| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * Arithmetic modulo 2^255 - 19 |
| 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 "f25519.h" |
| 50 | |
| 51 | /*----- Basic setup -------------------------------------------------------*/ |
| 52 | |
| 53 | typedef f25519_piece piece; |
| 54 | |
| 55 | /* Elements x of GF(2^255 - 19) are represented by ten signed integers x_i: x |
| 56 | * = SUM_{0<=i<10} x_i 2^ceil(51i/2), mostly following Bernstein's original |
| 57 | * paper. |
| 58 | */ |
| 59 | |
| 60 | typedef int64 dblpiece; |
| 61 | typedef uint32 upiece; typedef uint64 udblpiece; |
| 62 | #define P p26 |
| 63 | #define PIECEWD(i) ((i)%2 ? 25 : 26) |
| 64 | #define NPIECE 10 |
| 65 | |
| 66 | #define M26 0x03ffffffu |
| 67 | #define M25 0x01ffffffu |
| 68 | #define B25 0x02000000u |
| 69 | #define B24 0x01000000u |
| 70 | |
| 71 | #define PIECES(v) v##0, v##1, v##2, v##3, v##4, v##5, v##6, v##7, v##8, v##9 |
| 72 | #define FETCH(v, w) do { \ |
| 73 | v##0 = (w)->P[0]; v##1 = (w)->P[1]; \ |
| 74 | v##2 = (w)->P[2]; v##3 = (w)->P[3]; \ |
| 75 | v##4 = (w)->P[4]; v##5 = (w)->P[5]; \ |
| 76 | v##6 = (w)->P[6]; v##7 = (w)->P[7]; \ |
| 77 | v##8 = (w)->P[8]; v##9 = (w)->P[9]; \ |
| 78 | } while (0) |
| 79 | #define STASH(w, v) do { \ |
| 80 | (w)->P[0] = v##0; (w)->P[1] = v##1; \ |
| 81 | (w)->P[2] = v##2; (w)->P[3] = v##3; \ |
| 82 | (w)->P[4] = v##4; (w)->P[5] = v##5; \ |
| 83 | (w)->P[6] = v##6; (w)->P[7] = v##7; \ |
| 84 | (w)->P[8] = v##8; (w)->P[9] = v##9; \ |
| 85 | } while (0) |
| 86 | |
| 87 | /*----- Debugging machinery -----------------------------------------------*/ |
| 88 | |
| 89 | #if defined(F25519_DEBUG) |
| 90 | |
| 91 | #include <stdio.h> |
| 92 | |
| 93 | #include "mp.h" |
| 94 | #include "mptext.h" |
| 95 | |
| 96 | static mp *get_2p255m91(void) |
| 97 | { |
| 98 | mpw w19 = 19; |
| 99 | mp *p = MP_NEW, m19; |
| 100 | |
| 101 | p = mp_setbit(p, MP_ZERO, 255); |
| 102 | mp_build(&m19, &w19, &w19 + 1); |
| 103 | p = mp_sub(p, p, &m19); |
| 104 | return (p); |
| 105 | } |
| 106 | |
| 107 | DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 32, get_2p255m91()) |
| 108 | |
| 109 | #endif |
| 110 | |
| 111 | /*----- Loading and storing -----------------------------------------------*/ |
| 112 | |
| 113 | /* --- @f25519_load@ --- * |
| 114 | * |
| 115 | * Arguments: @f25519 *z@ = where to store the result |
| 116 | * @const octet xv[32]@ = source to read |
| 117 | * |
| 118 | * Returns: --- |
| 119 | * |
| 120 | * Use: Reads an element of %$\gf{2^{255} - 19}$% in external |
| 121 | * representation from @xv@ and stores it in @z@. |
| 122 | * |
| 123 | * External representation is little-endian base-256. Some |
| 124 | * elements have multiple encodings, which are not produced by |
| 125 | * correct software; use of noncanonical encodings is not an |
| 126 | * error, and toleration of them is considered a performance |
| 127 | * feature. |
| 128 | */ |
| 129 | |
| 130 | void f25519_load(f25519 *z, const octet xv[32]) |
| 131 | { |
| 132 | |
| 133 | uint32 xw0 = LOAD32_L(xv + 0), xw1 = LOAD32_L(xv + 4), |
| 134 | xw2 = LOAD32_L(xv + 8), xw3 = LOAD32_L(xv + 12), |
| 135 | xw4 = LOAD32_L(xv + 16), xw5 = LOAD32_L(xv + 20), |
| 136 | xw6 = LOAD32_L(xv + 24), xw7 = LOAD32_L(xv + 28); |
| 137 | piece PIECES(x), b, c; |
| 138 | |
| 139 | /* First, split the 32-bit words into the irregularly-sized pieces we need |
| 140 | * for the field representation. These pieces are all positive. We'll do |
| 141 | * the sign correction afterwards. |
| 142 | * |
| 143 | * It may be that the top bit of the input is set. Avoid trouble by |
| 144 | * folding that back round into the bottom piece of the representation. |
| 145 | * |
| 146 | * Here, we briefly have 0 <= x_0 < 2^26 + 19, but will resolve this later. |
| 147 | * Otherwise, we have 0 <= x_{2i} < 2^26, and 0 <= x_{2i+1} < 2^25. |
| 148 | */ |
| 149 | x0 = ((xw0 << 0)&0x03ffffff) + 19*((xw7 >> 31)&0x00000001); |
| 150 | x1 = ((xw1 << 6)&0x01ffffc0) | ((xw0 >> 26)&0x0000003f); |
| 151 | x2 = ((xw2 << 13)&0x03ffe000) | ((xw1 >> 19)&0x00001fff); |
| 152 | x3 = ((xw3 << 19)&0x01f80000) | ((xw2 >> 13)&0x0007ffff); |
| 153 | x4 = ((xw3 >> 6)&0x03ffffff); |
| 154 | x5 = (xw4 << 0)&0x01ffffff; |
