| 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 secnet. |
| 11 | * See README for full list of copyright holders. |
| 12 | * |
| 13 | * secnet is free software; you can redistribute it and/or modify it |
| 14 | * under the terms of the GNU General Public License as published by |
| 15 | * the Free Software Foundation; either version d of the License, or |
| 16 | * (at your option) any later version. |
| 17 | * |
| 18 | * secnet is distributed in the hope that it will be useful, but |
| 19 | * WITHOUT ANY WARRANTY; without even the implied warranty of |
| 20 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| 21 | * General Public License for more details. |
| 22 | * |
| 23 | * You should have received a copy of the GNU General Public License |
| 24 | * version 3 along with secnet; if not, see |
| 25 | * https://www.gnu.org/licenses/gpl.html. |
| 26 | * |
| 27 | * This file was originally part of Catacomb, but has been automatically |
| 28 | * modified for incorporation into secnet: see `import-catacomb-crypto' |
| 29 | * for details. |
| 30 | * |
| 31 | * Catacomb is free software; you can redistribute it and/or modify |
| 32 | * it under the terms of the GNU Library General Public License as |
| 33 | * published by the Free Software Foundation; either version 2 of the |
| 34 | * License, or (at your option) any later version. |
| 35 | * |
| 36 | * Catacomb is distributed in the hope that it will be useful, |
| 37 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 38 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| 39 | * GNU Library General Public License for more details. |
| 40 | * |
| 41 | * You should have received a copy of the GNU Library General Public |
| 42 | * License along with Catacomb; if not, write to the Free |
| 43 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
| 44 | * MA 02111-1307, USA. |
| 45 | */ |
| 46 | |
| 47 | /*----- Header files ------------------------------------------------------*/ |
| 48 | |
| 49 | #include "fgoldi.h" |
| 50 | |
| 51 | /*----- Basic setup -------------------------------------------------------* |
| 52 | * |
| 53 | * Let φ = 2^224; then p = φ^2 - φ - 1, and, in GF(p), we have φ^2 = φ + 1 |
| 54 | * (hence the name). |
| 55 | */ |
| 56 | |
| 57 | typedef fgoldi_piece piece; |
| 58 | |
| 59 | /* We represent an element of GF(p) as 16 28-bit signed integer pieces x_i: |
| 60 | * x = SUM_{0<=i<16} x_i 2^(28i). |
| 61 | */ |
| 62 | |
| 63 | typedef int64 dblpiece; |
| 64 | typedef uint32 upiece; typedef uint64 udblpiece; |
| 65 | #define PIECEWD(i) 28 |
| 66 | #define NPIECE 16 |
| 67 | #define P p28 |
| 68 | |
| 69 | #define B27 0x08000000u |
| 70 | #define M28 0x0fffffffu |
| 71 | #define M32 0xffffffffu |
| 72 | |
| 73 | /*----- Debugging machinery -----------------------------------------------*/ |
| 74 | |
| 75 | #if defined(FGOLDI_DEBUG) |
| 76 | |
| 77 | #include <stdio.h> |
| 78 | |
| 79 | #include "mp.h" |
| 80 | #include "mptext.h" |
| 81 | |
| 82 | static mp *get_pgoldi(void) |
| 83 | { |
| 84 | mp *p = MP_NEW, *t = MP_NEW; |
| 85 | |
| 86 | p = mp_setbit(p, MP_ZERO, 448); |
| 87 | t = mp_setbit(t, MP_ZERO, 224); |
| 88 | p = mp_sub(p, p, t); |
| 89 | p = mp_sub(p, p, MP_ONE); |
| 90 | mp_drop(t); |
| 91 | return (p); |
| 92 | } |
| 93 | |
| 94 | DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 56, get_pgoldi()) |
| 95 | |
| 96 | #endif |
| 97 | |
| 98 | /*----- Loading and storing -----------------------------------------------*/ |
| 99 | |
| 100 | /* --- @fgoldi_load@ --- * |
| 101 | * |
| 102 | * Arguments: @fgoldi *z@ = where to store the result |
| 103 | * @const octet xv[56]@ = source to read |
