| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * Arithmetic in the Goldilocks field GF(2^448 - 2^224 - 1) |
| 4 | * |
| 5 | * (c) 2017 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 "config.h" |
| 31 | |
| 32 | #include "ct.h" |
| 33 | #include "fgoldi.h" |
| 34 | |
| 35 | /*----- Basic setup -------------------------------------------------------* |
| 36 | * |
| 37 | * Let φ = 2^224; then p = φ^2 - φ - 1, and, in GF(p), we have φ^2 = φ + 1 |
| 38 | * (hence the name). |
| 39 | */ |
| 40 | |
| 41 | typedef fgoldi_piece piece; |
| 42 | |
| 43 | #if FGOLDI_IMPL == 28 |
| 44 | /* We represent an element of GF(p) as 16 28-bit signed integer pieces x_i: |
| 45 | * x = SUM_{0<=i<16} x_i 2^(28i). |
| 46 | */ |
| 47 | |
| 48 | typedef int64 dblpiece; |
| 49 | typedef uint32 upiece; typedef uint64 udblpiece; |
| 50 | #define PIECEWD(i) 28 |
| 51 | #define NPIECE 16 |
| 52 | #define P p28 |
| 53 | |
| 54 | #define B28 0x10000000u |
| 55 | #define B27 0x08000000u |
| 56 | #define M28 0x0fffffffu |
| 57 | #define M27 0x07ffffffu |
| 58 | #define M32 0xffffffffu |
| 59 | |
| 60 | #elif FGOLDI_IMPL == 12 |
| 61 | /* We represent an element of GF(p) as 40 signed integer pieces x_i: x = |
| 62 | * SUM_{0<=i<40} x_i 2^ceil(224i/20). Pieces i with i == 0 (mod 5) are 12 |
| 63 | * bits wide; the others are 11 bits wide, so they form eight groups of 56 |
| 64 | * bits. |
| 65 | */ |
| 66 | |
| 67 | typedef int32 dblpiece; |
| 68 | typedef uint16 upiece; typedef uint32 udblpiece; |
| 69 | #define PIECEWD(i) ((i)%5 ? 11 : 12) |
| 70 | #define NPIECE 40 |
| 71 | #define P p12 |
| 72 | |
| 73 | #define B12 0x1000u |
| 74 | #define B11 0x0800u |
| 75 | #define B10 0x0400u |
| 76 | #define M12 0xfffu |
| 77 | #define M11 0x7ffu |
| 78 | #define M10 0x3ffu |
| 79 | #define M8 0xffu |
| 80 | |
| 81 | #endif |
| 82 | |
| 83 | /*----- Debugging machinery -----------------------------------------------*/ |
| 84 | |
| 85 | #if defined(FGOLDI_DEBUG) || defined(TEST_RIG) |
| 86 | |
| 87 | #include <stdio.h> |
| 88 | |
| 89 | #include "mp.h" |
| 90 | #include "mptext.h" |
| 91 | |
| 92 | static mp *get_pgoldi(void) |
| 93 | { |
| 94 | mp *p = MP_NEW, *t = MP_NEW; |
| 95 | |
| 96 | p = mp_setbit(p, MP_ZERO, 448); |
| 97 | t = mp_setbit(t, MP_ZERO, 224); |
| 98 | p = mp_sub(p, p, t); |
| 99 | p = mp_sub(p, p, MP_ONE); |
| 100 | mp_drop(t); |
| 101 | return (p); |
| 102 | } |
| 103 | |
| 104 | DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 56, get_pgoldi()) |
| 105 | |
| 106 | #endif |
| 107 | |
| 108 | /*----- Loading and storing -----------------------------------------------*/ |
| 109 | |
| 110 | /* --- @fgoldi_load@ --- * |
| 111 | * |
| 112 | * Arguments: @fgoldi *z@ = where to store the result |
| 113 | * @const octet xv[56]@ = source to read |
| 114 | * |
| 115 | * Returns: --- |
| 116 | * |
| 117 | * Use: Reads an element of %$\gf{2^{448} - 2^{224} - 19}$% in |
| 118 | * external representation from @xv@ and stores it in @z@. |
| 119 | * |
| 120 | * External representation is little-endian base-256. Some |
| 121 | * elements have multiple encodings, which are not produced by |
| 122 | * correct software; use of noncanonical encodings is not an |
| 123 | * error, and toleration of them is considered a performance |
| 124 | * feature. |
| 125 | */ |
| 126 | |
| 127 | void fgoldi_load(fgoldi *z, const octet xv[56]) |
| 128 | { |
| 129 | #if FGOLDI_IMPL == 28 |
| 130 | |
| 131 | unsigned i; |
| 132 | uint32 xw[14]; |
| 133 | piece b, c; |
| 134 | |
| 135 | /* First, read the input value as words. */ |
| 136 | for (i = 0; i < 14; i++) xw[i] = LOAD32_L(xv + 4*i); |
| 137 | |
| 138 | /* Extract unsigned 28-bit pieces from the words. */ |
| 139 | z->P[ 0] = (xw[ 0] >> 0)&M28; |
| 140 | z->P[ 7] = (xw[ 6] >> 4)&M28; |
| 141 | z->P[ 8] = (xw[ 7] >> 0)&M28; |
| 142 | z->P[15] = (xw[13] >> 4)&M28; |
| 143 | for (i = 1; i < 7; i++) { |
| 144 | z->P[i + 0] = ((xw[i + 0] << (4*i)) | (xw[i - 1] >> (32 - 4*i)))&M28; |
| 145 | z->P[i + 8] = ((xw[i + 7] << (4*i)) | (xw[i + 6] >> (32 - 4*i)))&M28; |
| 146 | } |
| 147 | |
| 148 | /* Convert the nonnegative pieces into a balanced signed representation, so |
| 149 | * each piece ends up in the interval |z_i| <= 2^27. For each piece, if |
| 150 | * its top bit is set, lend a bit leftwards; in the case of z_15, reduce |
| 151 | * this bit by adding it onto z_0 and z_8, since this is the φ^2 bit, and |
| 152 | * φ^2 = φ + 1. We delay this carry until after all of the pieces have |
| 153 | * been balanced. If we don't do this, then we have to do a more expensive |
| 154 | * test for nonzeroness to decide whether to lend a bit leftwards rather |
| 155 | * than just testing a single bit. |
| 156 | * |
| 157 | * Note that we don't try for a canonical representation here: both upper |
| 158 | * and lower bounds are achievable. |
| 159 | */ |
| 160 | b = z->P[15]&B27; z->P[15] -= b << 1; c = b >> 27; |
| 161 | for (i = NPIECE - 1; i--; ) |
| 162 | { b = z->P[i]&B27; z->P[i] -= b << 1; z->P[i + 1] += b >> 27; } |
| 163 | z->P[0] += c; z->P[8] += c; |
| 164 | |
| 165 | #elif FGOLDI_IMPL == 12 |
| 166 | |
| 167 | unsigned i, j, n, w, b; |
| 168 | uint32 a; |
| 169 | int c; |
| 170 | |
| 171 | /* First, convert the bytes into nonnegative pieces. */ |
| 172 | for (i = j = a = n = 0, w = PIECEWD(0); i < 56; i++) { |
| 173 | a |= (uint32)xv[i] << n; n += 8; |
| 174 | if (n >= w) { |
| 175 | z->P[j++] = a&MASK(w); |
| 176 | a >>= w; n -= w; w = PIECEWD(j); |
| 177 | } |
| 178 | } |
| 179 | |
| 180 | /* Convert the nonnegative pieces into a balanced signed representation, so |
| 181 | * each piece ends up in the interval |z_i| <= 2^11 + 1. |
| 182 | */ |
| 183 | b = z->P[39]&B10; z->P[39] -= b << 1; c = b >> 10; |
| 184 | for (i = NPIECE - 1; i--; ) { |
| 185 | w = PIECEWD(i) - 1; |
| 186 | b = z->P[i]&BIT(w); |
| 187 | z->P[i] -= b << 1; |