| 155 | x6 = ((xw5 << 7)&0x03ffff80) | ((xw4 >> 25)&0x0000007f); |
| 156 | x7 = ((xw6 << 13)&0x01ffe000) | ((xw5 >> 19)&0x00001fff); |
| 157 | x8 = ((xw7 << 20)&0x03f00000) | ((xw6 >> 12)&0x000fffff); |
| 158 | x9 = ((xw7 >> 6)&0x01ffffff); |
| 159 | |
| 160 | /* Next, we convert these pieces into a roughly balanced signed |
| 161 | * representation. For each piece, check to see if its top bit is set. If |
| 162 | * it is, then lend a bit to the next piece over. For x_9, this needs to |
| 163 | * be carried around, which is a little fiddly. In particular, we delay |
| 164 | * the carry until after all of the pieces have been balanced. If we don't |
| 165 | * do this, then we have to do a more expensive test for nonzeroness to |
| 166 | * decide whether to lend a bit leftwards rather than just testing a single |
| 167 | * bit. |
| 168 | * |
| 169 | * This fixes up the anomalous size of x_0: the loan of a bit becomes an |
| 170 | * actual carry if x_0 >= 2^26. By the end, then, we have: |
| 171 | * |
| 172 | * { 2^25 if i even |
| 173 | * |x_i| <= { |
| 174 | * { 2^24 if i odd |
| 175 | * |
| 176 | * Note that we don't try for a canonical representation here: both upper |
| 177 | * and lower bounds are achievable. |
| 178 | * |
| 179 | * All of the x_i at this point are positive, so we don't need to do |
| 180 | * anything wierd when masking them. |
| 181 | */ |
| 182 | b = x9&B24; c = 19&((b >> 19) - (b >> 24)); x9 -= b << 1; |
| 183 | b = x8&B25; x9 += b >> 25; x8 -= b << 1; |
| 184 | b = x7&B24; x8 += b >> 24; x7 -= b << 1; |
| 185 | b = x6&B25; x7 += b >> 25; x6 -= b << 1; |
| 186 | b = x5&B24; x6 += b >> 24; x5 -= b << 1; |
| 187 | b = x4&B25; x5 += b >> 25; x4 -= b << 1; |
| 188 | b = x3&B24; x4 += b >> 24; x3 -= b << 1; |
| 189 | b = x2&B25; x3 += b >> 25; x2 -= b << 1; |
| 190 | b = x1&B24; x2 += b >> 24; x1 -= b << 1; |
| 191 | b = x0&B25; x1 += (b >> 25) + (x0 >> 26); x0 = (x0&M26) - (b << 1); |
| 192 | x0 += c; |
| 193 | |
| 194 | /* And with that, we're done. */ |
| 195 | STASH(z, x); |
| 196 | } |
| 197 | |
| 198 | /* --- @f25519_store@ --- * |
| 199 | * |
| 200 | * Arguments: @octet zv[32]@ = where to write the result |
| 201 | * @const f25519 *x@ = the field element to write |
| 202 | * |
| 203 | * Returns: --- |
| 204 | * |
| 205 | * Use: Stores a field element in the given octet vector in external |
| 206 | * representation. A canonical encoding is always stored, so, |
| 207 | * in particular, the top bit of @xv[31]@ is always left clear. |
| 208 | */ |
| 209 | |
| 210 | void f25519_store(octet zv[32], const f25519 *x) |
| 211 | { |
| 212 | |
| 213 | piece PIECES(x), PIECES(y), c, d; |
| 214 | uint32 zw0, zw1, zw2, zw3, zw4, zw5, zw6, zw7; |
| 215 | mask32 m; |
| 216 | |
| 217 | FETCH(x, x); |
| 218 | |
| 219 | /* First, propagate the carries throughout the pieces. By the end of this, |
| 220 | * we'll have all of the pieces canonically sized and positive, and maybe |
| 221 | * there'll be (signed) carry out. The carry c is in { -1, 0, +1 }, and |
| 222 | * the remaining value will be in the half-open interval [0, 2^255). The |
| 223 | * whole represented value is then x + 2^255 c. |
| 224 | * |
| 225 | * It's worth paying careful attention to the bounds. We assume that we |
| 226 | * start out with |x_i| <= 2^30. We start by cutting off and reducing the |
| 227 | * carry c_9 from the topmost piece, x_9. This leaves 0 <= x_9 < 2^25; and |
| 228 | * we'll have |c_9| <= 2^5. We multiply this by 19 and we'll add it onto |
| 229 | * x_0 and propagate the carries: but what bounds can we calculate on x |
| 230 | * before this? |
| 231 | * |
| 232 | * Let o_i = floor(51 i/2). We have X_i = SUM_{0<=j<i} x_j 2^{o_i}, so |
| 233 | * x = X_10. We see, inductively, that |X_i| < 2^{31+o_{i-1}}: X_0 = 0; |
| 234 | * |x_i| <= 2^30; and |X_{i+1}| = |X_i + x_i 2^{o_i}| <= |X_i| + 2^{30+o_i} |
| 235 | * < 2^{31+o_i}. Then x = X_9 + 2^230 x_9, and we have better bounds for |
| 236 | * x_9, so |
| 237 | * |
| 238 | * -2^235 < x + 19 c_9 < 2^255 + 2^235 |
| 239 | * |
| 240 | * Here, the x_i are signed, so we must be cautious about bithacking them. |
| 241 | */ |
| 242 | c = ASR(piece, x9, 25); x9 = (upiece)x9&M25; |
