1 // The MIT License (MIT) 2 // 3 // Copyright (c) 2015-2016 the fiat-crypto authors (see the AUTHORS file). 4 // 5 // Permission is hereby granted, free of charge, to any person obtaining a copy 6 // of this software and associated documentation files (the "Software"), to deal 7 // in the Software without restriction, including without limitation the rights 8 // to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 // copies of the Software, and to permit persons to whom the Software is 10 // furnished to do so, subject to the following conditions: 11 // 12 // The above copyright notice and this permission notice shall be included in all 13 // copies or substantial portions of the Software. 14 // 15 // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 // IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 // FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 // AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 // LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 // OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 // SOFTWARE. 22 23 // The field arithmetic code is generated by Fiat 24 // (https://github.com/mit-plv/fiat-crypto), which is MIT licensed. 25 // 26 // An implementation of the NIST P-256 elliptic curve point multiplication. 27 // 256-bit Montgomery form, generated using fiat-crypto, for 64 and 32-bit. 28 // Field operations with inputs in [0,p) return outputs in [0,p). 29 30 #include <openssl/base.h> 31 32 #include <openssl/bn.h> 33 #include <openssl/ec.h> 34 #include <openssl/err.h> 35 #include <openssl/mem.h> 36 #include <openssl/type_check.h> 37 38 #include <assert.h> 39 #include <string.h> 40 41 #include "../../crypto/fipsmodule/delocate.h" 42 #include "../../crypto/fipsmodule/ec/internal.h" 43 #include "../../crypto/internal.h" 44 45 46 // MSVC does not implement uint128_t, and crashes with intrinsics 47 #if defined(BORINGSSL_HAS_UINT128) 48 #define BORINGSSL_NISTP256_64BIT 1 49 #include "p256_64.h" 50 #else 51 #include "p256_32.h" 52 #endif 53 54 55 // utility functions, handwritten 56 57 #define NBYTES 32 58 59 #if defined(BORINGSSL_NISTP256_64BIT) 60 61 #define NLIMBS 4 62 typedef uint64_t limb_t; 63 typedef uint64_t fe[NLIMBS]; 64 #else // 64BIT; else 32BIT 65 66 #define NLIMBS 8 67 typedef uint32_t limb_t; 68 typedef uint32_t fe[NLIMBS]; 69 70 #endif // 64BIT 71 72 #define fe_add fiat_p256_add 73 #define fe_sub fiat_p256_sub 74 #define fe_opp fiat_p256_opp 75 76 #define fe_mul fiat_p256_mul 77 #define fe_sqr fiat_p256_square 78 79 #define fe_tobytes fiat_p256_to_bytes 80 #define fe_frombytes fiat_p256_from_bytes 81 82 static limb_t fe_nz(const limb_t in1[NLIMBS]) { 83 limb_t ret; 84 fiat_p256_nonzero(&ret, in1); 85 return ret; 86 } 87 88 static void fe_copy(limb_t out[NLIMBS], const limb_t in1[NLIMBS]) { 89 for (int i = 0; i < NLIMBS; i++) { 90 out[i] = in1[i]; 91 } 92 } 93 94 static void fe_cmovznz(limb_t out[NLIMBS], limb_t t, const limb_t z[NLIMBS], 95 const limb_t nz[NLIMBS]) { 96 fiat_p256_selectznz(out, !!t, z, nz); 97 } 98 99 static void fe_from_montgomery(fe x) { 100 