| 188 | z->P[i + 1] += b >> w; |
| 189 | } |
| 190 | z->P[0] += c; z->P[20] += c; |
| 191 | |
| 192 | #endif |
| 193 | } |
| 194 | |
| 195 | /* --- @fgoldi_store@ --- * |
| 196 | * |
| 197 | * Arguments: @octet zv[56]@ = where to write the result |
| 198 | * @const fgoldi *x@ = the field element to write |
| 199 | * |
| 200 | * Returns: --- |
| 201 | * |
| 202 | * Use: Stores a field element in the given octet vector in external |
| 203 | * representation. A canonical encoding is always stored. |
| 204 | */ |
| 205 | |
| 206 | void fgoldi_store(octet zv[56], const fgoldi *x) |
| 207 | { |
| 208 | #if FGOLDI_IMPL == 28 |
| 209 | |
| 210 | piece y[NPIECE], yy[NPIECE], c, d; |
| 211 | uint32 u, v; |
| 212 | mask32 m; |
| 213 | unsigned i; |
| 214 | |
| 215 | for (i = 0; i < NPIECE; i++) y[i] = x->P[i]; |
| 216 | |
| 217 | /* First, propagate the carries. By the end of this, we'll have all of the |
| 218 | * the pieces canonically sized and positive, and maybe there'll be |
| 219 | * (signed) carry out. The carry c is in { -1, 0, +1 }, and the remaining |
| 220 | * value will be in the half-open interval [0, φ^2). The whole represented |
| 221 | * value is then y + φ^2 c. |
| 222 | * |
| 223 | * Assume that we start out with |y_i| <= 2^30. We start off by cutting |
| 224 | * off and reducing the carry c_15 from the topmost piece, y_15. This |
| 225 | * leaves 0 <= y_15 < 2^28; and we'll have |c_15| <= 4. We'll add this |
| 226 | * onto y_0 and y_8, and propagate the carries. It's very clear that we'll |
| 227 | * end up with |y + (φ + 1) c_15 - φ^2/2| << φ^2. |
| 228 | * |
| 229 | * Here, the y_i are signed, so we must be cautious about bithacking them. |
| 230 | */ |
| 231 | c = ASR(piece, y[15], 28); y[15] = (upiece)y[15]&M28; y[8] += c; |
| 232 | for (i = 0; i < NPIECE; i++) |
| 233 | { y[i] += c; c = ASR(piece, y[i], 28); y[i] = (upiece)y[i]&M28; } |
| 234 | |
| 235 | /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and |
| 236 | * y >= p, then we should subtract p from the whole value; if c = -1 then |
| 237 | * we should add p; and otherwise we should do nothing. |
| 238 | * |
| 239 | * But conditional behaviour is bad, m'kay. So here's what we do instead. |
| 240 | * |
| 241 | * The first job is to sort out what we wanted to do. If c = -1 then we |
| 242 | * want to (a) invert the constant addend and (b) feed in a carry-in; |
| 243 | * otherwise, we don't. |
| 244 | */ |
| 245 | m = SIGN(c)&M28; |
| 246 | d = m&1; |
| 247 | |
| 248 | /* Now do the addition/subtraction. Remember that all of the y_i are |
| 249 | * nonnegative, so shifting and masking are safe and easy. |
| 250 | */ |
| 251 | d += y[0] + (1 ^ m); yy[0] = d&M28; d >>= 28; |
| 252 | for (i = 1; i < 8; i++) |
| 253 | { d += y[i] + m; yy[i] = d&M28; d >>= 28; } |
| 254 | d += y[8] + (1 ^ m); yy[8] = d&M28; d >>= 28; |
| 255 | for (i = 9; i < 16; i++) |
| 256 | { d += y[i] + m; yy[i] = d&M28; d >>= 28; } |
| 257 | |
| 258 | /* The final carry-out is in d; since we only did addition, and the y_i are |
| 259 | * nonnegative, then d is in { 0, 1 }. We want to keep y', rather than y, |
| 260 | * if (a) c /= 0 (in which case we know that the old value was |
| 261 | * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that |
| 262 | * the subtraction didn't cause a borrow, so we must be in the case where |
| 263 | * p <= y < φ^2. |
| 264 | */ |
| 265 | m = NONZEROP(c) | ~NONZEROP(d - 1); |
| 266 | for (i = 0; i < NPIECE; i++) y[i] = (yy[i]&m) | (y[i]&~m); |
| 267 | |
| 268 | /* Extract 32-bit words from the value. */ |
| 269 | for (i = 0; i < 7; i++) { |
| 270 | u = ((y[i + 0] >> (4*i)) | ((uint32)y[i + 1] << (28 - 4*i)))&M32; |
| 271 | v = ((y[i + 8] >> (4*i)) | ((uint32)y[i + 9] << (28 - 4*i)))&M32; |
| 272 | STORE32_L(zv + 4*i, u); |
| 273 | STORE32_L(zv + 4*i + 28, v); |
| 274 | } |
| 275 | |
| 276 | #elif FGOLDI_IMPL == 12 |
| 277 | |
| 278 | piece y[NPIECE], yy[NPIECE], c, d; |
| 279 | uint32 a; |
| 280 | mask32 m, mm; |
| 281 | unsigned i, j, n, w; |
| 282 | |
| 283 | for (i = 0; i < NPIECE; i++) y[i] = x->P[i]; |
| 284 | |
| 285 | /* First, propagate the carries. By the end of this, we'll have all of the |
| 286 | * the pieces canonically sized and positive, and maybe there'll be |
| 287 | * (signed) carry out. The carry c is in { -1, 0, +1 }, and the remaining |
| 288 | * value will be in the half-open interval [0, φ^2). The whole represented |
| 289 | * value is then y + φ^2 c. |
| 290 | * |
| 291 | * Assume that we start out with |y_i| <= 2^14. We start off by cutting |
| 292 | * off and reducing the carry c_39 from the topmost piece, y_39. This |
| 293 | * leaves 0 <= y_39 < 2^11; and we'll have |c_39| <= 16. We'll add this |
| 294 | * onto y_0 and y_20, and propagate the carries. It's very clear that |
| 295 | * we'll end up with |y + (φ + 1) c_39 - φ^2/2| << φ^2. |
| 296 | * |
| 297 | * Here, the y_i are signed, so we must be cautious about bithacking them. |
| 298 | */ |
| 299 | c = ASR(piece, y[39], 11); y[39] = (piece)y[39]&M11; y[20] += c; |
| 300 | for (i = 0; i < NPIECE; i++) { |
| 301 | w = PIECEWD(i); m = (1 << w) - 1; |
| 302 | y[i] += c; c = ASR(piece, y[i], w); y[i] = (upiece)y[i]&m; |
| 303 | } |
| 304 | |
| 305 | /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and |
| 306 | * y >= p, then we should subtract p from the whole value; if c = -1 then |
| 307 | * we should add p; and otherwise we should do nothing. |
| 308 | * |
| 309 | * But conditional behaviour is bad, m'kay. So here's what we do instead. |
| 310 | * |
| 311 | * The first job is to sort out what we wanted to do. If c = -1 then we |
| 312 | * want to (a) invert the constant addend and (b) feed in a carry-in; |
| 313 | * otherwise, we don't. |
| 314 | */ |
| 315 | mm = SIGN(c); |
| 316 | d = m&1; |
| 317 | |
| 318 | /* Now do the addition/subtraction. Remember that all of the y_i are |