| 243 | x0 += 19*c; c = ASR(piece, x0, 26); x0 = (upiece)x0&M26; |
| 244 | x1 += c; c = ASR(piece, x1, 25); x1 = (upiece)x1&M25; |
| 245 | x2 += c; c = ASR(piece, x2, 26); x2 = (upiece)x2&M26; |
| 246 | x3 += c; c = ASR(piece, x3, 25); x3 = (upiece)x3&M25; |
| 247 | x4 += c; c = ASR(piece, x4, 26); x4 = (upiece)x4&M26; |
| 248 | x5 += c; c = ASR(piece, x5, 25); x5 = (upiece)x5&M25; |
| 249 | x6 += c; c = ASR(piece, x6, 26); x6 = (upiece)x6&M26; |
| 250 | x7 += c; c = ASR(piece, x7, 25); x7 = (upiece)x7&M25; |
| 251 | x8 += c; c = ASR(piece, x8, 26); x8 = (upiece)x8&M26; |
| 252 | x9 += c; c = ASR(piece, x9, 25); x9 = (upiece)x9&M25; |
| 253 | |
| 254 | /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and |
| 255 | * x >= 2^255 - 19, then we should subtract 2^255 - 19 from the whole |
| 256 | * value; if c = -1 then we should add 2^255 - 19; and otherwise we should |
| 257 | * do nothing. |
| 258 | * |
| 259 | * But conditional behaviour is bad, m'kay. So here's what we do instead. |
| 260 | * |
| 261 | * The first job is to sort out what we wanted to do. If c = -1 then we |
| 262 | * want to (a) invert the constant addend and (b) feed in a carry-in; |
| 263 | * otherwise, we don't. |
| 264 | */ |
| 265 | m = SIGN(c); |
| 266 | d = m&1; |
| 267 | |
| 268 | /* Now do the addition/subtraction. Remember that all of the x_i are |
| 269 | * nonnegative, so shifting and masking are safe and easy. |
| 270 | */ |
| 271 | d += x0 + (19 ^ (M26&m)); y0 = d&M26; d >>= 26; |
| 272 | d += x1 + (M25&m); y1 = d&M25; d >>= 25; |
| 273 | d += x2 + (M26&m); y2 = d&M26; d >>= 26; |
| 274 | d += x3 + (M25&m); y3 = d&M25; d >>= 25; |
| 275 | d += x4 + (M26&m); y4 = d&M26; d >>= 26; |
| 276 | d += x5 + (M25&m); y5 = d&M25; d >>= 25; |
| 277 | d += x6 + (M26&m); y6 = d&M26; d >>= 26; |
| 278 | d += x7 + (M25&m); y7 = d&M25; d >>= 25; |
| 279 | d += x8 + (M26&m); y8 = d&M26; d >>= 26; |
| 280 | d += x9 + (M25&m); y9 = d&M25; d >>= 25; |
| 281 | |
| 282 | /* The final carry-out is in d; since we only did addition, and the x_i are |
| 283 | * nonnegative, then d is in { 0, 1 }. We want to keep y, rather than x, |
| 284 | * if (a) c /= 0 (in which case we know that the old value was |
| 285 | * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that |
| 286 | * the subtraction didn't cause a borrow, so we must be in the case where |
| 287 | * 2^255 - 19 <= x < 2^255). |
| 288 | */ |
| 289 | m = NONZEROP(c) | ~NONZEROP(d - 1); |
| 290 | x0 = (y0&m) | (x0&~m); x1 = (y1&m) | (x1&~m); |
| 291 | x2 = (y2&m) | (x2&~m); x3 = (y3&m) | (x3&~m); |
| 292 | x4 = (y4&m) | (x4&~m); x5 = (y5&m) | (x5&~m); |
| 293 | x6 = (y6&m) | (x6&~m); x7 = (y7&m) | (x7&~m); |
| 294 | x8 = (y8&m) | (x8&~m); x9 = (y9&m) | (x9&~m); |
| 295 | |
| 296 | /* Extract 32-bit words from the value. */ |
| 297 | zw0 = ((x0 >> 0)&0x03ffffff) | (((uint32)x1 << 26)&0xfc000000); |
| 298 | zw1 = ((x1 >> 6)&0x0007ffff) | (((uint32)x2 << 19)&0xfff80000); |
| 299 | zw2 = ((x2 >> 13)&0x00001fff) | (((uint32)x3 << 13)&0xffffe000); |
| 300 | zw3 = ((x3 >> 19)&0x0000003f) | (((uint32)x4 << 6)&0xffffffc0); |
| 301 | zw4 = ((x5 >> 0)&0x01ffffff) | (((uint32)x6 << 25)&0xfe000000); |
| 302 | zw5 = ((x6 >> 7)&0x0007ffff) | (((uint32)x7 << 19)&0xfff80000); |
| 303 | zw6 = ((x7 >> 13)&0x00000fff) | (((uint32)x8 << 12)&0xfffff000); |
| 304 | zw7 = ((x8 >> 20)&0x0000003f) | (((uint32)x9 << 6)&0x7fffffc0); |
| 305 | |
| 306 | /* Store the result as an octet string. */ |
| 307 | STORE32_L(zv + 0, zw0); STORE32_L(zv + 4, zw1); |
| 308 | STORE32_L(zv + 8, zw2); STORE32_L(zv + 12, zw3); |
| 309 | STORE32_L(zv + 16, zw4); STORE32_L(zv + 20, zw5); |
| 310 | STORE32_L(zv + 24, zw6); STORE32_L(zv + 28, zw7); |
| 311 | } |
| 312 | |
| 313 | /* --- @f25519_set@ --- * |
| 314 | * |
| 315 | * Arguments: @f25519 *z@ = where to write the result |
| 316 | * @int a@ = a small-ish constant |
| 317 | * |
| 318 | * Returns: --- |
| 319 | * |
| 320 | * Use: Sets @z@ to equal @a@. |
| 321 | */ |
| 322 | |
| 323 | void f25519_set(f25519 *x, int a) |
| 324 | { |
| 325 | unsigned i; |
| 326 | |
| 327 | x->P[0] = a; |
| 328 | for (i = 1; i < NPIECE; i++) x->P[i] = 0; |
| 329 | } |
| 330 | |
| 331 | /*----- Basic arithmetic --------------------------------------------------*/ |