| 104 | * |
| 105 | * Returns: --- |
| 106 | * |
| 107 | * Use: Reads an element of %$\gf{2^{448} - 2^{224} - 19}$% in |
| 108 | * external representation from @xv@ and stores it in @z@. |
| 109 | * |
| 110 | * External representation is little-endian base-256. Some |
| 111 | * elements have multiple encodings, which are not produced by |
| 112 | * correct software; use of noncanonical encodings is not an |
| 113 | * error, and toleration of them is considered a performance |
| 114 | * feature. |
| 115 | */ |
| 116 | |
| 117 | void fgoldi_load(fgoldi *z, const octet xv[56]) |
| 118 | { |
| 119 | |
| 120 | unsigned i; |
| 121 | uint32 xw[14]; |
| 122 | piece b, c; |
| 123 | |
| 124 | /* First, read the input value as words. */ |
| 125 | for (i = 0; i < 14; i++) xw[i] = LOAD32_L(xv + 4*i); |
| 126 | |
| 127 | /* Extract unsigned 28-bit pieces from the words. */ |
| 128 | z->P[ 0] = (xw[ 0] >> 0)&M28; |
| 129 | z->P[ 7] = (xw[ 6] >> 4)&M28; |
| 130 | z->P[ 8] = (xw[ 7] >> 0)&M28; |
| 131 | z->P[15] = (xw[13] >> 4)&M28; |
| 132 | for (i = 1; i < 7; i++) { |
| 133 | z->P[i + 0] = ((xw[i + 0] << (4*i)) | (xw[i - 1] >> (32 - 4*i)))&M28; |
| 134 | z->P[i + 8] = ((xw[i + 7] << (4*i)) | (xw[i + 6] >> (32 - 4*i)))&M28; |
| 135 | } |
| 136 | |
| 137 | /* Convert the nonnegative pieces into a balanced signed representation, so |
| 138 | * each piece ends up in the interval |z_i| <= 2^27. For each piece, if |
| 139 | * its top bit is set, lend a bit leftwards; in the case of z_15, reduce |
| 140 | * this bit by adding it onto z_0 and z_8, since this is the φ^2 bit, and |
| 141 | * φ^2 = φ + 1. We delay this carry until after all of the pieces have |
| 142 | * been balanced. If we don't do this, then we have to do a more expensive |
| 143 | * test for nonzeroness to decide whether to lend a bit leftwards rather |
| 144 | * than just testing a single bit. |
| 145 | * |
| 146 | * Note that we don't try for a canonical representation here: both upper |
| 147 | * and lower bounds are achievable. |
| 148 | */ |
| 149 | b = z->P[15]&B27; z->P[15] -= b << 1; c = b >> 27; |
| 150 | for (i = NPIECE - 1; i--; ) |
| 151 | { b = z->P[i]&B27; z->P[i] -= b << 1; z->P[i + 1] += b >> 27; } |
| 152 | z->P[0] += c; z->P[8] += c; |
| 153 | } |
| 154 | |
| 155 | /* --- @fgoldi_store@ --- * |
| 156 | * |
| 157 | * Arguments: @octet zv[56]@ = where to write the result |
| 158 | * @const fgoldi *x@ = the field element to write |
| 159 | * |
| 160 | * Returns: --- |
| 161 | * |
| 162 | * Use: Stores a field element in the given octet vector in external |
| 163 | * representation. A canonical encoding is always stored. |
| 164 | */ |
| 165 | |
| 166 | void fgoldi_store(octet zv[56], const fgoldi *x) |
| 167 | { |
| 168 | |
| 169 | piece y[NPIECE], yy[NPIECE], c, d; |
| 170 | uint32 u, v; |
| 171 | mask32 m; |
| 172 | unsigned i; |
| 173 | |
| 174 | for (i = 0; i < NPIECE; i++) y[i] = x->P[i]; |
| 175 | |
| 176 | /* First, propagate the carries. By the end of this, we'll have all of the |
| 177 | * the pieces canonically sized and positive, and maybe there'll be |
| 178 | * (signed) carry out. The carry c is in { -1, 0, +1 }, and the remaining |
| 179 | * value will be in the half-open interval [0, φ^2). The whole represented |
| 180 | * value is then y + φ^2 c. |
| 181 | * |
| 182 | * Assume that we start out with |y_i| <= 2^30. We start off by cutting |
| 183 | * off and reducing the carry c_15 from the topmost piece, y_15. This |
| 184 | * leaves 0 <= y_15 < 2^28; and we'll have |c_15| <= 4. We'll add this |
| 185 | * onto y_0 and y_8, and propagate the carries. It's very clear that we'll |
| 186 | * end up with |y + (φ + 1) c_15 - φ^2/2| << φ^2. |