| 319 | * nonnegative, so shifting and masking are safe and easy. |
| 320 | */ |
| 321 | d += y[ 0] + (1 ^ (mm&M12)); yy[ 0] = d&M12; d >>= 12; |
| 322 | for (i = 1; i < 20; i++) { |
| 323 | w = PIECEWD(i); m = MASK(w); |
| 324 | d += y[ i] + (mm&m); yy[ i] = d&m; d >>= w; |
| 325 | } |
| 326 | d += y[20] + (1 ^ (mm&M12)); yy[20] = d&M12; d >>= 12; |
| 327 | for (i = 21; i < 40; i++) { |
| 328 | w = PIECEWD(i); m = MASK(w); |
| 329 | d += y[ i] + (mm&m); yy[ i] = d&m; d >>= w; |
| 330 | } |
| 331 | |
| 332 | /* The final carry-out is in d; since we only did addition, and the y_i are |
| 333 | * nonnegative, then d is in { 0, 1 }. We want to keep y', rather than y, |
| 334 | * if (a) c /= 0 (in which case we know that the old value was |
| 335 | * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that |
| 336 | * the subtraction didn't cause a borrow, so we must be in the case where |
| 337 | * p <= y < φ^2. |
| 338 | */ |
| 339 | m = NONZEROP(c) | ~NONZEROP(d - 1); |
| 340 | for (i = 0; i < NPIECE; i++) y[i] = (yy[i]&m) | (y[i]&~m); |
| 341 | |
| 342 | /* Convert that back into octets. */ |
| 343 | for (i = j = a = n = 0; i < NPIECE; i++) { |
| 344 | a |= (uint32)y[i] << n; n += PIECEWD(i); |
| 345 | while (n >= 8) { zv[j++] = a&M8; a >>= 8; n -= 8; } |
| 346 | } |
| 347 | |
| 348 | #endif |
| 349 | } |
| 350 | |
| 351 | /* --- @fgoldi_set@ --- * |
| 352 | * |
| 353 | * Arguments: @fgoldi *z@ = where to write the result |
| 354 | * @int a@ = a small-ish constant |
| 355 | * |
| 356 | * Returns: --- |
| 357 | * |
| 358 | * Use: Sets @z@ to equal @a@. |
| 359 | */ |
| 360 | |
| 361 | void fgoldi_set(fgoldi *x, int a) |
| 362 | { |
| 363 | unsigned i; |
| 364 | |
| 365 | x->P[0] = a; |
| 366 | for (i = 1; i < NPIECE; i++) x->P[i] = 0; |
| 367 | } |
| 368 | |
| 369 | /*----- Basic arithmetic --------------------------------------------------*/ |
| 370 | |
| 371 | /* --- @fgoldi_add@ --- * |
| 372 | * |
| 373 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 374 | * @const fgoldi *x, *y@ = two operands |
| 375 | * |
| 376 | * Returns: --- |
| 377 | * |
| 378 | * Use: Set @z@ to the sum %$x + y$%. |
| 379 | */ |
| 380 | |
| 381 | void fgoldi_add(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 382 | { |
| 383 | unsigned i; |
| 384 | for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] + y->P[i]; |
| 385 | } |
| 386 | |
| 387 | /* --- @fgoldi_sub@ --- * |
| 388 | * |
| 389 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 390 | * @const fgoldi *x, *y@ = two operands |
| 391 | * |
| 392 | * Returns: --- |
| 393 | * |
| 394 | * Use: Set @z@ to the difference %$x - y$%. |
| 395 | */ |
| 396 | |
| 397 | void fgoldi_sub(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 398 | { |
| 399 | unsigned i; |
| 400 | for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] - y->P[i]; |
| 401 | } |
| 402 | |
| 403 | /* --- @fgoldi_neg@ --- * |
| 404 | * |
| 405 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 406 | * @const fgoldi *x@ = an operand |
| 407 | * |
| 408 | * Returns: --- |
| 409 | * |
| 410 | * Use: Set @z = -x@. |
| 411 | */ |
| 412 | |
| 413 | void fgoldi_neg(fgoldi *z, const fgoldi *x) |
| 414 | { |
| 415 | unsigned i; |
| 416 | for (i = 0; i < NPIECE; i++) z->P[i] = -x->P[i]; |
| 417 | } |
| 418 | |
| 419 | /*----- Constant-time utilities -------------------------------------------*/ |
| 420 | |
| 421 | /* --- @fgoldi_pick2@ --- * |
| 422 | * |
| 423 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 424 | * @const fgoldi *x, *y@ = two operands |
| 425 | * @uint32 m@ = a mask |
| 426 | * |
| 427 | * Returns: --- |
| 428 | * |
| 429 | * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set |
| 430 | * @z = x@. If @m@ has some other value, then scramble @z@ in |
| 431 | * an unhelpful way. |
| 432 | */ |
| 433 | |
| 434 | void fgoldi_pick2(fgoldi *z, const fgoldi *x, const fgoldi *y, uint32 m) |
| 435 | { |
| 436 | mask32 mm = FIX_MASK32(m); |
| 437 | unsigned i; |
| 438 | for (i = 0; i < NPIECE; i++) z->P[i] = PICK2(x->P[i], y->P[i], mm); |
| 439 | } |
| 440 | |
| 441 | /* --- @fgoldi_pickn@ --- * |
| 442 | * |
| 443 | * Arguments: @fgoldi *z@ = where to put the result |
| 444 | * @const fgoldi *v@ = a table of entries |
| 445 | * @size_t n@ = the number of entries in @v@ |
| 446 | * @size_t i@ = an index |
| 447 | * |
| 448 | * Returns: --- |
| 449 | * |
| 450 | * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then |
| 451 | * do something unhelpful; otherwise, if @i >= n@ then set @z@ |
| 452 | * to zero. |
| 453 | */ |
| 454 | |
| 455 | void fgoldi_pickn(fgoldi *z, const fgoldi *v, size_t n, size_t i) |
| 456 | { |
| 457 | uint32 b = (uint32)1 << (31 - i); |
| 458 | mask32 m; |
| 459 | unsigned j; |
| 460 | |
| 461 | for (j = 0; j < NPIECE; j++) z->P[j] = 0; |
| 462 | while (n--) { |
| 463 | m = SIGN(b); |
| 464 | for (j = 0; j < NPIECE; j++) CONDPICK(z->P[j], v->P[j], m); |
| 465 | v++; b <<= 1; |
| 466 | } |
| 467 | } |
| 468 | |
| 469 | /* --- @fgoldi_condswap@ --- * |
| 470 | * |
| 471 | * Arguments: @fgoldi *x, *y@ = two operands |
| 472 | * @uint32 m@ = a mask |
| 473 | * |
| 474 | * Returns: --- |
| 475 | * |
| 476 | * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then |
| 477 | * exchange @x@ and @y@. If @m@ has some other value, then |
| 478 | * scramble @x@ and @y@ in an unhelpful way. |
| 479 | */ |
| 480 | |
| 481 | void fgoldi_condswap(fgoldi *x, fgoldi *y, uint32 m) |
| 482 | { |
| 483 | unsigned i; |
| 484 | mask32 mm = FIX_MASK32(m); |
| 485 | |
| 486 | for (i = 0; i < NPIECE; i++) CONDSWAP(x->P[i], y->P[i], mm); |
| 487 | } |
| 488 | |
| 489 | /* --- @fgoldi_condneg@ --- * |
| 490 | * |
| 491 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 492 | * @const fgoldi *x@ = an operand |
| 493 | * @uint32 m@ = a mask |
| 494 | * |
| 495 | * Returns: --- |
| 496 | * |