| 332 | |
| 333 | /* --- @f25519_add@ --- * |
| 334 | * |
| 335 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) |
| 336 | * @const f25519 *x, *y@ = two operands |
| 337 | * |
| 338 | * Returns: --- |
| 339 | * |
| 340 | * Use: Set @z@ to the sum %$x + y$%. |
| 341 | */ |
| 342 | |
| 343 | void f25519_add(f25519 *z, const f25519 *x, const f25519 *y) |
| 344 | { |
| 345 | z->P[0] = x->P[0] + y->P[0]; z->P[1] = x->P[1] + y->P[1]; |
| 346 | z->P[2] = x->P[2] + y->P[2]; z->P[3] = x->P[3] + y->P[3]; |
| 347 | z->P[4] = x->P[4] + y->P[4]; z->P[5] = x->P[5] + y->P[5]; |
| 348 | z->P[6] = x->P[6] + y->P[6]; z->P[7] = x->P[7] + y->P[7]; |
| 349 | z->P[8] = x->P[8] + y->P[8]; z->P[9] = x->P[9] + y->P[9]; |
| 350 | } |
| 351 | |
| 352 | /* --- @f25519_sub@ --- * |
| 353 | * |
| 354 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) |
| 355 | * @const f25519 *x, *y@ = two operands |
| 356 | * |
| 357 | * Returns: --- |
| 358 | * |
| 359 | * Use: Set @z@ to the difference %$x - y$%. |
| 360 | */ |
| 361 | |
| 362 | void f25519_sub(f25519 *z, const f25519 *x, const f25519 *y) |
| 363 | { |
| 364 | z->P[0] = x->P[0] - y->P[0]; z->P[1] = x->P[1] - y->P[1]; |
| 365 | z->P[2] = x->P[2] - y->P[2]; z->P[3] = x->P[3] - y->P[3]; |
| 366 | z->P[4] = x->P[4] - y->P[4]; z->P[5] = x->P[5] - y->P[5]; |
| 367 | z->P[6] = x->P[6] - y->P[6]; z->P[7] = x->P[7] - y->P[7]; |
| 368 | z->P[8] = x->P[8] - y->P[8]; z->P[9] = x->P[9] - y->P[9]; |
| 369 | } |
| 370 | |
| 371 | /* --- @f25519_neg@ --- * |
| 372 | * |
| 373 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) |
| 374 | * @const f25519 *x@ = an operand |
| 375 | * |
| 376 | * Returns: --- |
| 377 | * |
| 378 | * Use: Set @z = -x@. |
| 379 | */ |
| 380 | |
| 381 | void f25519_neg(f25519 *z, const f25519 *x) |
| 382 | { |
| 383 | z->P[0] = -x->P[0]; z->P[1] = -x->P[1]; |
| 384 | z->P[2] = -x->P[2]; z->P[3] = -x->P[3]; |
| 385 | z->P[4] = -x->P[4]; z->P[5] = -x->P[5]; |
| 386 | z->P[6] = -x->P[6]; z->P[7] = -x->P[7]; |
| 387 | z->P[8] = -x->P[8]; z->P[9] = -x->P[9]; |
| 388 | } |
| 389 | |
| 390 | /*----- Constant-time utilities -------------------------------------------*/ |
| 391 | |
| 392 | /* --- @f25519_pick2@ --- * |
| 393 | * |
| 394 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) |
| 395 | * @const f25519 *x, *y@ = two operands |
| 396 | * @uint32 m@ = a mask |
| 397 | * |
| 398 | * Returns: --- |
| 399 | * |
| 400 | * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set |
| 401 | * @z = x@. If @m@ has some other value, then scramble @z@ in |
| 402 | * an unhelpful way. |
| 403 | */ |
| 404 | |
| 405 | void f25519_pick2(f25519 *z, const f25519 *x, const f25519 *y, uint32 m) |
| 406 | { |
| 407 | mask32 mm = FIX_MASK32(m); |
| 408 | |
| 409 | z->P[0] = PICK2(x->P[0], y->P[0], mm); |
| 410 | z->P[1] = PICK2(x->P[1], y->P[1], mm); |
| 411 | z->P[2] = PICK2(x->P[2], y->P[2], mm); |
| 412 | z->P[3] = PICK2(x->P[3], y->P[3], mm); |
| 413 | z->P[4] = PICK2(x->P[4], y->P[4], mm); |
| 414 | z->P[5] = PICK2(x->P[5], y->P[5], mm); |
| 415 | z->P[6] = PICK2(x->P[6], y->P[6], mm); |
| 416 | z->P[7] = PICK2(x->P[7], y->P[7], mm); |
| 417 | z->P[8] = PICK2(x->P[8], y->P[8], mm); |
| 418 | z->P[9] = PICK2(x->P[9], y->P[9], mm); |
| 419 | } |
| 420 | |
| 421 | /* --- @f25519_pickn@ --- * |
| 422 | * |
| 423 | * Arguments: @f25519 *z@ = where to put the result |
| 424 | * @const f25519 *v@ = a table of entries |
| 425 | * @size_t n@ = the number of entries in @v@ |
| 426 | * @size_t i@ = an index |
| 427 | * |
| 428 | * Returns: --- |
| 429 | * |
| 430 | * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then |
| 431 | * do something unhelpful; otherwise, if @i >= n@ then set @z@ |
| 432 | * to zero. |
| 433 | */ |
| 434 | |
| 435 | void f25519_pickn(f25519 *z, const f25519 *v, size_t n, size_t i) |
| 436 | { |
| 437 | uint32 b = (uint32)1 << (31 - i); |
| 438 | mask32 m; |
| 439 | |
| 440 | z->P[0] = z->P[1] = z->P[2] = z->P[3] = z->P[4] = |
| 441 | z->P[5] = z->P[6] = z->P[7] = z->P[8] = z->P[9] = 0; |
| 442 | while (n--) { |
| 443 | m = SIGN(b); |
| 444 | CONDPICK(z->P[0], v->P[0], m); |
| 445 | CONDPICK(z->P[1], v->P[1], m); |
| 446 | CONDPICK(z->P[2], v->P[2], m); |
| 447 | CONDPICK(z->P[3], v->P[3], m); |