fiat_p256_from_montgomery(x, x); 101 } 102 103 static void fe_from_generic(fe out, const EC_FELEM *in) { 104 fe_frombytes(out, in->bytes); 105 } 106 107 static void fe_to_generic(EC_FELEM *out, const fe in) { 108 // This works because 256 is a multiple of 64, so there are no excess bytes to 109 // zero when rounding up to |BN_ULONG|s. 110 OPENSSL_STATIC_ASSERT( 111 256 / 8 == sizeof(BN_ULONG) * ((256 + BN_BITS2 - 1) / BN_BITS2), 112 "fe_tobytes leaves bytes uninitialized"); 113 fe_tobytes(out->bytes, in); 114 } 115 116 // fe_inv calculates |out| = |in|^{-1} 117 // 118 // Based on Fermat's Little Theorem: 119 // a^p = a (mod p) 120 // a^{p-1} = 1 (mod p) 121 // a^{p-2} = a^{-1} (mod p) 122 static void fe_inv(fe out, const fe in) { 123 fe ftmp, ftmp2; 124 // each e_I will hold |in|^{2^I - 1} 125 fe e2, e4, e8, e16, e32, e64; 126 127 fe_sqr(ftmp, in); // 2^1 128 fe_mul(ftmp, in, ftmp); // 2^2 - 2^0 129 fe_copy(e2, ftmp); 130 fe_sqr(ftmp, ftmp); // 2^3 - 2^1 131 fe_sqr(ftmp, ftmp); // 2^4 - 2^2 132 fe_mul(ftmp, ftmp, e2); // 2^4 - 2^0 133 fe_copy(e4, ftmp); 134 fe_sqr(ftmp, ftmp); // 2^5 - 2^1 135 fe_sqr(ftmp, ftmp); // 2^6 - 2^2 136 fe_sqr(ftmp, ftmp); // 2^7 - 2^3 137 fe_sqr(ftmp, ftmp); // 2^8 - 2^4 138 fe_mul(ftmp, ftmp, e4); // 2^8 - 2^0 139 fe_copy(e8, ftmp); 140 for (size_t i = 0; i < 8; i++) { 141 fe_sqr(ftmp, ftmp); 142 } // 2^16 - 2^8 143 fe_mul(ftmp, ftmp, e8); // 2^16 - 2^0 144 fe_copy(e16, ftmp); 145 for (size_t i = 0; i < 16; i++) { 146 fe_sqr(ftmp, ftmp); 147 } // 2^32 - 2^16 148 fe_mul(ftmp, ftmp, e16); // 2^32 - 2^0 149 fe_copy(e32, ftmp); 150 for (size_t i = 0; i < 32; i++) { 151 fe_sqr(ftmp, ftmp); 152 } // 2^64 - 2^32 153 fe_copy(e64, ftmp); 154 fe_mul(ftmp, ftmp, in); // 2^64 - 2^32 + 2^0 155 for (size_t i = 0; i < 192; i++) { 156 fe_sqr(ftmp, ftmp); 157 } // 2^256 - 2^224 + 2^192 158 159 fe_mul(ftmp2, e64, e32); // 2^64 - 2^0 160 for (size_t i = 0; i < 16; i++) { 161 fe_sqr(ftmp2, ftmp2); 162 } // 2^80 - 2^16 163 fe_mul(ftmp2, ftmp2, e16); // 2^80 - 2^0 164 for (size_t i = 0; i < 8; i++) { 165 fe_sqr(ftmp2, ftmp2); 166 } // 2^88 - 2^8 167 fe_mul(ftmp2, ftmp2, e8); // 2^88 - 2^0 168 for (size_t i = 0; i < 4; i++) { 169 fe_sqr(ftmp2, ftmp2); 170 } // 2^92 - 2^4 171 fe_mul(ftmp2, ftmp2, e4); // 2^92 - 2^0 172 fe_sqr(ftmp2, ftmp2); // 2^93 - 2^1 173 fe_sqr(ftmp2, ftmp2); // 2^94 - 2^2 174 fe_mul(ftmp2, ftmp2, e2); // 2^94 - 2^0 175 fe_sqr(ftmp2, ftmp2); // 2^95 - 2^1 176 fe_sqr(ftmp2, ftmp2); // 2^96 - 2^2 177 fe_mul(ftmp2, ftmp2, in); // 2^96 - 3 178 179 fe_mul(out, ftmp2, ftmp); // 2^256 - 2^224 + 2^192 + 2^96 - 3 180 } 181 182 // Group operations 183 // ---------------- 184 // 185 // Building on top of the field operations we have the operations on the 186 // elliptic curve group itself. Points on the curve are represented in Jacobian 187 // coordinates. 