| 497 | * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set |
| 498 | * @z = -x@. If @m@ has some other value then scramble @z@ in |
| 499 | * an unhelpful way. |
| 500 | */ |
| 501 | |
| 502 | void fgoldi_condneg(fgoldi *z, const fgoldi *x, uint32 m) |
| 503 | { |
| 504 | #ifdef NEG_TWOC |
| 505 | mask32 m_xor = FIX_MASK32(m); |
| 506 | piece m_add = m&1; |
| 507 | # define CONDNEG(x) (((x) ^ m_xor) + m_add) |
| 508 | #else |
| 509 | int s = PICK2(-1, +1, m); |
| 510 | # define CONDNEG(x) (s*(x)) |
| 511 | #endif |
| 512 | |
| 513 | unsigned i; |
| 514 | for (i = 0; i < NPIECE; i++) z->P[i] = CONDNEG(x->P[i]); |
| 515 | |
| 516 | #undef CONDNEG |
| 517 | } |
| 518 | |
| 519 | /*----- Multiplication ----------------------------------------------------*/ |
| 520 | |
| 521 | #if FGOLDI_IMPL == 28 |
| 522 | |
| 523 | /* Let B = 2^63 - 1 be the largest value such that +B and -B can be |
| 524 | * represented in a double-precision piece. On entry, it must be the case |
| 525 | * that |X_i| <= M <= B - 2^27 for some M. If this is the case, then, on |
| 526 | * exit, we will have |Z_i| <= 2^27 + M/2^27. |
| 527 | */ |
| 528 | #define CARRY_REDUCE(z, x) do { \ |
| 529 | dblpiece _t[NPIECE], _c; \ |
| 530 | unsigned _i; \ |
| 531 | \ |
| 532 | /* Bias the input pieces. This keeps the carries and so on centred \ |
| 533 | * around zero rather than biased positive. \ |
| 534 | */ \ |
| 535 | for (_i = 0; _i < NPIECE; _i++) _t[_i] = (x)[_i] + B27; \ |
| 536 | \ |
| 537 | /* Calculate the reduced pieces. Careful with the bithacking. */ \ |
| 538 | _c = ASR(dblpiece, _t[15], 28); \ |
| 539 | (z)[0] = (dblpiece)((udblpiece)_t[0]&M28) - B27 + _c; \ |
| 540 | for (_i = 1; _i < NPIECE; _i++) { \ |
| 541 | (z)[_i] = (dblpiece)((udblpiece)_t[_i]&M28) - B27 + \ |
| 542 | ASR(dblpiece, _t[_i - 1], 28); \ |
| 543 | } \ |
| 544 | (z)[8] += _c; \ |
| 545 | } while (0) |
| 546 | |
| 547 | #elif FGOLDI_IMPL == 12 |
| 548 | |
| 549 | static void carry_reduce(dblpiece x[NPIECE]) |
| 550 | { |
| 551 | /* Initial bounds: we assume |x_i| < 2^31 - 2^27. */ |
| 552 | |
| 553 | unsigned i, j; |
| 554 | dblpiece c; |
| 555 | |
| 556 | /* The result is nearly canonical, because we do sequential carry |
| 557 | * propagation, because smaller processors are more likely to prefer the |
| 558 | * smaller working set than the instruction-level parallelism. |
| 559 | * |
| 560 | * Start at x_37; truncate it to 10 bits, and propagate the carry to x_38. |
| 561 | * Truncate x_38 to 10 bits, and add the carry onto x_39. Truncate x_39 to |
| 562 | * 10 bits, and add the carry onto x_0 and x_20. And so on. |
| 563 | * |
| 564 | * Once we reach x_37 for the second time, we start with |x_37| <= 2^10. |
| 565 | * The carry into x_37 is at most 2^21; so the carry out into x_38 has |
| 566 | * magnitude at most 2^10. In turn, |x_38| <= 2^10 before the carry, so is |
| 567 | * now no more than 2^11 in magnitude, and the carry out into x_39 is at |
| 568 | * most 1. This leaves |x_39| <= 2^10 + 1 after carry propagation. |
| 569 | * |
| 570 | * Be careful with the bit hacking because the quantities involved are |
| 571 | * signed. |
| 572 | */ |
| 573 | |
| 574 | /* For each piece, we bias it so that floor division (as done by an |
| 575 | * arithmetic right shift) and modulus (as done by bitwise-AND) does the |
| 576 | * right thing. |
| 577 | */ |
| 578 | #define CARRY(i, wd, b, m) do { \ |
| 579 | x[i] += (b); \ |
| 580 | c = ASR(dblpiece, x[i], (wd)); \ |
| 581 | x[i] = (dblpiece)((udblpiece)x[i]&(m)) - (b); \ |
| 582 | } while (0) |
| 583 | |
| 584 | { CARRY(37, 11, B10, M11); } |
| 585 | { x[38] += c; CARRY(38, 11, B10, M11); } |
| 586 | { x[39] += c; CARRY(39, 11, B10, M11); } |
| 587 | x[20] += c; |
| 588 | for (i = 0; i < 35; ) { |
| 589 | { x[i] += c; CARRY( i, 12, B11, M12); i++; } |
| 590 | for (j = i + 4; i < j; ) { x[i] += c; CARRY( i, 11, B10, M11); i++; } |
| 591 | } |
| 592 | { x[i] += c; CARRY( i, 12, B11, M12); i++; } |
| 593 | while (i < 39) { x[i] += c; CARRY( i, 11, B10, M11); i++; } |
| 594 | x[39] += c; |
| 595 | } |
| 596 | |
| 597 | #endif |
| 598 | |
| 599 | /* --- @fgoldi_mulconst@ --- * |
| 600 | * |
| 601 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 602 | * @const fgoldi *x@ = an operand |
| 603 | * @long a@ = a small-ish constant; %$|a| < 2^{20}$%. |
| 604 | * |
| 605 | * Returns: --- |
| 606 | * |
| 607 | * Use: Set @z@ to the product %$a x$%. |
| 608 | */ |
| 609 | |
| 610 | void fgoldi_mulconst(fgoldi *z, const fgoldi *x, long a) |
| 611 | { |
| 612 | unsigned i; |
| 613 | dblpiece zz[NPIECE], aa = a; |
| 614 | |
| 615 | for (i = 0; i < NPIECE; i++) zz[i] = aa*x->P[i]; |
| 616 | #if FGOLDI_IMPL == 28 |
| 617 | CARRY_REDUCE(z->P, zz); |
| 618 | #elif FGOLDI_IMPL == 12 |
| 619 | carry_reduce(zz); |
| 620 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; |
| 621 | #endif |
| 622 | } |
| 623 | |
| 624 | /* --- @fgoldi_mul@ --- * |
| 625 | * |
| 626 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 627 | * @const fgoldi *x, *y@ = two operands |
| 628 | * |
| 629 | * Returns: --- |
| 630 | * |
| 631 | * Use: Set @z@ to the product %$x y$%. |
| 632 | */ |
| 633 | |
| 634 | void fgoldi_mul(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 635 | { |
| 636 | dblpiece zz[NPIECE], u[NPIECE]; |
| 637 | piece ab[NPIECE/2], cd[NPIECE/2]; |
| 638 | const piece |
| 639 | *a = x->P + NPIECE/2, *b = x->P, |
| 640 | *c = y->P + NPIECE/2, *d = y->P; |
| 641 | unsigned i, j; |
| 642 | |
| 643 | #if FGOLDI_IMPL == 28 |
| 644 | |
| 645 | # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) |
| 646 | |
| 647 | #elif FGOLDI_IMPL == 12 |
| 648 | |
| 649 | static const unsigned short off[39] = { |
| 650 | 0, 12, 23, 34, 45, 56, 68, 79, 90, 101, |
| 651 | 112, 124, 135, 146, 157, 168, 180, 191, 202, 213, |
| 652 | 224, 236, 247, 258, 269, 280, 292, 303, 314, 325, |