| 448 | CONDPICK(z->P[4], v->P[4], m); |
| 449 | CONDPICK(z->P[5], v->P[5], m); |
| 450 | CONDPICK(z->P[6], v->P[6], m); |
| 451 | CONDPICK(z->P[7], v->P[7], m); |
| 452 | CONDPICK(z->P[8], v->P[8], m); |
| 453 | CONDPICK(z->P[9], v->P[9], m); |
| 454 | v++; b <<= 1; |
| 455 | } |
| 456 | } |
| 457 | |
| 458 | /* --- @f25519_condswap@ --- * |
| 459 | * |
| 460 | * Arguments: @f25519 *x, *y@ = two operands |
| 461 | * @uint32 m@ = a mask |
| 462 | * |
| 463 | * Returns: --- |
| 464 | * |
| 465 | * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then |
| 466 | * exchange @x@ and @y@. If @m@ has some other value, then |
| 467 | * scramble @x@ and @y@ in an unhelpful way. |
| 468 | */ |
| 469 | |
| 470 | void f25519_condswap(f25519 *x, f25519 *y, uint32 m) |
| 471 | { |
| 472 | mask32 mm = FIX_MASK32(m); |
| 473 | |
| 474 | CONDSWAP(x->P[0], y->P[0], mm); |
| 475 | CONDSWAP(x->P[1], y->P[1], mm); |
| 476 | CONDSWAP(x->P[2], y->P[2], mm); |
| 477 | CONDSWAP(x->P[3], y->P[3], mm); |
| 478 | CONDSWAP(x->P[4], y->P[4], mm); |
| 479 | CONDSWAP(x->P[5], y->P[5], mm); |
| 480 | CONDSWAP(x->P[6], y->P[6], mm); |
| 481 | CONDSWAP(x->P[7], y->P[7], mm); |
| 482 | CONDSWAP(x->P[8], y->P[8], mm); |
| 483 | CONDSWAP(x->P[9], y->P[9], mm); |
| 484 | } |
| 485 | |
| 486 | /* --- @f25519_condneg@ --- * |
| 487 | * |
| 488 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) |
| 489 | * @const f25519 *x@ = an operand |
| 490 | * @uint32 m@ = a mask |
| 491 | * |
| 492 | * Returns: --- |
| 493 | * |
| 494 | * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set |
| 495 | * @z = -x@. If @m@ has some other value then scramble @z@ in |
| 496 | * an unhelpful way. |
| 497 | */ |
| 498 | |
| 499 | void f25519_condneg(f25519 *z, const f25519 *x, uint32 m) |
| 500 | { |
| 501 | mask32 m_xor = FIX_MASK32(m); |
| 502 | piece m_add = m&1; |
| 503 | # define CONDNEG(x) (((x) ^ m_xor) + m_add) |
| 504 | |
| 505 | z->P[0] = CONDNEG(x->P[0]); |
| 506 | z->P[1] = CONDNEG(x->P[1]); |
| 507 | z->P[2] = CONDNEG(x->P[2]); |
| 508 | z->P[3] = CONDNEG(x->P[3]); |
| 509 | z->P[4] = CONDNEG(x->P[4]); |
| 510 | z->P[5] = CONDNEG(x->P[5]); |
| 511 | z->P[6] = CONDNEG(x->P[6]); |
| 512 | z->P[7] = CONDNEG(x->P[7]); |
| 513 | z->P[8] = CONDNEG(x->P[8]); |
| 514 | z->P[9] = CONDNEG(x->P[9]); |
| 515 | |
| 516 | #undef CONDNEG |
| 517 | } |
| 518 | |
| 519 | /*----- Multiplication ----------------------------------------------------*/ |
| 520 | |
| 521 | /* Let B = 2^63 - 1 be the largest value such that +B and -B can be |
| 522 | * represented in a double-precision piece. On entry, it must be the case |
| 523 | * that |X_i| <= M <= B - 2^25 for some M. If this is the case, then, on |
| 524 | * exit, we will have |Z_i| <= 2^25 + 19 M/2^25. |
| 525 | */ |
| 526 | #define CARRYSTEP(z, x, m, b, f, xx, n) do { \ |
| 527 | (z) = (dblpiece)((udblpiece)(x)&(m)) - (b) + \ |
| 528 | (f)*ASR(dblpiece, (xx), (n)); \ |
| 529 | } while (0) |
| 530 | #define CARRY_REDUCE(z, x) do { \ |
| 531 | dblpiece PIECES(_t); \ |
| 532 | \ |
| 533 | /* Bias the input pieces. This keeps the carries and so on centred \ |
| 534 | * around zero rather than biased positive. \ |
| 535 | */ \ |
| 536 | _t0 = (x##0) + B25; _t1 = (x##1) + B24; \ |
| 537 | _t2 = (x##2) + B25; _t3 = (x##3) + B24; \ |
| 538 | _t4 = (x##4) + B25; _t5 = (x##5) + B24; \ |
| 539 | _t6 = (x##6) + B25; _t7 = (x##7) + B24; \ |
| 540 | _t8 = (x##8) + B25; _t9 = (x##9) + B24; \ |
| 541 | \ |
| 542 | /* Calculate the reduced pieces. Careful with the bithacking. */ \ |
| 543 | CARRYSTEP(z##0, _t0, M26, B25, 19, _t9, 25); \ |
| 544 | CARRYSTEP(z##1, _t1, M25, B24, 1, _t0, 26); \ |
| 545 | CARRYSTEP(z##2, _t2, M26, B25, 1, _t1, 25); \ |
| 546 | CARRYSTEP(z##3, _t3, M25, B24, 1, _t2, 26); \ |
| 547 | CARRYSTEP(z##4, _t4, M26, B25, 1, _t3, 25); \ |
| 548 | CARRYSTEP(z##5, _t5, M25, B24, 1, _t4, 26); \ |
| 549 | CARRYSTEP(z##6, _t6, M26, B25, 1, _t5, 25); \ |
| 550 | CARRYSTEP(z##7, _t7, M25, B24, 1, _t6, 26); \ |
| 551 | CARRYSTEP(z##8, _t8, M26, B25, 1, _t7, 25); \ |
| 552 | CARRYSTEP(z##9, _t9, M25, B24, 1, _t8, 26); \ |
| 553 | } while (0) |
| 554 | |
| 555 | /* --- @f25519_mulconst@ --- * |
| 556 | * |
| 557 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) |