188 // 189 // Both operations were transcribed to Coq and proven to correspond to naive 190 // implementations using Affine coordinates, for all suitable fields. In the 191 // Coq proofs, issues of constant-time execution and memory layout (aliasing) 192 // conventions were not considered. Specification of affine coordinates: 193 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Spec/WeierstrassCurve.v#L28> 194 // As a sanity check, a proof that these points form a commutative group: 195 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/AffineProofs.v#L33> 196 197 // point_double calculates 2*(x_in, y_in, z_in) 198 // 199 // The method is taken from: 200 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b 201 // 202 // Coq transcription and correctness proof: 203 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L93> 204 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L201> 205 // 206 // Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. 207 // while x_out == y_in is not (maybe this works, but it's not tested). 208 static void point_double(fe x_out, fe y_out, fe z_out, 209 const fe x_in, const fe y_in, const fe z_in) { 210 fe delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta; 211 // delta = z^2 212 fe_sqr(delta, z_in); 213 // gamma = y^2 214 fe_sqr(gamma, y_in); 215 // beta = x*gamma 216 fe_mul(beta, x_in, gamma); 217 218 // alpha = 3*(x-delta)*(x+delta) 219 fe_sub(ftmp, x_in, delta); 220 fe_add(ftmp2, x_in, delta); 221 222 fe_add(tmptmp, ftmp2, ftmp2); 223 fe_add(ftmp2, ftmp2, tmptmp); 224 fe_mul(alpha, ftmp, ftmp2); 225 226 // x' = alpha^2 - 8*beta 227 fe_sqr(x_out, alpha); 228 fe_add(fourbeta, beta, beta); 229 fe_add(fourbeta, fourbeta, fourbeta); 230 fe_add(tmptmp, fourbeta, fourbeta); 231 fe_sub(x_out, x_out, tmptmp); 232 233 // z' = (y + z)^2 - gamma - delta 234 fe_add(delta, gamma, delta); 235 fe_add(ftmp, y_in, z_in); 236 fe_sqr(z_out, ftmp); 237 fe_sub(z_out, z_out, delta); 238 239 // y' = alpha*(4*beta - x') - 8*gamma^2 240 fe_sub(y_out, fourbeta, x_out); 241 fe_add(gamma, gamma, gamma); 242 fe_sqr(gamma, gamma); 243 fe_mul(y_out, alpha, y_out); 244 fe_add(gamma, gamma, gamma); 245 fe_sub(y_out, y_out, gamma); 246 } 247 248 // point_add calcuates (x1, y1, z1) + (x2, y2, z2) 249 // 250 // The method is taken from: 251 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, 252 // adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). 253 // 254 // Coq transcription and correctness proof: 255 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L135> 256 // <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L205> 257 // 258 // This function includes a branch for checking whether the two input points 259 // are equal, (while not equal to the point at infinity). This case never 260 // happens during single point multiplication, so there is no timing leak for 261 // ECDH or ECDSA signing. 262 static void point_add(fe x3, fe y3, fe z3, const fe x1, 263 const fe y1, const fe z1, const int mixed, 264 const fe x2, const fe y2, const fe z2) { 265 fe x_out, y_out, z_out; 266 limb_t z1nz = fe_nz(z1); 267 limb_t z2nz = fe_nz(z2); 268 269 // z1z1 = z1z1 = z1**2 270 fe z1z1; fe_sqr(z1z1, z1); 271 272 fe u1, s1, two_z1z2; 273 if (!mixed) { 274 // z2z2 = z2**2 275 fe z2z2; fe_sqr(z2z2, z2); 276 277 // u1 = x1*z2z2 278 fe_mul(u1, x1, z2z2); 279 280 // two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 281 fe_add(two_z1z2, z1, z2); 282 fe_sqr(two_z1z2, two_z1z2); 283 fe_sub(two_z1z2, two_z1z2, z1z1); 284 fe_sub(two_z1z2, two_z1z2, z2z2); 285 286 // s1 = y1 * z2**3 287 fe_mul(s1, z2, z2z2); 288 fe_mul(s1, s1, y1); 289 } else { 290 // We'll assume z2 = 1 (special case z2 = 0 is handled later). 