| 187 | * |
| 188 | * Here, the y_i are signed, so we must be cautious about bithacking them. |
| 189 | */ |
| 190 | c = ASR(piece, y[15], 28); y[15] = (upiece)y[15]&M28; y[8] += c; |
| 191 | for (i = 0; i < NPIECE; i++) |
| 192 | { y[i] += c; c = ASR(piece, y[i], 28); y[i] = (upiece)y[i]&M28; } |
| 193 | |
| 194 | /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and |
| 195 | * y >= p, then we should subtract p from the whole value; if c = -1 then |
| 196 | * we should add p; and otherwise we should do nothing. |
| 197 | * |
| 198 | * But conditional behaviour is bad, m'kay. So here's what we do instead. |
| 199 | * |
| 200 | * The first job is to sort out what we wanted to do. If c = -1 then we |
| 201 | * want to (a) invert the constant addend and (b) feed in a carry-in; |
| 202 | * otherwise, we don't. |
| 203 | */ |
| 204 | m = SIGN(c)&M28; |
| 205 | d = m&1; |
| 206 | |
| 207 | /* Now do the addition/subtraction. Remember that all of the y_i are |
| 208 | * nonnegative, so shifting and masking are safe and easy. |
| 209 | */ |
| 210 | d += y[0] + (1 ^ m); yy[0] = d&M28; d >>= 28; |
| 211 | for (i = 1; i < 8; i++) |
| 212 | { d += y[i] + m; yy[i] = d&M28; d >>= 28; } |
| 213 | d += y[8] + (1 ^ m); yy[8] = d&M28; d >>= 28; |
| 214 | for (i = 9; i < 16; i++) |
| 215 | { d += y[i] + m; yy[i] = d&M28; d >>= 28; } |
| 216 | |
| 217 | /* The final carry-out is in d; since we only did addition, and the y_i are |
| 218 | * nonnegative, then d is in { 0, 1 }. We want to keep y', rather than y, |
| 219 | * if (a) c /= 0 (in which case we know that the old value was |
| 220 | * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that |
| 221 | * the subtraction didn't cause a borrow, so we must be in the case where |
| 222 | * p <= y < φ^2. |
| 223 | */ |
| 224 | m = NONZEROP(c) | ~NONZEROP(d - 1); |
| 225 | for (i = 0; i < NPIECE; i++) y[i] = (yy[i]&m) | (y[i]&~m); |
| 226 | |
| 227 | /* Extract 32-bit words from the value. */ |
| 228 | for (i = 0; i < 7; i++) { |
| 229 | u = ((y[i + 0] >> (4*i)) | ((uint32)y[i + 1] << (28 - 4*i)))&M32; |
| 230 | v = ((y[i + 8] >> (4*i)) | ((uint32)y[i + 9] << (28 - 4*i)))&M32; |
| 231 | STORE32_L(zv + 4*i, u); |
| 232 | STORE32_L(zv + 4*i + 28, v); |
| 233 | } |
| 234 | } |
| 235 | |
| 236 | /* --- @fgoldi_set@ --- * |
| 237 | * |
| 238 | * Arguments: @fgoldi *z@ = where to write the result |
| 239 | * @int a@ = a small-ish constant |
| 240 | * |
| 241 | * Returns: --- |
| 242 | * |
| 243 | * Use: Sets @z@ to equal @a@. |
| 244 | */ |
| 245 | |
| 246 | void fgoldi_set(fgoldi *x, int a) |
| 247 | { |
| 248 | unsigned i; |
| 249 | |
| 250 | x->P[0] = a; |
| 251 | for (i = 1; i < NPIECE; i++) x->P[i] = 0; |
| 252 | } |
| 253 | |
| 254 | /*----- Basic arithmetic --------------------------------------------------*/ |
| 255 | |
| 256 | /* --- @fgoldi_add@ --- * |
| 257 | * |
| 258 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 259 | * @const fgoldi *x, *y@ = two operands |
| 260 | * |
| 261 | * Returns: --- |
| 262 | * |
| 263 | * Use: Set @z@ to the sum %$x + y$%. |
| 264 | */ |
| 265 | |
| 266 | void fgoldi_add(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 267 | { |
| 268 | unsigned i; |
| 269 | for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] + y->P[i]; |
| 270 | } |
| 271 | |
| 272 | /* --- @fgoldi_sub@ --- * |
| 273 | * |
| 274 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 275 | * @const fgoldi *x, *y@ = two operands |
| 276 | * |
| 277 | * Returns: --- |
| 278 | * |
| 279 | * Use: Set @z@ to the difference %$x - y$%. |
| 280 | */ |
| 281 | |
| 282 | void fgoldi_sub(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 283 | { |