| 558 | * @const f25519 *x@ = an operand |
| 559 | * @long a@ = a small-ish constant; %$|a| < 2^{20}$%. |
| 560 | * |
| 561 | * Returns: --- |
| 562 | * |
| 563 | * Use: Set @z@ to the product %$a x$%. |
| 564 | */ |
| 565 | |
| 566 | void f25519_mulconst(f25519 *z, const f25519 *x, long a) |
| 567 | { |
| 568 | |
| 569 | piece PIECES(x); |
| 570 | dblpiece PIECES(z), aa = a; |
| 571 | |
| 572 | FETCH(x, x); |
| 573 | |
| 574 | /* Suppose that |x_i| <= 2^27, and |a| <= 2^23. Then we'll have |
| 575 | * |z_i| <= 2^50. |
| 576 | */ |
| 577 | z0 = aa*x0; z1 = aa*x1; z2 = aa*x2; z3 = aa*x3; z4 = aa*x4; |
| 578 | z5 = aa*x5; z6 = aa*x6; z7 = aa*x7; z8 = aa*x8; z9 = aa*x9; |
| 579 | |
| 580 | /* Following `CARRY_REDUCE', we'll have |z_i| <= 2^26. */ |
| 581 | CARRY_REDUCE(z, z); |
| 582 | STASH(z, z); |
| 583 | } |
| 584 | |
| 585 | /* --- @f25519_mul@ --- * |
| 586 | * |
| 587 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) |
| 588 | * @const f25519 *x, *y@ = two operands |
| 589 | * |
| 590 | * Returns: --- |
| 591 | * |
| 592 | * Use: Set @z@ to the product %$x y$%. |
| 593 | */ |
| 594 | |
| 595 | void f25519_mul(f25519 *z, const f25519 *x, const f25519 *y) |
| 596 | { |
| 597 | |
| 598 | piece PIECES(x), PIECES(y); |
| 599 | dblpiece PIECES(z); |
| 600 | unsigned i; |
| 601 | |
| 602 | FETCH(x, x); FETCH(y, y); |
| 603 | |
| 604 | /* Suppose that |x_i|, |y_i| <= 2^27. Then we'll have |
| 605 | * |
| 606 | * |z_0| <= 267*2^54 |
| 607 | * |z_1| <= 154*2^54 |
| 608 | * |z_2| <= 213*2^54 |
| 609 | * |z_3| <= 118*2^54 |
| 610 | * |z_4| <= 159*2^54 |
| 611 | * |z_5| <= 82*2^54 |
| 612 | * |z_6| <= 105*2^54 |
| 613 | * |z_7| <= 46*2^54 |
| 614 | * |z_8| <= 51*2^54 |
| 615 | * |z_9| <= 10*2^54 |
| 616 | * |
| 617 | * all of which are less than 2^63 - 2^25. |
| 618 | */ |
| 619 | |
| 620 | #define M(a, b) ((dblpiece)(a)*(b)) |
| 621 | z0 = M(x0, y0) + |
| 622 | 19*(M(x2, y8) + M(x4, y6) + M(x6, y4) + M(x8, y2)) + |
| 623 | 38*(M(x1, y9) + M(x3, y7) + M(x5, y5) + M(x7, y3) + M(x9, y1)); |
| 624 | z1 = M(x0, y1) + M(x1, y0) + |
| 625 | 19*(M(x2, y9) + M(x3, y8) + M(x4, y7) + M(x5, y6) + |
| 626 | M(x6, y5) + M(x7, y4) + M(x8, y3) + M(x9, y2)); |
| 627 | z2 = M(x0, y2) + M(x2, y0) + |
| 628 | 2* M(x1, y1) + |
| 629 | 19*(M(x4, y8) + M(x6, y6) + M(x8, y4)) + |
| 630 | 38*(M(x3, y9) + M(x5, y7) + M(x7, y5) + M(x9, y3)); |
| 631 | z3 = M(x0, y3) + M(x1, y2) + M(x2, y1) + M(x3, y0) + |
| 632 | 19*(M(x4, y9) + M(x5, y8) + M(x6, y7) + |
| 633 | M(x7, y6) + M(x8, y5) + M(x9, y4)); |
| 634 | z4 = M(x0, y4) + M(x2, y2) + M(x4, y0) + |
| 635 | 2*(M(x1, y3) + M(x3, y1)) + |
| 636 | 19*(M(x6, y8) + M(x8, y6)) + |
| 637 | 38*(M(x5, y9) + M(x7, y7) + M(x9, y5)); |
| 638 | z5 = M(x0, y5) + M(x1, y4) + M(x2, y3) + |
| 639 | M(x3, y2) + M(x4, y1) + M(x5, y0) + |
| 640 | 19*(M(x6, y9) + M(x7, y8) + M(x8, y7) + M(x9, y6)); |
| 641 | z6 = M(x0, y6) + M(x2, y4) + M(x4, y2) + M(x6, y0) + |
| 642 | 2*(M(x1, y5) + M(x3, y3) + M(x5, y1)) + |
| 643 | 19* M(x8, y8) + |
| 644 | 38*(M(x7, y9) + M(x9, y7)); |
| 645 | z7 = M(x0, y7) + M(x1, y6) + M(x2, y5) + M(x3, y4) + |
| 646 | M(x4, y3) + M(x5, y2) + M(x6, y1) + M(x7, y0) + |
| 647 | 19*(M(x8, y9) + M(x9, y8)); |
| 648 | z8 = M(x0, y8) + M(x2, y6) + M(x4, y4) + M(x6, y2) + M(x8, y0) + |
| 649 | 2*(M(x1, y7) + M(x3, y5) + M(x5, y3) + M(x7, y1)) + |
| 650 | 38* M(x9, y9); |
| 651 | z9 = M(x0, y9) + M(x1, y8) + M(x2, y7) + M(x3, y6) + M(x4, y5) + |
| 652 | M(x5, y4) + M(x6, y3) + M(x7, y2) + M(x8, y1) + M(x9, y0); |
| 653 | #undef M |
| 654 | |
| 655 | /* From above, we have |z_i| <= 2^63 - 2^25. A pass of `CARRY_REDUCE' will |
| 656 | * leave |z_i| <= 2^38 + 2^25; and a second pass will leave |z_i| <= 2^25 + |
| 657 | * 2^13, which is comfortable for an addition prior to the next |
| 658 | * multiplication. |
| 659 | */ |
| 660 | for (i = 0; i < 2; i++) CARRY_REDUCE(z, z); |
| 661 | STASH(z, z); |
| 662 | } |
| 663 | |
| 664 | /* --- @f25519_sqr@ --- * |
| 665 | * |
| 666 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) |
| 667 | * @const f25519 *x@ = an operand |
| 668 | * |
| 669 | * Returns: --- |
| 670 | * |
| 671 | * Use: Set @z@ to the square %$x^2$%. |
| 672 | */ |
| 673 | |