291 292 // u1 = x1*z2z2 293 fe_copy(u1, x1); 294 // two_z1z2 = 2z1z2 295 fe_add(two_z1z2, z1, z1); 296 // s1 = y1 * z2**3 297 fe_copy(s1, y1); 298 } 299 300 // u2 = x2*z1z1 301 fe u2; fe_mul(u2, x2, z1z1); 302 303 // h = u2 - u1 304 fe h; fe_sub(h, u2, u1); 305 306 limb_t xneq = fe_nz(h); 307 308 // z_out = two_z1z2 * h 309 fe_mul(z_out, h, two_z1z2); 310 311 // z1z1z1 = z1 * z1z1 312 fe z1z1z1; fe_mul(z1z1z1, z1, z1z1); 313 314 // s2 = y2 * z1**3 315 fe s2; fe_mul(s2, y2, z1z1z1); 316 317 // r = (s2 - s1)*2 318 fe r; 319 fe_sub(r, s2, s1); 320 fe_add(r, r, r); 321 322 limb_t yneq = fe_nz(r); 323 324 if (!xneq && !yneq && z1nz && z2nz) { 325 point_double(x3, y3, z3, x1, y1, z1); 326 return; 327 } 328 329 // I = (2h)**2 330 fe i; 331 fe_add(i, h, h); 332 fe_sqr(i, i); 333 334 // J = h * I 335 fe j; fe_mul(j, h, i); 336 337 // V = U1 * I 338 fe v; fe_mul(v, u1, i); 339 340 // x_out = r**2 - J - 2V 341 fe_sqr(x_out, r); 342 fe_sub(x_out, x_out, j); 343 fe_sub(x_out, x_out, v); 344 fe_sub(x_out, x_out, v); 345 346 // y_out = r(V-x_out) - 2 * s1 * J 347 fe_sub(y_out, v, x_out); 348 fe_mul(y_out, y_out, r); 349 fe s1j; 350 fe_mul(s1j, s1, j); 351 fe_sub(y_out, y_out, s1j); 352 fe_sub(y_out, y_out, s1j); 353 354 fe_cmovznz(x_out, z1nz, x2, x_out); 355 fe_cmovznz(x3, z2nz, x1, x_out); 356 fe_cmovznz(y_out, z1nz, y2, y_out); 357 fe_cmovznz(y3, z2nz, y1, y_out); 358 fe_cmovznz(z_out, z1nz, z2, z_out); 359 fe_cmovznz(z3, z2nz, z1, z_out); 360 } 361 362 // Base point pre computation 363 // -------------------------- 364 // 365 // Two different sorts of precomputed tables are used in the following code. 366 // Each contain various points on the curve, where each point is three field 367 // elements (x, y, z). 368 // 369 // For the base point table, z is usually 1 (0 for the point at infinity). 370 // This table has 2 * 16 elements, starting with the following: 371 // index | bits | point 372 // ------+---------+------------------------------ 373 // 0 | 0 0 0 0 | 0G 374 // 1 | 0 0 0 1 | 1G 375 // 2 | 0 0 1 0 | 2^64G 376 // 3 | 0 0 1 1 | (2^64 + 1)G 377 // 4 | 0 1 0 0 | 2^128G 378 // 5 | 0 1 0 1 | (2^128 + 1)G 379 // 6 | 0 1 1 0 | (2^128 + 2^64)G 380 // 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G 381 // 8 | 1 0 0 0 | 2^192G 382 // 9 | 1 0 0 1 | (2^192 + 1)G 383 // 10 | 1 0 1 0 | (2^192 + 2^64)G 384 // 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G 385 // 12 | 1 1 0 0 | (2^192 + 2^128)G 386 // 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G 387 // 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G 388 // 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G 389 // followed by a copy of this with each element multiplied by 2^32. 