| 653 | 336, 348, 359, 370, 381, 392, 404, 415, 426 |
| 654 | }; |
| 655 | |
| 656 | #define M(x,i, y,j) \ |
| 657 | (((dblpiece)(x)[i]*(y)[j]) << (off[i] + off[j] - off[(i) + (j)])) |
| 658 | |
| 659 | #endif |
| 660 | |
| 661 | /* Behold the magic. |
| 662 | * |
| 663 | * Write x = a φ + b, and y = c φ + d. Then x y = a c φ^2 + |
| 664 | * (a d + b c) φ + b d. Karatsuba and Ofman observed that a d + b c = |
| 665 | * (a + b) (c + d) - a c - b d, saving a multiplication, and Hamburg chose |
| 666 | * the prime p so that φ^2 = φ + 1. So |
| 667 | * |
| 668 | * x y = ((a + b) (c + d) - b d) φ + a c + b d |
| 669 | */ |
| 670 | |
| 671 | for (i = 0; i < NPIECE; i++) zz[i] = 0; |
| 672 | |
| 673 | /* Our first job will be to calculate (1 - φ) b d, and write the result |
| 674 | * into z. As we do this, an interesting thing will happen. Write |
| 675 | * b d = u φ + v; then (1 - φ) b d = u φ + v - u φ^2 - v φ = (1 - φ) v - u. |
| 676 | * So, what we do is to write the product end-swapped and negated, and then |
| 677 | * we'll subtract the (negated, remember) high half from the low half. |
| 678 | */ |
| 679 | for (i = 0; i < NPIECE/2; i++) { |
| 680 | for (j = 0; j < NPIECE/2 - i; j++) |
| 681 | zz[i + j + NPIECE/2] -= M(b,i, d,j); |
| 682 | for (; j < NPIECE/2; j++) |
| 683 | zz[i + j - NPIECE/2] -= M(b,i, d,j); |
| 684 | } |
| 685 | for (i = 0; i < NPIECE/2; i++) |
| 686 | zz[i] -= zz[i + NPIECE/2]; |
| 687 | |
| 688 | /* Next, we add on a c. There are no surprises here. */ |
| 689 | for (i = 0; i < NPIECE/2; i++) |
| 690 | for (j = 0; j < NPIECE/2; j++) |
| 691 | zz[i + j] += M(a,i, c,j); |
| 692 | |
| 693 | /* Now, calculate a + b and c + d. */ |
| 694 | for (i = 0; i < NPIECE/2; i++) |
| 695 | { ab[i] = a[i] + b[i]; cd[i] = c[i] + d[i]; } |
| 696 | |
| 697 | /* Finally (for the multiplication) we must add on (a + b) (c + d) φ. |
| 698 | * Write (a + b) (c + d) as u φ + v; then we actually want u φ^2 + v φ = |
| 699 | * v φ + (1 + φ) u. We'll store u in a temporary place and add it on |
| 700 | * twice. |
| 701 | */ |
| 702 | for (i = 0; i < NPIECE; i++) u[i] = 0; |
| 703 | for (i = 0; i < NPIECE/2; i++) { |
| 704 | for (j = 0; j < NPIECE/2 - i; j++) |
| 705 | zz[i + j + NPIECE/2] += M(ab,i, cd,j); |
| 706 | for (; j < NPIECE/2; j++) |
| 707 | u[i + j - NPIECE/2] += M(ab,i, cd,j); |
| 708 | } |
| 709 | for (i = 0; i < NPIECE/2; i++) |
| 710 | { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } |
| 711 | |
| 712 | #undef M |
| 713 | |
| 714 | #if FGOLDI_IMPL == 28 |
| 715 | /* That wraps it up for the multiplication. Let's figure out some bounds. |
| 716 | * Fortunately, Karatsuba is a polynomial identity, so all of the pieces |
| 717 | * end up the way they'd be if we'd done the thing the easy way, which |
| 718 | * simplifies the analysis. Suppose we started with |x_i|, |y_i| <= 9/5 |
| 719 | * 2^28. The overheads in the result are given by the coefficients of |
| 720 | * |
| 721 | * ((u^16 - 1)/(u - 1))^2 mod u^16 - u^8 - 1 |
| 722 | * |
| 723 | * the greatest of which is 38. So |z_i| <= 38*81/25*2^56 < 2^63. |
| 724 | * |
| 725 | * Anyway, a round of `CARRY_REDUCE' will leave us with |z_i| < 2^27 + |
| 726 | * 2^36; and a second round will leave us with |z_i| < 2^27 + 512. |
| 727 | */ |
| 728 | for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); |
| 729 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; |
| 730 | #elif FGOLDI_IMPL == 12 |
| 731 | carry_reduce(zz); |
| 732 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; |
| 733 | #endif |
| 734 | } |
| 735 | |
| 736 | /* --- @fgoldi_sqr@ --- * |
| 737 | * |
| 738 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 739 | * @const fgoldi *x@ = an operand |
| 740 | * |
| 741 | * Returns: --- |
| 742 | * |
| 743 | * Use: Set @z@ to the square %$x^2$%. |
| 744 | */ |
| 745 | |
| 746 | void fgoldi_sqr(fgoldi *z, const fgoldi *x) |
| 747 | { |
| 748 | #if FGOLDI_IMPL == 28 |
| 749 | |
| 750 | dblpiece zz[NPIECE], u[NPIECE]; |
| 751 | piece ab[NPIECE]; |
| 752 | const piece *a = x->P + NPIECE/2, *b = x->P; |
| 753 | unsigned i, j; |
| 754 | |
| 755 | # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) |
| 756 | |
| 757 | /* The magic is basically the same as `fgoldi_mul' above. We write |
| 758 | * x = a φ + b and use Karatsuba and the special prime shape. This time, |
| 759 | * we have |
| 760 | * |
| 761 | * x^2 = ((a + b)^2 - b^2) φ + a^2 + b^2 |
| 762 | */ |
| 763 | |
| 764 | for (i = 0; i < NPIECE; i++) zz[i] = 0; |
| 765 | |
| 766 | /* Our first job will be to calculate (1 - φ) b^2, and write the result |
| 767 | * into z. Again, this interacts pleasantly with the prime shape. |
| 768 | */ |
| 769 | for (i = 0; i < NPIECE/4; i++) { |
| 770 | zz[2*i + NPIECE/2] -= M(b,i, b,i); |
| 771 | for (j = i + 1; j < NPIECE/2 - i; j++) |
| 772 | zz[i + j + NPIECE/2] -= 2*M(b,i, b,j); |
| 773 | for (; j < NPIECE/2; j++) |
| 774 | zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); |
| 775 | } |
| 776 | for (; i < NPIECE/2; i++) { |
| 777 | zz[2*i - NPIECE/2] -= M(b,i, b,i); |
| 778 | for (j = i + 1; j < NPIECE/2; j++) |
| 779 | zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); |
| 780 | } |
| 781 | for (i = 0; i < NPIECE/2; i++) |
| 782 | zz[i] -= zz[i + NPIECE/2]; |
| 783 | |
| 784 | /* Next, we add on a^2. There are no surprises here. */ |
| 785 | for (i = 0; i < NPIECE/2; i++) { |
| 786 | zz[2*i] += M(a,i, a,i); |
| 787 | for (j = i + 1; j < NPIECE/2; j++) |
| 788 | zz[i + j] += 2*M(a,i, a,j); |
| 789 | } |
| 790 | |
| 791 | /* Now, calculate a + b. */ |
| 792 | for (i = 0; i < NPIECE/2; i++) |
| 793 | ab[i] = a[i] + b[i]; |
| 794 | |
| 795 | /* Finally (for the multiplication) we must add on (a + b)^2 φ. |
| 796 | * Write (a + b)^2 as u φ + v; then we actually want (u + v) φ + u. We'll |
| 797 | * store u in a temporary place and add it on twice. |
| 798 | */ |
| 799 | for (i = 0; i < NPIECE; i++) u[i] = 0; |