| 284 | unsigned i; |
| 285 | for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] - y->P[i]; |
| 286 | } |
| 287 | |
| 288 | /* --- @fgoldi_neg@ --- * |
| 289 | * |
| 290 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 291 | * @const fgoldi *x@ = an operand |
| 292 | * |
| 293 | * Returns: --- |
| 294 | * |
| 295 | * Use: Set @z = -x@. |
| 296 | */ |
| 297 | |
| 298 | void fgoldi_neg(fgoldi *z, const fgoldi *x) |
| 299 | { |
| 300 | unsigned i; |
| 301 | for (i = 0; i < NPIECE; i++) z->P[i] = -x->P[i]; |
| 302 | } |
| 303 | |
| 304 | /*----- Constant-time utilities -------------------------------------------*/ |
| 305 | |
| 306 | /* --- @fgoldi_pick2@ --- * |
| 307 | * |
| 308 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 309 | * @const fgoldi *x, *y@ = two operands |
| 310 | * @uint32 m@ = a mask |
| 311 | * |
| 312 | * Returns: --- |
| 313 | * |
| 314 | * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set |
| 315 | * @z = x@. If @m@ has some other value, then scramble @z@ in |
| 316 | * an unhelpful way. |
| 317 | */ |
| 318 | |
| 319 | void fgoldi_pick2(fgoldi *z, const fgoldi *x, const fgoldi *y, uint32 m) |
| 320 | { |
| 321 | mask32 mm = FIX_MASK32(m); |
| 322 | unsigned i; |
| 323 | for (i = 0; i < NPIECE; i++) z->P[i] = PICK2(x->P[i], y->P[i], mm); |
| 324 | } |
| 325 | |
| 326 | /* --- @fgoldi_pickn@ --- * |
| 327 | * |
| 328 | * Arguments: @fgoldi *z@ = where to put the result |
| 329 | * @const fgoldi *v@ = a table of entries |
| 330 | * @size_t n@ = the number of entries in @v@ |
| 331 | * @size_t i@ = an index |
| 332 | * |
| 333 | * Returns: --- |
| 334 | * |
| 335 | * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then |
| 336 | * do something unhelpful; otherwise, if @i >= n@ then set @z@ |
| 337 | * to zero. |
| 338 | */ |
| 339 | |
| 340 | void fgoldi_pickn(fgoldi *z, const fgoldi *v, size_t n, size_t i) |
| 341 | { |
| 342 | uint32 b = (uint32)1 << (31 - i); |
| 343 | mask32 m; |
| 344 | unsigned j; |
| 345 | |
| 346 | for (j = 0; j < NPIECE; j++) z->P[j] = 0; |
| 347 | while (n--) { |
| 348 | m = SIGN(b); |
| 349 | for (j = 0; j < NPIECE; j++) CONDPICK(z->P[j], v->P[j], m); |
| 350 | v++; b <<= 1; |
| 351 | } |
| 352 | } |
| 353 | |
| 354 | /* --- @fgoldi_condswap@ --- * |
| 355 | * |
| 356 | * Arguments: @fgoldi *x, *y@ = two operands |
| 357 | * @uint32 m@ = a mask |
| 358 | * |
| 359 | * Returns: --- |
| 360 | * |
| 361 | * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then |
| 362 | * exchange @x@ and @y@. If @m@ has some other value, then |
| 363 | * scramble @x@ and @y@ in an unhelpful way. |
| 364 | */ |
| 365 | |
| 366 | void fgoldi_condswap(fgoldi *x, fgoldi *y, uint32 m) |
| 367 | { |
| 368 | unsigned i; |
| 369 | mask32 mm = FIX_MASK32(m); |
| 370 | |
| 371 | for (i = 0; i < NPIECE; i++) CONDSWAP(x->P[i], y->P[i], mm); |
| 372 | } |
| 373 | |
| 374 | /* --- @fgoldi_condneg@ --- * |
| 375 | * |
| 376 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 377 | * @const fgoldi *x@ = an operand |
| 378 | * @uint32 m@ = a mask |
| 379 | * |
| 380 | * Returns: --- |
| 381 | * |
| 382 | * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set |
| 383 | * @z = -x@. If @m@ has some other value then scramble @z@ in |
| 384 | * an unhelpful way. |
| 385 | */ |
| 386 | |
| 387 | void fgoldi_condneg(fgoldi *z, const fgoldi *x, uint32 m) |
| 388 | { |
| 389 | mask32 m_xor = FIX_MASK32(m); |
| 390 | piece m_add = m&1; |
| 391 | # define CONDNEG(x) (((x) ^ m_xor) + m_add) |
| 392 | |
| 393 | unsigned i; |
| 394 | for (i = 0; i < NPIECE; i++) z->P[i] = CONDNEG(x->P[i]); |
| 395 | |
| 396 | #undef CONDNEG |
| 397 | } |
| 398 | |