390 // 391 // The reason for this is so that we can clock bits into four different 392 // locations when doing simple scalar multiplies against the base point, 393 // and then another four locations using the second 16 elements. 394 // 395 // Tables for other points have table[i] = iG for i in 0 .. 16. 396 397 // g_pre_comp is the table of precomputed base points 398 #if defined(BORINGSSL_NISTP256_64BIT) 399 static const fe g_pre_comp[2][16][3] = { 400 {{{0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}}, 401 {{0x79e730d418a9143c, 0x75ba95fc5fedb601, 0x79fb732b77622510, 402 0x18905f76a53755c6}, 403 {0xddf25357ce95560a, 0x8b4ab8e4ba19e45c, 0xd2e88688dd21f325, 404 0x8571ff1825885d85}, 405 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 406 {{0x4f922fc516a0d2bb, 0xd5cc16c1a623499, 0x9241cf3a57c62c8b, 407 0x2f5e6961fd1b667f}, 408 {0x5c15c70bf5a01797, 0x3d20b44d60956192, 0x4911b37071fdb52, 409 0xf648f9168d6f0f7b}, 410 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 411 {{0x9e566847e137bbbc, 0xe434469e8a6a0bec, 0xb1c4276179d73463, 412 0x5abe0285133d0015}, 413 {0x92aa837cc04c7dab, 0x573d9f4c43260c07, 0xc93156278e6cc37, 414 0x94bb725b6b6f7383}, 415 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 416 {{0x62a8c244bfe20925, 0x91c19ac38fdce867, 0x5a96a5d5dd387063, 417 0x61d587d421d324f6}, 418 {0xe87673a2a37173ea, 0x2384800853778b65, 0x10f8441e05bab43e, 419 0xfa11fe124621efbe}, 420 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 421 {{0x1c891f2b2cb19ffd, 0x1ba8d5bb1923c23, 0xb6d03d678ac5ca8e, 422 0x586eb04c1f13bedc}, 423 {0xc35c6e527e8ed09, 0x1e81a33c1819ede2, 0x278fd6c056c652fa, 424 0x19d5ac0870864f11}, 425 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 426 {{0x62577734d2b533d5, 0x673b8af6a1bdddc0, 0x577e7c9aa79ec293, 427 0xbb6de651c3b266b1}, 428 {0xe7e9303ab65259b3, 0xd6a0afd3d03a7480, 0xc5ac83d19b3cfc27, 429 0x60b4619a5d18b99b}, 430 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 431 {{0xbd6a38e11ae5aa1c, 0xb8b7652b49e73658, 0xb130014ee5f87ed, 432 0x9d0f27b2aeebffcd}, 433 {0xca9246317a730a55, 0x9c955b2fddbbc83a, 0x7c1dfe0ac019a71, 434 0x244a566d356ec48d}, 435 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 436 {{0x56f8410ef4f8b16a, 0x97241afec47b266a, 0xa406b8e6d9c87c1, 437 0x803f3e02cd42ab1b}, 438 {0x7f0309a804dbec69, 0xa83b85f73bbad05f, 0xc6097273ad8e197f, 439 0xc097440e5067adc1}, 440 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 441 {{0x846a56f2c379ab34, 0xa8ee068b841df8d1, 0x20314459176c68ef, 442 0xf1af32d5915f1f30}, 443 {0x99c375315d75bd50, 0x837cffbaf72f67bc, 0x613a41848d7723f, 444 0x23d0f130e2d41c8b}, 445 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 446 {{0xed93e225d5be5a2b, 0x6fe799835934f3c6, 0x4314092622626ffc, 447 0x50bbb4d97990216a}, 448 {0x378191c6e57ec63e, 0x65422c40181dcdb2, 0x41a8099b0236e0f6, 449 0x2b10011801fe49c3}, 450 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 451 {{0xfc68b5c59b391593, 0xc385f5a2598270fc, 0x7144f3aad19adcbb, 452 0xdd55899983fbae0c}, 453 {0x93b88b8e74b82ff4, 0xd2e03c4071e734c9, 0x9a7a9eaf43c0322a, 454 0xe6e4c551149d6041}, 455 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 456 {{0x5fe14bfe80ec21fe, 0xf6ce116ac255be82, 