| 800 | for (i = 0; i < NPIECE/4; i++) { |
| 801 | zz[2*i + NPIECE/2] += M(ab,i, ab,i); |
| 802 | for (j = i + 1; j < NPIECE/2 - i; j++) |
| 803 | zz[i + j + NPIECE/2] += 2*M(ab,i, ab,j); |
| 804 | for (; j < NPIECE/2; j++) |
| 805 | u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); |
| 806 | } |
| 807 | for (; i < NPIECE/2; i++) { |
| 808 | u[2*i - NPIECE/2] += M(ab,i, ab,i); |
| 809 | for (j = i + 1; j < NPIECE/2; j++) |
| 810 | u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); |
| 811 | } |
| 812 | for (i = 0; i < NPIECE/2; i++) |
| 813 | { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } |
| 814 | |
| 815 | #undef M |
| 816 | |
| 817 | /* Finally, carrying. */ |
| 818 | for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); |
| 819 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; |
| 820 | |
| 821 | #elif FGOLDI_IMPL == 12 |
| 822 | fgoldi_mul(z, x, x); |
| 823 | #endif |
| 824 | } |
| 825 | |
| 826 | /*----- More advanced operations ------------------------------------------*/ |
| 827 | |
| 828 | /* --- @fgoldi_inv@ --- * |
| 829 | * |
| 830 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 831 | * @const fgoldi *x@ = an operand |
| 832 | * |
| 833 | * Returns: --- |
| 834 | * |
| 835 | * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If |
| 836 | * %$x = 0$% then @z@ is set to zero. This is considered a |
| 837 | * feature. |
| 838 | */ |
| 839 | |
| 840 | void fgoldi_inv(fgoldi *z, const fgoldi *x) |
| 841 | { |
| 842 | fgoldi t, u; |
| 843 | unsigned i; |
| 844 | |
| 845 | #define SQRN(z, x, n) do { \ |
| 846 | fgoldi_sqr((z), (x)); \ |
| 847 | for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \ |
| 848 | } while (0) |
| 849 | |
| 850 | /* Calculate x^-1 = x^(p - 2) = x^(2^448 - 2^224 - 3), which also handles |
| 851 | * x = 0 as intended. The addition chain is home-made. |
| 852 | */ /* step | value */ |
| 853 | fgoldi_sqr(&u, x); /* 1 | 2 */ |
| 854 | fgoldi_mul(&t, &u, x); /* 2 | 3 */ |
| 855 | SQRN(&u, &t, 2); /* 4 | 12 */ |
| 856 | fgoldi_mul(&t, &u, &t); /* 5 | 15 */ |
| 857 | SQRN(&u, &t, 4); /* 9 | 240 */ |
| 858 | fgoldi_mul(&u, &u, &t); /* 10 | 255 = 2^8 - 1 */ |
| 859 | SQRN(&u, &u, 4); /* 14 | 2^12 - 16 */ |
| 860 | fgoldi_mul(&t, &u, &t); /* 15 | 2^12 - 1 */ |
| 861 | SQRN(&u, &t, 12); /* 27 | 2^24 - 2^12 */ |
| 862 | fgoldi_mul(&u, &u, &t); /* 28 | 2^24 - 1 */ |
| 863 | SQRN(&u, &u, 12); /* 40 | 2^36 - 2^12 */ |
| 864 | fgoldi_mul(&t, &u, &t); /* 41 | 2^36 - 1 */ |
| 865 | fgoldi_sqr(&t, &t); /* 42 | 2^37 - 2 */ |
| 866 | fgoldi_mul(&t, &t, x); /* 43 | 2^37 - 1 */ |
| 867 | SQRN(&u, &t, 37); /* 80 | 2^74 - 2^37 */ |
| 868 | fgoldi_mul(&u, &u, &t); /* 81 | 2^74 - 1 */ |
| 869 | SQRN(&u, &u, 37); /* 118 | 2^111 - 2^37 */ |
| 870 | fgoldi_mul(&t, &u, &t); /* 119 | 2^111 - 1 */ |
| 871 | SQRN(&u, &t, 111); /* 230 | 2^222 - 2^111 */ |
| 872 | fgoldi_mul(&t, &u, &t); /* 231 | 2^222 - 1 */ |
| 873 | fgoldi_sqr(&u, &t); /* 232 | 2^223 - 2 */ |
| 874 | fgoldi_mul(&u, &u, x); /* 233 | 2^223 - 1 */ |
| 875 | SQRN(&u, &u, 223); /* 456 | 2^446 - 2^223 */ |
| 876 | fgoldi_mul(&t, &u, &t); /* 457 | 2^446 - 2^222 - 1 */ |
| 877 | SQRN(&t, &t, 2); /* 459 | 2^448 - 2^224 - 4 */ |
| 878 | fgoldi_mul(z, &t, x); /* 460 | 2^448 - 2^224 - 3 */ |
| 879 | |
| 880 | #undef SQRN |
| 881 | } |
| 882 | |
| 883 | /* --- @fgoldi_quosqrt@ --- * |
| 884 | * |
| 885 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 886 | * @const fgoldi *x, *y@ = two operands |
| 887 | * |
| 888 | * Returns: Zero if successful, @-1@ if %$x/y$% is not a square. |
| 889 | * |
| 890 | * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%. |
| 891 | * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x |
| 892 | * \ne 0$% then the operation fails. If you wanted a specific |
| 893 | * square root then you'll have to pick it yourself. |
| 894 | */ |
| 895 | |
| 896 | int fgoldi_quosqrt(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 897 | { |
| 898 | fgoldi t, u, v; |
| 899 | octet xb[56], b0[56]; |
| 900 | int32 rc = -1; |
| 901 | mask32 m; |
| 902 | unsigned i; |
| 903 | |
| 904 | #define SQRN(z, x, n) do { \ |
| 905 | fgoldi_sqr((z), (x)); \ |
| 906 | for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \ |
| 907 | } while (0) |
| 908 | |
| 909 | /* This is, fortunately, significantly easier than the equivalent problem |
| 910 | * in GF(2^255 - 19), since p == 3 (mod 4). |
| 911 | * |
| 912 | * If x/y is square, then so is v = y^2 x/y = x y, and therefore u has |
| 913 | * order r = (p - 1)/2. Let w = v^{(p-3)/4}. Then w^2 = v^{(p-3)/2} = |
| 914 | * u^{r-1} = 1/v = 1/x y. Clearly, then, (x w)^2 = x^2/x y = x/y, so x w |
| 915 | * is one of the square roots we seek. |
| 916 | * |
| 917 | * The addition chain, then, is a prefix of the previous one. |
| 918 | */ |
| 919 | fgoldi_mul(&v, x, y); |
| 920 | |
| 921 | fgoldi_sqr(&u, &v); /* 1 | 2 */ |
| 922 | fgoldi_mul(&t, &u, &v); /* 2 | 3 */ |
| 923 | SQRN(&u, &t, 2); /* 4 | 12 */ |
| 924 | fgoldi_mul(&t, &u, &t); /* 5 | 15 */ |
| 925 | SQRN(&u, &t, 4); /* 9 | 240 */ |
| 926 | fgoldi_mul(&u, &u, &t); /* 10 | 255 = 2^8 - 1 */ |
| 927 | SQRN(&u, &u, 4); /* 14 | 2^12 - 16 */ |
| 928 | fgoldi_mul(&t, &u, &t); /* 15 | 2^12 - 1 */ |
| 929 | SQRN(&u, &t, 12); /* 27 | 2^24 - 2^12 */ |
| 930 | fgoldi_mul(&u, &u, &t); /* 28 | 2^24 - 1 */ |
| 931 | SQRN(&u, &u, 12); /* 40 | 2^36 - 2^12 */ |
| 932 | fgoldi_mul(&t, &u, &t); /* 41 | 2^36 - 1 */ |
| 933 | fgoldi_sqr(&t, &t); /* 42 | 2^37 - 2 */ |
| 934 | fgoldi_mul(&t, &t, &v); /* 43 | 2^37 - 1 */ |
| 935 | SQRN(&u, &t, 37); /* 80 | 2^74 - 2^37 */ |
| 936 | fgoldi_mul(&u, &u, &t); /* 81 | 2^74 - 1 */ |
| 937 | SQRN(&u, &u, 37); /* 118 | 2^111 - 2^37 */ |
| 938 | fgoldi_mul(&t, &u, &t); /* 119 | 2^111 - 1 */ |
| 939 | SQRN(&u, &t, 111); /* 230 | 2^222 - 2^111 */ |
| 940 | fgoldi_mul(&t, &u, &t); /* 231 | 2^222 - 1 */ |