| 399 | /*----- Multiplication ----------------------------------------------------*/ |
| 400 | |
| 401 | /* Let B = 2^63 - 1 be the largest value such that +B and -B can be |
| 402 | * represented in a double-precision piece. On entry, it must be the case |
| 403 | * that |X_i| <= M <= B - 2^27 for some M. If this is the case, then, on |
| 404 | * exit, we will have |Z_i| <= 2^27 + M/2^27. |
| 405 | */ |
| 406 | #define CARRY_REDUCE(z, x) do { \ |
| 407 | dblpiece _t[NPIECE], _c; \ |
| 408 | unsigned _i; \ |
| 409 | \ |
| 410 | /* Bias the input pieces. This keeps the carries and so on centred \ |
| 411 | * around zero rather than biased positive. \ |
| 412 | */ \ |
| 413 | for (_i = 0; _i < NPIECE; _i++) _t[_i] = (x)[_i] + B27; \ |
| 414 | \ |
| 415 | /* Calculate the reduced pieces. Careful with the bithacking. */ \ |
| 416 | _c = ASR(dblpiece, _t[15], 28); \ |
| 417 | (z)[0] = (dblpiece)((udblpiece)_t[0]&M28) - B27 + _c; \ |
| 418 | for (_i = 1; _i < NPIECE; _i++) { \ |
| 419 | (z)[_i] = (dblpiece)((udblpiece)_t[_i]&M28) - B27 + \ |
| 420 | ASR(dblpiece, _t[_i - 1], 28); \ |
| 421 | } \ |
| 422 | (z)[8] += _c; \ |
| 423 | } while (0) |
| 424 | |
| 425 | /* --- @fgoldi_mulconst@ --- * |
| 426 | * |
| 427 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 428 | * @const fgoldi *x@ = an operand |
| 429 | * @long a@ = a small-ish constant; %$|a| < 2^{20}$%. |
| 430 | * |
| 431 | * Returns: --- |
| 432 | * |
| 433 | * Use: Set @z@ to the product %$a x$%. |
| 434 | */ |
| 435 | |
| 436 | void fgoldi_mulconst(fgoldi *z, const fgoldi *x, long a) |
| 437 | { |
| 438 | unsigned i; |
| 439 | dblpiece zz[NPIECE], aa = a; |
| 440 | |
| 441 | for (i = 0; i < NPIECE; i++) zz[i] = aa*x->P[i]; |
| 442 | CARRY_REDUCE(z->P, zz); |
| 443 | } |
| 444 | |
| 445 | /* --- @fgoldi_mul@ --- * |
| 446 | * |
| 447 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 448 | * @const fgoldi *x, *y@ = two operands |
| 449 | * |
| 450 | * Returns: --- |
| 451 | * |
| 452 | * Use: Set @z@ to the product %$x y$%. |
| 453 | */ |
| 454 | |
| 455 | void fgoldi_mul(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 456 | { |
| 457 | dblpiece zz[NPIECE], u[NPIECE]; |
| 458 | piece ab[NPIECE/2], cd[NPIECE/2]; |
| 459 | const piece |
| 460 | *a = x->P + NPIECE/2, *b = x->P, |
| 461 | *c = y->P + NPIECE/2, *d = y->P; |
| 462 | unsigned i, j; |
| 463 | |
| 464 | # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) |
| 465 | |
| 466 | /* Behold the magic. |
| 467 | * |
| 468 | * Write x = a φ + b, and y = c φ + d. Then x y = a c φ^2 + |
| 469 | * (a d + b c) φ + b d. Karatsuba and Ofman observed that a d + b c = |
| 470 | * (a + b) (c + d) - a c - b d, saving a multiplication, and Hamburg chose |
| 471 | * the prime p so that φ^2 = φ + 1. So |
| 472 | * |
| 473 | * x y = ((a + b) (c + d) - b d) φ + a c + b d |
| 474 | */ |
| 475 | |
| 476 | for (i = 0; i < NPIECE; i++) zz[i] = 0; |
| 477 | |
| 478 | /* Our first job will be to calculate (1 - φ) b d, and write the result |
| 479 | * into z. As we do this, an interesting thing will happen. Write |
| 480 | * b d = u φ + v; then (1 - φ) b d = u φ + v - u φ^2 - v φ = (1 - φ) v - u. |
| 481 | * So, what we do is to write the product end-swapped and negated, and then |
| 482 | * we'll subtract the (negated, remember) high half from the low half. |
| 483 | */ |
| 484 | for (i = 0; i < NPIECE/2; i++) { |
| 485 | for (j = 0; j < NPIECE/2 - i; j++) |
| 486 | zz[i + j + NPIECE/2] -= M(b,i, d,j); |
| 487 | for (; j < NPIECE/2; j++) |
| 488 | zz[i + j - NPIECE/2] -= M(b,i, d,j); |
| 489 | } |
| 490 | for (i = 0; i < NPIECE/2; i++) |
| 491 | zz[i] -= zz[i + NPIECE/2]; |
| 492 | |