0x98bc5a072f4a5d67, 457 0xfad27148db7e63af}, 458 {0x90c0b6ac29ab05b3, 0x37a9a83c4e251ae6, 0xa7dc875c2aade7d, 459 0x77387de39f0e1a84}, 460 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 461 {{0x1e9ecc49a56c0dd7, 0xa5cffcd846086c74, 0x8f7a1408f505aece, 462 0xb37b85c0bef0c47e}, 463 {0x3596b6e4cc0e6a8f, 0xfd6d4bbf6b388f23, 0xaba453fac39cef4e, 464 0x9c135ac8f9f628d5}, 465 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 466 {{0xa1c729495c8f8be, 0x2961c4803bf362bf, 0x9e418403df63d4ac, 467 0xc109f9cb91ece900}, 468 {0xc2d095d058945705, 0xb9083d96ddeb85c0, 0x84692b8d7a40449b, 469 0x9bc3344f2eee1ee1}, 470 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 471 {{0xd5ae35642913074, 0x55491b2748a542b1, 0x469ca665b310732a, 472 0x29591d525f1a4cc1}, 473 {0xe76f5b6bb84f983f, 0xbe7eef419f5f84e1, 0x1200d49680baa189, 474 0x6376551f18ef332c}, 475 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}}, 476 {{{0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}, {0x0, 0x0, 0x0, 0x0}}, 477 {{0x202886024147519a, 0xd0981eac26b372f0, 0xa9d4a7caa785ebc8, 478 0xd953c50ddbdf58e9}, 479 {0x9d6361ccfd590f8f, 0x72e9626b44e6c917, 0x7fd9611022eb64cf, 480 0x863ebb7e9eb288f3}, 481 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 482 {{0x4fe7ee31b0e63d34, 0xf4600572a9e54fab, 0xc0493334d5e7b5a4, 483 0x8589fb9206d54831}, 484 {0xaa70f5cc6583553a, 0x879094ae25649e5, 0xcc90450710044652, 485 0xebb0696d02541c4f}, 486 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 487 {{0xabbaa0c03b89da99, 0xa6f2d79eb8284022, 0x27847862b81c05e8, 488 0x337a4b5905e54d63}, 489 {0x3c67500d21f7794a, 0x207005b77d6d7f61, 0xa5a378104cfd6e8, 490 0xd65e0d5f4c2fbd6}, 491 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 492 {{0xd433e50f6d3549cf, 0x6f33696ffacd665e, 0x695bfdacce11fcb4, 493 0x810ee252af7c9860}, 494 {0x65450fe17159bb2c, 0xf7dfbebe758b357b, 0x2b057e74d69fea72, 495 0xd485717a92731745}, 496 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 497 {{0xce1f69bbe83f7669, 0x9f8ae8272877d6b, 0x9548ae543244278d, 498 0x207755dee3c2c19c}, 499 {0x87bd61d96fef1945, 0x18813cefb12d28c3, 0x9fbcd1d672df64aa, 500 0x48dc5ee57154b00d}, 501 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 502 {{0xef0f469ef49a3154, 0x3e85a5956e2b2e9a, 0x45aaec1eaa924a9c, 503 0xaa12dfc8a09e4719}, 504 {0x26f272274df69f1d, 0xe0e4c82ca2ff5e73, 0xb9d8ce73b7a9dd44, 505 0x6c036e73e48ca901}, 506 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 507 {{0xe1e421e1a47153f0, 0xb86c3b79920418c9, 0x93bdce87705d7672, 508 0xf25ae793cab79a77}, 509 {0x1f3194a36d869d0c, 0x9d55c8824986c264, 0x49fb5ea3096e945e, 510 0x39b8e65313db0a3e}, 511 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 512 {{0xe3417bc035d0b34a, 0x440b386b8327c0a7, 0x8fb7262dac0362d1, 513 0x2c41114ce0cdf943}, 514 {0x2ba5cef1ad95a0b1, 0xc09b37a867d54362, 0x26d6cdd201e486c9, 515 0x20477abf42ff9297}, 516 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 517 {{0xf121b41bc0a67d2, 0x62d4760a444d248a, 0xe044f1d659b4737, 518 0x8fde365250bb4a8}, 519 {0xaceec3da848bf287, 0xc2a62182d3369d6e, 0x3582dfdc92449482, 520 0x2f7e2fd2565d6cd7}, 521 