| 941 | fgoldi_sqr(&u, &t); /* 232 | 2^223 - 2 */ |
| 942 | fgoldi_mul(&u, &u, &v); /* 233 | 2^223 - 1 */ |
| 943 | SQRN(&u, &u, 223); /* 456 | 2^446 - 2^223 */ |
| 944 | fgoldi_mul(&t, &u, &t); /* 457 | 2^446 - 2^222 - 1 */ |
| 945 | |
| 946 | #undef SQRN |
| 947 | |
| 948 | /* Now we must decide whether the answer was right. We should have z^2 = |
| 949 | * x/y, so y z^2 = x. |
| 950 | * |
| 951 | * The easiest way to compare is to encode. This isn't as wasteful as it |
| 952 | * sounds: the hard part is normalizing the representations, which we have |
| 953 | * to do anyway. |
| 954 | */ |
| 955 | fgoldi_mul(z, x, &t); |
| 956 | fgoldi_sqr(&t, z); |
| 957 | fgoldi_mul(&t, &t, y); |
| 958 | fgoldi_store(xb, x); |
| 959 | fgoldi_store(b0, &t); |
| 960 | m = -ct_memeq(xb, b0, 56); |
| 961 | rc = PICK2(0, rc, m); |
| 962 | return (rc); |
| 963 | } |
| 964 | |
| 965 | /*----- Test rig ----------------------------------------------------------*/ |
| 966 | |
| 967 | #ifdef TEST_RIG |
| 968 | |
| 969 | #include <mLib/report.h> |
| 970 | #include <mLib/str.h> |
| 971 | #include <mLib/testrig.h> |
| 972 | |
| 973 | static void fixdstr(dstr *d) |
| 974 | { |
| 975 | if (d->len > 56) |
| 976 | die(1, "invalid length for fgoldi"); |
| 977 | else if (d->len < 56) { |
| 978 | dstr_ensure(d, 56); |
| 979 | memset(d->buf + d->len, 0, 56 - d->len); |
| 980 | d->len = 56; |
| 981 | } |
| 982 | } |
| 983 | |
| 984 | static void cvt_fgoldi(const char *buf, dstr *d) |
| 985 | { |
| 986 | dstr dd = DSTR_INIT; |
| 987 | |
| 988 | type_hex.cvt(buf, &dd); fixdstr(&dd); |
| 989 | dstr_ensure(d, sizeof(fgoldi)); d->len = sizeof(fgoldi); |
| 990 | fgoldi_load((fgoldi *)d->buf, (const octet *)dd.buf); |
| 991 | dstr_destroy(&dd); |
| 992 | } |
| 993 | |
| 994 | static void dump_fgoldi(dstr *d, FILE *fp) |
| 995 | { fdump(stderr, "???", (const piece *)d->buf); } |
| 996 | |
| 997 | static void cvt_fgoldi_ref(const char *buf, dstr *d) |
| 998 | { type_hex.cvt(buf, d); fixdstr(d); } |
| 999 | |
| 1000 | static void dump_fgoldi_ref(dstr *d, FILE *fp) |
| 1001 | { |
| 1002 | fgoldi x; |
| 1003 | |
| 1004 | fgoldi_load(&x, (const octet *)d->buf); |
| 1005 | fdump(stderr, "???", x.P); |
| 1006 | } |
| 1007 | |
| 1008 | static int eq(const fgoldi *x, dstr *d) |
| 1009 | { octet b[56]; fgoldi_store(b, x); return (memcmp(b, d->buf, 56) == 0); } |
| 1010 | |
| 1011 | static const test_type |
| 1012 | type_fgoldi = { cvt_fgoldi, dump_fgoldi }, |
| 1013 | type_fgoldi_ref = { cvt_fgoldi_ref, dump_fgoldi_ref }; |
| 1014 | |
| 1015 | #define TEST_UNOP(op) \ |
| 1016 | static int vrf_##op(dstr dv[]) \ |
| 1017 | { \ |
| 1018 | fgoldi *x = (fgoldi *)dv[0].buf; \ |
| 1019 | fgoldi z, zz; \ |
| 1020 | int ok = 1; \ |
| 1021 | \ |
| 1022 | fgoldi_##op(&z, x); \ |
| 1023 | if (!eq(&z, &dv[1])) { \ |
| 1024 | ok = 0; \ |
| 1025 | fprintf(stderr, "failed!\n"); \ |
| 1026 | fdump(stderr, "x", x->P); \ |
| 1027 | fdump(stderr, "calc", z.P); \ |
| 1028 | fgoldi_load(&zz, (const octet *)dv[1].buf); \ |
| 1029 | fdump(stderr, "z", zz.P); \ |
| 1030 | } \ |
| 1031 | \ |
| 1032 | return (ok); \ |
| 1033 | } |
| 1034 | |
| 1035 | TEST_UNOP(sqr) |
| 1036 | TEST_UNOP(inv) |
| 1037 | TEST_UNOP(neg) |
| 1038 | |
| 1039 | #define TEST_BINOP(op) \ |
| 1040 | static int vrf_##op(dstr dv[]) \ |
| 1041 | { \ |
| 1042 | fgoldi *x = (fgoldi *)dv[0].buf, *y = (fgoldi *)dv[1].buf; \ |
| 1043 | fgoldi z, zz; \ |
| 1044 | int ok = 1; \ |
| 1045 | \ |
| 1046 | fgoldi_##op(&z, x, y); \ |
| 1047 | if (!eq(&z, &dv[2])) { \ |
| 1048 | ok = 0; \ |
| 1049 | fprintf(stderr, "failed!\n"); \ |
| 1050 | fdump(stderr, "x", x->P); \ |
| 1051 | fdump(stderr, "y", y->P); \ |
| 1052 | fdump(stderr, "calc", z.P); \ |
| 1053 | fgoldi_load(&zz, (const octet *)dv[2].buf); \ |
| 1054 | fdump(stderr, "z", zz.P); \ |
| 1055 | } \ |
| 1056 | \ |
| 1057 | return (ok); \ |
| 1058 | } |
| 1059 | |
| 1060 | TEST_BINOP(add) |
| 1061 | TEST_BINOP(sub) |
| 1062 | TEST_BINOP(mul) |
| 1063 | |
| 1064 | static int vrf_mulc(dstr dv[]) |
| 1065 | { |
| 1066 | fgoldi *x = (fgoldi *)dv[0].buf; |
| 1067 | long a = *(const long *)dv[1].buf; |
| 1068 | fgoldi z, zz; |
| 1069 | int ok = 1; |
| 1070 | |
| 1071 | fgoldi_mulconst(&z, x, a); |
| 1072 | if (!eq(&z, &dv[2])) { |
| 1073 | ok = 0; |
| 1074 | fprintf(stderr, "failed!\n"); |
| 1075 | fdump(stderr, "x", x->P); |
| 1076 | fprintf(stderr, "a = %ld\n", a); |
| 1077 | fdump(stderr, "calc", z.P); |
| 1078 | fgoldi_load(&zz, (const octet *)dv[2].buf); |
| 1079 | fdump(stderr, "z", zz.P); |
| 1080 | } |
| 1081 | |
| 1082 | return (ok); |
| 1083 | } |
| 1084 | |
| 1085 | static int vrf_condneg(dstr dv[]) |
| 1086 | { |
| 1087 | fgoldi *x = (fgoldi *)dv[0].buf; |
| 1088 | uint32 m = *(uint32 *)dv[1].buf; |
| 1089 | fgoldi z; |
| 1090 | int ok = 1; |
| 1091 | |
| 1092 | fgoldi_condneg(&z, x, m); |
| 1093 | if (!eq(&z, &dv[2])) { |
| 1094 | ok = 0; |
| 1095 | fprintf(stderr, "failed!\n"); |
| 1096 | fdump(stderr, "x", x->P); |
| 1097 | fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m); |
| 1098 | fdump(stderr, "calc z", z.P); |
| 1099 | fgoldi_load(&z, (const octet *)dv[1].buf); |
| 1100 | fdump(stderr, "want z", z.P); |
| 1101 | } |
| 1102 | |
| 1103 | return (ok); |
| 1104 | } |
| 1105 | |
| 1106 | static int vrf_pick2(dstr dv[]) |
| 1107 | { |
| 1108 | fgoldi *x = (fgoldi *)dv[0].buf, *y = (fgoldi *)dv[1].buf; |
| 1109 | uint32 m = *(uint32 *)dv[2].buf; |
| 1110 | fgoldi z; |
| 1111 | int ok = 1; |
| 1112 | |
| 1113 | fgoldi_pick2(&z, x, y, m); |
| 1114 | if (!eq(&z, &dv[3])) { |
| 1115 | ok = 0; |
| 1116 | fprintf(stderr, "failed!\n"); |
| 1117 | fdump(stderr, "x", x->P); |
| 1118 | fdump(stderr, "y", y->P); |
| 1119 | fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m); |
| 1120 | fdump(stderr, "calc z", z.P); |
| 1121 | fgoldi_load(&z, (const octet *)dv[3].buf); |
| 1122 | fdump(stderr, "want z", z.P); |