| 493 | /* Next, we add on a c. There are no surprises here. */ |
| 494 | for (i = 0; i < NPIECE/2; i++) |
| 495 | for (j = 0; j < NPIECE/2; j++) |
| 496 | zz[i + j] += M(a,i, c,j); |
| 497 | |
| 498 | /* Now, calculate a + b and c + d. */ |
| 499 | for (i = 0; i < NPIECE/2; i++) |
| 500 | { ab[i] = a[i] + b[i]; cd[i] = c[i] + d[i]; } |
| 501 | |
| 502 | /* Finally (for the multiplication) we must add on (a + b) (c + d) φ. |
| 503 | * Write (a + b) (c + d) as u φ + v; then we actually want u φ^2 + v φ = |
| 504 | * v φ + (1 + φ) u. We'll store u in a temporary place and add it on |
| 505 | * twice. |
| 506 | */ |
| 507 | for (i = 0; i < NPIECE; i++) u[i] = 0; |
| 508 | for (i = 0; i < NPIECE/2; i++) { |
| 509 | for (j = 0; j < NPIECE/2 - i; j++) |
| 510 | zz[i + j + NPIECE/2] += M(ab,i, cd,j); |
| 511 | for (; j < NPIECE/2; j++) |
| 512 | u[i + j - NPIECE/2] += M(ab,i, cd,j); |
| 513 | } |
| 514 | for (i = 0; i < NPIECE/2; i++) |
| 515 | { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } |
| 516 | |
| 517 | #undef M |
| 518 | |
| 519 | /* That wraps it up for the multiplication. Let's figure out some bounds. |
| 520 | * Fortunately, Karatsuba is a polynomial identity, so all of the pieces |
| 521 | * end up the way they'd be if we'd done the thing the easy way, which |
| 522 | * simplifies the analysis. Suppose we started with |x_i|, |y_i| <= 9/5 |
| 523 | * 2^28. The overheads in the result are given by the coefficients of |
| 524 | * |
| 525 | * ((u^16 - 1)/(u - 1))^2 mod u^16 - u^8 - 1 |
| 526 | * |
| 527 | * the greatest of which is 38. So |z_i| <= 38*81/25*2^56 < 2^63. |
| 528 | * |
| 529 | * Anyway, a round of `CARRY_REDUCE' will leave us with |z_i| < 2^27 + |
| 530 | * 2^36; and a second round will leave us with |z_i| < 2^27 + 512. |
| 531 | */ |
| 532 | for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); |
| 533 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; |
| 534 | } |
| 535 | |
| 536 | /* --- @fgoldi_sqr@ --- * |
| 537 | * |
| 538 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 539 | * @const fgoldi *x@ = an operand |
| 540 | * |
| 541 | * Returns: --- |
| 542 | * |
| 543 | * Use: Set @z@ to the square %$x^2$%. |
| 544 | */ |
| 545 | |
| 546 | void fgoldi_sqr(fgoldi *z, const fgoldi *x) |
| 547 | { |
| 548 | |
| 549 | dblpiece zz[NPIECE], u[NPIECE]; |
| 550 | piece ab[NPIECE]; |
| 551 | const piece *a = x->P + NPIECE/2, *b = x->P; |
| 552 | unsigned i, j; |
| 553 | |
| 554 | # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) |
| 555 | |
| 556 | /* The magic is basically the same as `fgoldi_mul' above. We write |
| 557 | * x = a φ + b and use Karatsuba and the special prime shape. This time, |
| 558 | * we have |
| 559 | * |
| 560 | * x^2 = ((a + b)^2 - b^2) φ + a^2 + b^2 |
| 561 | */ |
| 562 | |
| 563 | for (i = 0; i < NPIECE; i++) zz[i] = 0; |
| 564 | |
| 565 | /* Our first job will be to calculate (1 - φ) b^2, and write the result |
| 566 | * into z. Again, this interacts pleasantly with the prime shape. |
| 567 | */ |
| 568 | for (i = 0; i < NPIECE/4; i++) { |
| 569 | zz[2*i + NPIECE/2] -= M(b,i, b,i); |
| 570 | for (j = i + 1; j < NPIECE/2 - i; j++) |
| 571 | zz[i + j + NPIECE/2] -= 2*M(b,i, b,j); |
| 572 | for (; j < NPIECE/2; j++) |
| 573 | zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); |
| 574 | } |
| 575 | for (; i < NPIECE/2; i++) { |
| 576 | zz[2*i - NPIECE/2] -= M(b,i, b,i); |
| 577 | for (j = i + 1; j < NPIECE/2; j++) |
| 578 | zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); |
| 579 | } |
| 580 | for (i = 0; i < NPIECE/2; i++) |
| 581 | zz[i] -= zz[i + NPIECE/2]; |
| 582 | |
| 583 | /* Next, we add on a^2. There are no surprises here. */ |
