 415 | * exponentiation. |
| 416 | */ |
| 417 | |
| 418 | extern size_t exp_simulnext(exp_simul */*e*/, size_t /*x*/); |
| 419 | |
| 420 | /*----- That's all, folks -------------------------------------------------*/ |
| 421 | |
| 422 | #ifdef __cplusplus |
| 423 | } |
| 424 | #endif |