| 674 | void f25519_sqr(f25519 *z, const f25519 *x) |
| 675 | { |
| 676 | |
| 677 | piece PIECES(x); |
| 678 | dblpiece PIECES(z); |
| 679 | unsigned i; |
| 680 | |
| 681 | FETCH(x, x); |
| 682 | |
| 683 | /* See `f25519_mul' for bounds. */ |
| 684 | |
| 685 | #define M(a, b) ((dblpiece)(a)*(b)) |
| 686 | z0 = M(x0, x0) + |
| 687 | 38*(M(x2, x8) + M(x4, x6) + M(x5, x5)) + |
| 688 | 76*(M(x1, x9) + M(x3, x7)); |
| 689 | z1 = 2* M(x0, x1) + |
| 690 | 38*(M(x2, x9) + M(x3, x8) + M(x4, x7) + M(x5, x6)); |
| 691 | z2 = 2*(M(x0, x2) + M(x1, x1)) + |
| 692 | 19* M(x6, x6) + |
| 693 | 38* M(x4, x8) + |
| 694 | 76*(M(x3, x9) + M(x5, x7)); |
| 695 | z3 = 2*(M(x0, x3) + M(x1, x2)) + |
| 696 | 38*(M(x4, x9) + M(x5, x8) + M(x6, x7)); |
| 697 | z4 = M(x2, x2) + |
| 698 | 2* M(x0, x4) + |
| 699 | 4* M(x1, x3) + |
| 700 | 38*(M(x6, x8) + M(x7, x7)) + |
| 701 | 76* M(x5, x9); |
| 702 | z5 = 2*(M(x0, x5) + M(x1, x4) + M(x2, x3)) + |
| 703 | 38*(M(x6, x9) + M(x7, x8)); |
| 704 | z6 = 2*(M(x0, x6) + M(x2, x4) + M(x3, x3)) + |
| 705 | 4* M(x1, x5) + |
| 706 | 19* M(x8, x8) + |
| 707 | 76* M(x7, x9); |
| 708 | z7 = 2*(M(x0, x7) + M(x1, x6) + M(x2, x5) + M(x3, x4)) + |
| 709 | 38* M(x8, x9); |
| 710 | z8 = M(x4, x4) + |
| 711 | 2*(M(x0, x8) + M(x2, x6)) + |
| 712 | 4*(M(x1, x7) + M(x3, x5)) + |
| 713 | 38* M(x9, x9); |
| 714 | z9 = 2*(M(x0, x9) + M(x1, x8) + M(x2, x7) + M(x3, x6) + M(x4, x5)); |
| 715 | #undef M |
| 716 | |
| 717 | /* See `f25519_mul' for details. */ |
| 718 | for (i = 0; i < 2; i++) CARRY_REDUCE(z, z); |
| 719 | STASH(z, z); |
| 720 | } |
| 721 | |
| 722 | /*----- More complicated things -------------------------------------------*/ |
| 723 | |
| 724 | /* --- @f25519_inv@ --- * |
| 725 | * |
| 726 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) |
| 727 | * @const f25519 *x@ = an operand |
| 728 | * |
| 729 | * Returns: --- |
| 730 | * |
| 731 | * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If |
| 732 | * %$x = 0$% then @z@ is set to zero. This is considered a |
| 733 | * feature. |
| 734 | */ |
| 735 | |
| 736 | void f25519_inv(f25519 *z, const f25519 *x) |
| 737 | { |
| 738 | f25519 t, u, t2, t11, t2p10m1, t2p50m1; |
| 739 | unsigned i; |
| 740 | |
| 741 | #define SQRN(z, x, n) do { \ |
| 742 | f25519_sqr((z), (x)); \ |
| 743 | for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \ |
| 744 | } while (0) |
| 745 | |
| 746 | /* Calculate x^-1 = x^(p - 2) = x^(2^255 - 21), which also handles x = 0 as |
| 747 | * intended. The addition chain here is from Bernstein's implementation; I |
| 748 | * couldn't find a better one. |
| 749 | */ /* step | value */ |
| 750 | f25519_sqr(&t2, x); /* 1 | 2 */ |
| 751 | SQRN(&u, &t2, 2); /* 3 | 8 */ |
| 752 | f25519_mul(&t, &u, x); /* 4 | 9 */ |
| 753 | f25519_mul(&t11, &t, &t2); /* 5 | 11 = 2^5 - 21 */ |
| 754 | f25519_sqr(&u, &t11); /* 6 | 22 */ |
| 755 | f25519_mul(&t, &t, &u); /* 7 | 31 = 2^5 - 1 */ |
| 756 | SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */ |
| 757 | f25519_mul(&t2p10m1, &t, &u); /* 13 | 2^10 - 1 */ |
| 758 | SQRN(&u, &t2p10m1, 10); /* 23 | 2^20 - 2^10 */ |
| 759 | f25519_mul(&t, &t2p10m1, &u); /* 24 | 2^20 - 1 */ |
| 760 | SQRN(&u, &t, 20); /* 44 | 2^40 - 2^20 */ |
| 761 | f25519_mul(&t, &t, &u); /* 45 | 2^40 - 1 */ |
| 762 | SQRN(&u, &t, 10); /* 55 | 2^50 - 2^10 */ |
| 763 | f25519_mul(&t2p50m1, &t2p10m1, &u); /* 56 | 2^50 - 1 */ |
| 764 | SQRN(&u, &t2p50m1, 50); /* 106 | 2^100 - 2^50 */ |
| 765 | f25519_mul(&t, &t2p50m1, &u); /* 107 | 2^100 - 1 */ |
| 766 | SQRN(&u, &t, 100); /* 207 | 2^200 - 2^100 */ |
| 767 | f25519_mul(&t, &t, &u); /* 208 | 2^200 - 1 */ |
| 768 | SQRN(&u, &t, 50); /* 258 | 2^250 - 2^50 */ |
| 769 | f25519_mul(&t, &t2p50m1, &u); /* 259 | 2^250 - 1 */ |
| 770 | SQRN(&u, &t, 5); /* 264 | 2^255 - 2^5 */ |
| 771 | f25519_mul(z, &u, &t11); /* 265 | 2^255 - 21 */ |
| 772 | |
| 773 | #undef SQRN |
| 774 | } |
| 775 | |
| 776 | /* --- @f25519_quosqrt@ --- * |
| 777 | * |
| 778 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) |
| 779 | * @const f25519 *x, *y@ = two operands |
| 780 | * |
| 781 | * Returns: Zero if successful, @-1@ if %$x/y$% is not a square. |
| 782 | * |
| 783 | * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%. |