| 584 | for (i = 0; i < NPIECE/2; i++) { |
| 585 | zz[2*i] += M(a,i, a,i); |
| 586 | for (j = i + 1; j < NPIECE/2; j++) |
| 587 | zz[i + j] += 2*M(a,i, a,j); |
| 588 | } |
| 589 | |
| 590 | /* Now, calculate a + b. */ |
| 591 | for (i = 0; i < NPIECE/2; i++) |
| 592 | ab[i] = a[i] + b[i]; |
| 593 | |
| 594 | /* Finally (for the multiplication) we must add on (a + b)^2 φ. |
| 595 | * Write (a + b)^2 as u φ + v; then we actually want (u + v) φ + u. We'll |
| 596 | * store u in a temporary place and add it on twice. |
| 597 | */ |
| 598 | for (i = 0; i < NPIECE; i++) u[i] = 0; |
| 599 | for (i = 0; i < NPIECE/4; i++) { |
| 600 | zz[2*i + NPIECE/2] += M(ab,i, ab,i); |
| 601 | for (j = i + 1; j < NPIECE/2 - i; j++) |
| 602 | zz[i + j + NPIECE/2] += 2*M(ab,i, ab,j); |
| 603 | for (; j < NPIECE/2; j++) |
| 604 | u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); |
| 605 | } |
| 606 | for (; i < NPIECE/2; i++) { |
| 607 | u[2*i - NPIECE/2] += M(ab,i, ab,i); |
| 608 | for (j = i + 1; j < NPIECE/2; j++) |
| 609 | u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); |
| 610 | } |
| 611 | for (i = 0; i < NPIECE/2; i++) |
| 612 | { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } |
| 613 | |
| 614 | #undef M |
| 615 | |
| 616 | /* Finally, carrying. */ |
| 617 | for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); |
| 618 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; |
| 619 | } |
| 620 | |
| 621 | /*----- More advanced operations ------------------------------------------*/ |
| 622 | |
| 623 | /* --- @fgoldi_inv@ --- * |
| 624 | * |
| 625 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) |
| 626 | * @const fgoldi *x@ = an operand |
| 627 | * |
| 628 | * Returns: --- |
| 629 | * |
| 630 | * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If |
| 631 | * %$x = 0$% then @z@ is set to zero. This is considered a |
| 632 | * feature. |
| 633 | */ |
| 634 | |
| 635 | void fgoldi_inv(fgoldi *z, const fgoldi *x) |
| 636 | { |
| 637 | fgoldi t, u; |
| 638 | unsigned i; |
| 639 | |
| 640 | #define SQRN(z, x, n) do { \ |
| 641 | fgoldi_sqr((z), (x)); \ |
| 642 | for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \ |
| 643 | } while (0) |
| 644 | |
| 645 | /* Calculate x^-1 = x^(p - 2) = x^(2^448 - 2^224 - 3), which also handles |
| 646 | * x = 0 as intended. The addition chain is home-made. |
| 647 | */ /* step | value */ |
| 648 | fgoldi_sqr(&u, x); /* 1 | 2 */ |
| 649 | fgoldi_mul(&t, &u, x); /* 2 | 3 */ |
| 650 | SQRN(&u, &t, 2); /* 4 | 12 */ |
| 651 | fgoldi_mul(&t, &u, &t); /* 5 | 15 */ |
| 652 | SQRN(&u, &t, 4); /* 9 | 240 */ |
| 653 | fgoldi_mul(&u, &u, &t); /* 10 | 255 = 2^8 - 1 */ |
| 654 | SQRN(&u, &u, 4); /* 14 | 2^12 - 16 */ |
| 655 | fgoldi_mul(&t, &u, &t); /* 15 | 2^12 - 1 */ |
| 656 | SQRN(&u, &t, 12); /* 27 | 2^24 - 2^12 */ |
| 657 | fgoldi_mul(&u, &u, &t); /* 28 | 2^24 - 1 */ |
| 658 | SQRN(&u, &u, 12); /* 40 | 2^36 - 2^12 */ |
| 659 | fgoldi_mul(&t, &u, &t); /* 41 | 2^36 - 1 */ |
| 660 | fgoldi_sqr(&t, &t); /* 42 | 2^37 - 2 */ |
| 661 | fgoldi_mul(&t, &t, x); /* 43 | 2^37 - 1 */ |
| 662 | SQRN(&u, &t, 37); /* 80 | 2^74 - 2^37 */ |
| 663 | fgoldi_mul(&u, &u, &t); /* 81 | 2^74 - 1 */ |
| 664 | SQRN(&u, &u, 37); /* 118 | 2^111 - 2^37 */ |
| 665 | fgoldi_mul(&t, &u, &t); /* 119 | 2^111 - 1 */ |
| 666 | SQRN(&u, &t, 111); /* 230 | 2^222 - 2^111 */ |
| 667 | fgoldi_mul(&t, &u, &t); /* 231 | 2^222 - 1 */ |
| 668 | fgoldi_sqr(&u, &t); /* 232 | 2^223 - 2 */ |
| 669 | fgoldi_mul(&u, &u, x); /* 233 | 2^223 - 1 */ |
| 670 | SQRN(&u, &u, 223); /* 456 | 2^446 - 2^223 */ |
| 671 | fgoldi_mul(&t, &u, &t); /* 457 | 2^446 - 2^222 - 1 */ |