| 1123 | } |
| 1124 | |
| 1125 | return (ok); |
| 1126 | } |
| 1127 | |
| 1128 | static int vrf_pickn(dstr dv[]) |
| 1129 | { |
| 1130 | dstr d = DSTR_INIT; |
| 1131 | fgoldi v[32], z; |
| 1132 | size_t i = *(uint32 *)dv[1].buf, j, n; |
| 1133 | const char *p; |
| 1134 | char *q; |
| 1135 | int ok = 1; |
| 1136 | |
| 1137 | for (q = dv[0].buf, n = 0; (p = str_qword(&q, 0)) != 0; n++) |
| 1138 | { cvt_fgoldi(p, &d); v[n] = *(fgoldi *)d.buf; } |
| 1139 | |
| 1140 | fgoldi_pickn(&z, v, n, i); |
| 1141 | if (!eq(&z, &dv[2])) { |
| 1142 | ok = 0; |
| 1143 | fprintf(stderr, "failed!\n"); |
| 1144 | for (j = 0; j < n; j++) { |
| 1145 | fprintf(stderr, "v[%2u]", (unsigned)j); |
| 1146 | fdump(stderr, "", v[j].P); |
| 1147 | } |
| 1148 | fprintf(stderr, "i = %u\n", (unsigned)i); |
| 1149 | fdump(stderr, "calc z", z.P); |
| 1150 | fgoldi_load(&z, (const octet *)dv[2].buf); |
| 1151 | fdump(stderr, "want z", z.P); |
| 1152 | } |
| 1153 | |
| 1154 | dstr_destroy(&d); |
| 1155 | return (ok); |
| 1156 | } |
| 1157 | |
| 1158 | static int vrf_condswap(dstr dv[]) |
| 1159 | { |
| 1160 | fgoldi *x = (fgoldi *)dv[0].buf, *y = (fgoldi *)dv[1].buf; |
| 1161 | fgoldi xx = *x, yy = *y; |
| 1162 | uint32 m = *(uint32 *)dv[2].buf; |
| 1163 | int ok = 1; |
| 1164 | |
| 1165 | fgoldi_condswap(&xx, &yy, m); |
| 1166 | if (!eq(&xx, &dv[3]) || !eq(&yy, &dv[4])) { |
| 1167 | ok = 0; |
| 1168 | fprintf(stderr, "failed!\n"); |
| 1169 | fdump(stderr, "x", x->P); |
| 1170 | fdump(stderr, "y", y->P); |
| 1171 | fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m); |
| 1172 | fdump(stderr, "calc xx", xx.P); |
| 1173 | fdump(stderr, "calc yy", yy.P); |
| 1174 | fgoldi_load(&xx, (const octet *)dv[3].buf); |
| 1175 | fgoldi_load(&yy, (const octet *)dv[4].buf); |
| 1176 | fdump(stderr, "want xx", xx.P); |
| 1177 | fdump(stderr, "want yy", yy.P); |
| 1178 | } |
| 1179 | |
| 1180 | return (ok); |
| 1181 | } |
| 1182 | |
| 1183 | static int vrf_quosqrt(dstr dv[]) |
| 1184 | { |
| 1185 | fgoldi *x = (fgoldi *)dv[0].buf, *y = (fgoldi *)dv[1].buf; |
| 1186 | fgoldi z, zz; |
| 1187 | int rc; |
| 1188 | int ok = 1; |
| 1189 | |
| 1190 | if (dv[2].len) { fixdstr(&dv[2]); fixdstr(&dv[3]); } |
| 1191 | rc = fgoldi_quosqrt(&z, x, y); |
| 1192 | if (!dv[2].len ? !rc : (rc || (!eq(&z, &dv[2]) && !eq(&z, &dv[3])))) { |
| 1193 | ok = 0; |
| 1194 | fprintf(stderr, "failed!\n"); |
| 1195 | fdump(stderr, "x", x->P); |
| 1196 | fdump(stderr, "y", y->P); |
| 1197 | if (rc) fprintf(stderr, "calc: FAIL\n"); |
| 1198 | else fdump(stderr, "calc", z.P); |
| 1199 | if (!dv[2].len) |
| 1200 | fprintf(stderr, "exp: FAIL\n"); |
| 1201 | else { |
| 1202 | fgoldi_load(&zz, (const octet *)dv[2].buf); |
| 1203 | fdump(stderr, "z", zz.P); |
| 1204 | fgoldi_load(&zz, (const octet *)dv[3].buf); |
| 1205 | fdump(stderr, "z'", zz.P); |
| 1206 | } |
| 1207 | } |
| 1208 | |
| 1209 | return (ok); |
| 1210 | } |
| 1211 | |
| 1212 | static int vrf_sub_mulc_add_sub_mul(dstr dv[]) |
| 1213 | { |
| 1214 | fgoldi *u = (fgoldi *)dv[0].buf, *v = (fgoldi *)dv[1].buf, |
| 1215 | *w = (fgoldi *)dv[3].buf, *x = (fgoldi *)dv[4].buf, |
| 1216 | *y = (fgoldi *)dv[5].buf; |
| 1217 | long a = *(const long *)dv[2].buf; |
| 1218 | fgoldi umv, aumv, wpaumv, xmy, z, zz; |
| 1219 | int ok = 1; |
| 1220 | |
| 1221 | fgoldi_sub(&umv, u, v); |
| 1222 | fgoldi_mulconst(&aumv, &umv, a); |
| 1223 | fgoldi_add(&wpaumv, w, &aumv); |
| 1224 | fgoldi_sub(&xmy, x, y); |
| 1225 | fgoldi_mul(&z, &wpaumv, &xmy); |
| 1226 | |
| 1227 | if (!eq(&z, &dv[6])) { |
| 1228 | ok = 0; |
| 1229 | fprintf(stderr, "failed!\n"); |
| 1230 | fdump(stderr, "u", u->P); |
| 1231 | fdump(stderr, "v", v->P); |
| 1232 | fdump(stderr, "u - v", umv.P); |
| 1233 | fprintf(stderr, "a = %ld\n", a); |
| 1234 | fdump(stderr, "a (u - v)", aumv.P); |
| 1235 | fdump(stderr, "w + a (u - v)", wpaumv.P); |
| 1236 | fdump(stderr, "x", x->P); |
| 1237 | fdump(stderr, "y", y->P); |
| 1238 | fdump(stderr, "x - y", xmy.P); |
| 1239 | fdump(stderr, "(x - y) (w + a (u - v))", z.P); |
| 1240 | fgoldi_load(&zz, (const octet *)dv[6].buf); fdump(stderr, "z", zz.P); |
| 1241 | } |
| 1242 | |
| 1243 | return (ok); |
| 1244 | } |
| 1245 | |
| 1246 | static test_chunk tests[] = { |
| 1247 | { "add", vrf_add, { &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } }, |
| 1248 | { "sub", vrf_sub, { &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } }, |
| 1249 | { "neg", vrf_neg, { &type_fgoldi, &type_fgoldi_ref } }, |
| 1250 | { "condneg", vrf_condneg, |
| 1251 | { &type_fgoldi, &type_uint32, &type_fgoldi_ref } }, |
| 1252 | { "mul", vrf_mul, { &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } }, |
| 1253 | { "mulconst", vrf_mulc, { &type_fgoldi, &type_long, &type_fgoldi_ref } }, |
| 1254 | { "pick2", vrf_pick2, |
| 1255 | { &type_fgoldi, &type_fgoldi, &type_uint32, &type_fgoldi_ref } }, |
| 1256 | { "pickn", vrf_pickn, |
| 1257 | { &type_string, &type_uint32, &type_fgoldi_ref } }, |
| 1258 | { "condswap", vrf_condswap, |
| 1259 | { &type_fgoldi, &type_fgoldi, &type_uint32, |
| 1260 | &type_fgoldi_ref, &type_fgoldi_ref } }, |
| 1261 | { "sqr", vrf_sqr, { &type_fgoldi, &type_fgoldi_ref } }, |
| 1262 | { "inv", vrf_inv, { &type_fgoldi, &type_fgoldi_ref } }, |
| 1263 | { "quosqrt", vrf_quosqrt, |
| 1264 | { &type_fgoldi, &type_fgoldi, &type_hex, &type_hex } }, |
| 1265 | { "sub-mulc-add-sub-mul", vrf_sub_mulc_add_sub_mul, |
| 1266 | { &type_fgoldi, &type_fgoldi, &type_long, &type_fgoldi, |
| 1267 | &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } }, |
| 1268 | { 0, 0, { 0 } } |
| 1269 | }; |
| 1270 | |
| 1271 | int main(int argc, char *argv[]) |
| 1272 | { |
| 1273 | test_run(argc, argv, tests, SRCDIR "/t/fgoldi"); |
| 1274 | return (0); |
| 1275 | } |
| 1276 | |
| 1277 | #endif |
| 1278 | |
| 1279 | /*----- That's all, folks -------------------------------------------------*/ |