| 784 | * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x |
| 785 | * \ne 0$% then the operation fails. If you wanted a specific |
| 786 | * square root then you'll have to pick it yourself. |
| 787 | */ |
| 788 | |
| 789 | static const piece sqrtm1_pieces[NPIECE] = { |
| 790 | -32595792, -7943725, 9377950, 3500415, 12389472, |
| 791 | -272473, -25146209, -2005654, 326686, 11406482 |
| 792 | }; |
| 793 | #define SQRTM1 ((const f25519 *)sqrtm1_pieces) |
| 794 | |
| 795 | int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y) |
| 796 | { |
| 797 | f25519 t, u, v, w, t15; |
| 798 | octet xb[32], b0[32], b1[32]; |
| 799 | int32 rc = -1; |
| 800 | mask32 m; |
| 801 | unsigned i; |
| 802 | |
| 803 | #define SQRN(z, x, n) do { \ |
| 804 | f25519_sqr((z), (x)); \ |
| 805 | for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \ |
| 806 | } while (0) |
| 807 | |
| 808 | /* This is a bit tricky; the algorithm is loosely based on Bernstein, Duif, |
| 809 | * Lange, Schwabe, and Yang, `High-speed high-security signatures', |
| 810 | * 2011-09-26, https://ed25519.cr.yp.to/ed25519-20110926.pdf. |
| 811 | */ |
| 812 | f25519_mul(&v, x, y); |
| 813 | |
| 814 | /* Now for an addition chain. */ /* step | value */ |
| 815 | f25519_sqr(&u, &v); /* 1 | 2 */ |
| 816 | f25519_mul(&t, &u, &v); /* 2 | 3 */ |
| 817 | SQRN(&u, &t, 2); /* 4 | 12 */ |
| 818 | f25519_mul(&t15, &u, &t); /* 5 | 15 */ |
| 819 | f25519_sqr(&u, &t15); /* 6 | 30 */ |
| 820 | f25519_mul(&t, &u, &v); /* 7 | 31 = 2^5 - 1 */ |
| 821 | SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */ |
| 822 | f25519_mul(&t, &u, &t); /* 13 | 2^10 - 1 */ |
| 823 | SQRN(&u, &t, 10); /* 23 | 2^20 - 2^10 */ |
| 824 | f25519_mul(&u, &u, &t); /* 24 | 2^20 - 1 */ |
| 825 | SQRN(&u, &u, 10); /* 34 | 2^30 - 2^10 */ |
| 826 | f25519_mul(&t, &u, &t); /* 35 | 2^30 - 1 */ |
| 827 | f25519_sqr(&u, &t); /* 36 | 2^31 - 2 */ |
| 828 | f25519_mul(&t, &u, &v); /* 37 | 2^31 - 1 */ |
| 829 | SQRN(&u, &t, 31); /* 68 | 2^62 - 2^31 */ |
| 830 | f25519_mul(&t, &u, &t); /* 69 | 2^62 - 1 */ |
| 831 | SQRN(&u, &t, 62); /* 131 | 2^124 - 2^62 */ |
| 832 | f25519_mul(&t, &u, &t); /* 132 | 2^124 - 1 */ |
| 833 | SQRN(&u, &t, 124); /* 256 | 2^248 - 2^124 */ |
| 834 | f25519_mul(&t, &u, &t); /* 257 | 2^248 - 1 */ |
| 835 | f25519_sqr(&u, &t); /* 258 | 2^249 - 2 */ |
| 836 | f25519_mul(&t, &u, &v); /* 259 | 2^249 - 1 */ |
| 837 | SQRN(&t, &t, 3); /* 262 | 2^252 - 8 */ |
| 838 | f25519_sqr(&u, &t); /* 263 | 2^253 - 16 */ |
| 839 | f25519_mul(&t, &u, &t); /* 264 | 3*2^252 - 24 */ |
| 840 | f25519_mul(&t, &t, &t15); /* 265 | 3*2^252 - 9 */ |
| 841 | f25519_mul(&w, &t, &v); /* 266 | 3*2^252 - 8 */ |
| 842 | |
| 843 | /* Awesome. Now let me explain. Let v be a square in GF(p), and let w = |
| 844 | * v^(3*2^252 - 8). In particular, let's consider |
| 845 | * |
| 846 | * v^2 w^4 = v^2 v^{3*2^254 - 32} = (v^{2^254 - 10})^3 |
| 847 | * |
| 848 | * But 2^254 - 10 = ((2^255 - 19) - 1)/2 = (p - 1)/2. Since v is a square, |
| 849 | * it has order dividing (p - 1)/2, and therefore v^2 w^4 = 1 and |
| 850 | * |
| 851 | * w^4 = 1/v^2 |
| 852 | * |
| 853 | * That in turn implies that w^2 = ±1/v. Now, recall that v = x y, and let |
| 854 | * w' = w x. Then w'^2 = ±x^2/v = ±x/y. If y w'^2 = x then we set |
| 855 | * z = w', since we have z^2 = x/y; otherwise let z = i w', where i^2 = -1, |
| 856 | * so z^2 = -w^2 = x/y, and we're done. |
| 857 | * |
| 858 | * The easiest way to compare is to encode. This isn't as wasteful as it |
| 859 | * sounds: the hard part is normalizing the representations, which we have |
| 860 | * to do anyway. |
| 861 | */ |
| 862 | f25519_mul(&w, &w, x); |
| 863 | f25519_sqr(&t, &w); |
| 864 | f25519_mul(&t, &t, y); |
| 865 | f25519_neg(&u, &t); |
| 866 | f25519_store(xb, x); |
| 867 | f25519_store(b0, &t); |
| 868 | f25519_store(b1, &u); |
| 869 | f25519_mul(&u, &w, SQRTM1); |
| 870 | |
| 871 | m = -consttime_memeq(b0, xb, 32); |
| 872 | rc = PICK2(0, rc, m); |
| 873 | f25519_pick2(z, &w, &u, m); |
| 874 | m = -consttime_memeq(b1, xb, 32); |
| 875 | rc = PICK2(0, rc, m); |
| 876 | |
| 877 | /* And we're done. */ |
| 878 | return (rc); |
| 879 | } |
| 880 | |
| 881 | /*----- That's all, folks -------------------------------------------------*/ |