| 672 | SQRN(&t, &t, 2); /* 459 | 2^448 - 2^224 - 4 */ |
| 673 | fgoldi_mul(z, &t, x); /* 460 | 2^448 - 2^224 - 3 */ |
| 674 | |
| 675 | #undef SQRN |
| 676 | } |
| 677 | |
| 678 | /* --- @fgoldi_quosqrt@ --- * |
| 679 | * |
| 680 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) |
| 681 | * @const fgoldi *x, *y@ = two operands |
| 682 | * |
| 683 | * Returns: Zero if successful, @-1@ if %$x/y$% is not a square. |
| 684 | * |
| 685 | * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%. |
| 686 | * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x |
| 687 | * \ne 0$% then the operation fails. If you wanted a specific |
| 688 | * square root then you'll have to pick it yourself. |
| 689 | */ |
| 690 | |
| 691 | int fgoldi_quosqrt(fgoldi *z, const fgoldi *x, const fgoldi *y) |
| 692 | { |
| 693 | fgoldi t, u, v; |
| 694 | octet xb[56], b0[56]; |
| 695 | int32 rc = -1; |
| 696 | mask32 m; |
| 697 | unsigned i; |
| 698 | |
| 699 | #define SQRN(z, x, n) do { \ |
| 700 | fgoldi_sqr((z), (x)); \ |
| 701 | for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \ |
| 702 | } while (0) |
| 703 | |
| 704 | /* This is, fortunately, significantly easier than the equivalent problem |
| 705 | * in GF(2^255 - 19), since p == 3 (mod 4). |
| 706 | * |
| 707 | * If x/y is square, then so is v = y^2 x/y = x y, and therefore u has |
| 708 | * order r = (p - 1)/2. Let w = v^{(p-3)/4}. Then w^2 = v^{(p-3)/2} = |
| 709 | * u^{r-1} = 1/v = 1/x y. Clearly, then, (x w)^2 = x^2/x y = x/y, so x w |
| 710 | * is one of the square roots we seek. |
| 711 | * |
| 712 | * The addition chain, then, is a prefix of the previous one. |
| 713 | */ |
| 714 | fgoldi_mul(&v, x, y); |
| 715 | |
| 716 | fgoldi_sqr(&u, &v); /* 1 | 2 */ |
| 717 | fgoldi_mul(&t, &u, &v); /* 2 | 3 */ |
| 718 | SQRN(&u, &t, 2); /* 4 | 12 */ |
| 719 | fgoldi_mul(&t, &u, &t); /* 5 | 15 */ |
| 720 | SQRN(&u, &t, 4); /* 9 | 240 */ |
| 721 | fgoldi_mul(&u, &u, &t); /* 10 | 255 = 2^8 - 1 */ |
| 722 | SQRN(&u, &u, 4); /* 14 | 2^12 - 16 */ |
| 723 | fgoldi_mul(&t, &u, &t); /* 15 | 2^12 - 1 */ |
| 724 | SQRN(&u, &t, 12); /* 27 | 2^24 - 2^12 */ |
| 725 | fgoldi_mul(&u, &u, &t); /* 28 | 2^24 - 1 */ |
| 726 | SQRN(&u, &u, 12); /* 40 | 2^36 - 2^12 */ |
| 727 | fgoldi_mul(&t, &u, &t); /* 41 | 2^36 - 1 */ |
| 728 | fgoldi_sqr(&t, &t); /* 42 | 2^37 - 2 */ |
| 729 | fgoldi_mul(&t, &t, &v); /* 43 | 2^37 - 1 */ |
| 730 | SQRN(&u, &t, 37); /* 80 | 2^74 - 2^37 */ |
| 731 | fgoldi_mul(&u, &u, &t); /* 81 | 2^74 - 1 */ |
| 732 | SQRN(&u, &u, 37); /* 118 | 2^111 - 2^37 */ |
| 733 | fgoldi_mul(&t, &u, &t); /* 119 | 2^111 - 1 */ |
| 734 | SQRN(&u, &t, 111); /* 230 | 2^222 - 2^111 */ |
| 735 | fgoldi_mul(&t, &u, &t); /* 231 | 2^222 - 1 */ |
| 736 | fgoldi_sqr(&u, &t); /* 232 | 2^223 - 2 */ |
| 737 | fgoldi_mul(&u, &u, &v); /* 233 | 2^223 - 1 */ |
| 738 | SQRN(&u, &u, 223); /* 456 | 2^446 - 2^223 */ |
| 739 | fgoldi_mul(&t, &u, &t); /* 457 | 2^446 - 2^222 - 1 */ |
| 740 | |
| 741 | #undef SQRN |
| 742 | |
| 743 | /* Now we must decide whether the answer was right. We should have z^2 = |
| 744 | * x/y, so y z^2 = x. |
| 745 | * |
| 746 | * The easiest way to compare is to encode. This isn't as wasteful as it |
| 747 | * sounds: the hard part is normalizing the representations, which we have |
| 748 | * to do anyway. |
| 749 | */ |
| 750 | fgoldi_mul(z, x, &t); |
| 751 | fgoldi_sqr(&t, z); |
| 752 | fgoldi_mul(&t, &t, y); |
| 753 | fgoldi_store(xb, x); |
| 754 | fgoldi_store(b0, &t); |
| 755 | m = -consttime_memeq(xb, b0, 56); |
| 756 | rc = PICK2(0, rc, m); |
| 757 | return (rc); |
| 758 | } |
| 759 | |
| 760 | /*----- That's all, folks -------------------------------------------------*/ |