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 522 {{0xa0122b5178a876b, 0x51ff96ff085104b4, 0x50b31ab14f29f76, 523 0x84abb28b5f87d4e6}, 524 {0xd5ed439f8270790a, 0x2d6cb59d85e3f46b, 0x75f55c1b6c1e2212, 525 0xe5436f6717655640}, 526 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 527 {{0xc2965ecc9aeb596d, 0x1ea03e7023c92b4, 0x4704b4b62e013961, 528 0xca8fd3f905ea367}, 529 {0x92523a42551b2b61, 0x1eb7a89c390fcd06, 0xe7f1d2be0392a63e, 530 0x96dca2644ddb0c33}, 531 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 532 {{0x231c210e15339848, 0xe87a28e870778c8d, 0x9d1de6616956e170, 533 0x4ac3c9382bb09c0b}, 534 {0x19be05516998987d, 0x8b2376c4ae09f4d6, 0x1de0b7651a3f933d, 535 0x380d94c7e39705f4}, 536 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 537 {{0x3685954b8c31c31d, 0x68533d005bf21a0c, 0xbd7626e75c79ec9, 538 0xca17754742c69d54}, 539 {0xcc6edafff6d2dbb2, 0xfd0d8cbd174a9d18, 0x875e8793aa4578e8, 540 0xa976a7139cab2ce6}, 541 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 542 {{0xce37ab11b43ea1db, 0xa7ff1a95259d292, 0x851b02218f84f186, 543 0xa7222beadefaad13}, 544 {0xa2ac78ec2b0a9144, 0x5a024051f2fa59c5, 0x91d1eca56147ce38, 545 0xbe94d523bc2ac690}, 546 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}, 547 {{0x2d8daefd79ec1a0f, 0x3bbcd6fdceb39c97, 0xf5575ffc58f61a95, 548 0xdbd986c4adf7b420}, 549 {0x81aa881415f39eb7, 0x6ee2fcf5b98d976c, 0x5465475dcf2f717d, 550 0x8e24d3c46860bbd0}, 551 {0x1, 0xffffffff00000000, 0xffffffffffffffff, 0xfffffffe}}}}; 552 #else 553 static const fe g_pre_comp[2][16][3] = { 554 {{{0x0,0x0, 0x0,0x0, 0x0,0x0, 0x0,0x0}, 555 {0x0,0x0, 0x0,0x0, 0x0,0x0, 0x0,0x0}, 556 {0x0,0x0, 0x0,0x0, 0x0,0x0, 0x0,0x0}}, 557 {{0x18a9143c,0x79e730d4, 0x5fedb601,0x75ba95fc, 0x77622510,0x79fb732b, 558 0xa53755c6,0x18905f76}, 559 {0xce95560a,0xddf25357, 0xba19e45c,0x8b4ab8e4, 0xdd21f325,0xd2e88688, 560 0x25885d85,0x8571ff18}, 561 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 562 {{0x16a0d2bb,0x4f922fc5, 0x1a623499,0xd5cc16c, 0x57c62c8b,0x9241cf3a, 563 0xfd1b667f,0x2f5e6961}, 564 {0xf5a01797,0x5c15c70b, 0x60956192,0x3d20b44d, 0x71fdb52,0x4911b37, 565 0x8d6f0f7b,0xf648f916}, 566 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 567 {{0xe137bbbc,0x9e566847, 0x8a6a0bec,0xe434469e, 0x79d73463,0xb1c42761, 568 0x133d0015,0x5abe0285}, 569 {0xc04c7dab,0x92aa837c, 0x43260c07,0x573d9f4c, 0x78e6cc37,0xc931562, 570 0x6b6f7383,0x94bb725b}, 571 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 572 {{0xbfe20925,0x62a8c244, 0x8fdce867,0x91c19ac3, 0xdd387063,0x5a96a5d5, 573 0x21d324f6,0x61d587d4}, 574 {0xa37173ea,0xe87673a2, 0x53778b65,0x23848008, 0x5bab43e,0x10f8441e, 575 0x4621efbe,0xfa11fe12}, 576 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 577 {{0x2cb19ffd,0x1c891f2b, 0xb1923c23,0x1ba8d5b, 0x8ac5ca8e,0xb6d03d67, 578 0x1f13bedc,0x586eb04c}, 579 {0x27e8ed09,0xc35c6e5, 0x1819ede2,0x1e81a33c, 0x56c652fa,0x278fd6c0, 580 0x70864f11,0x19d5ac08}, 581 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 582 {{0xd2b533d5,0x62577734, 0xa1bdddc0,0x673b8af6, 0xa79ec293,0x577e7c9a, 583 0xc3b266b1,0xbb6de651}, 584 {0xb65259b3,0xe7e9303a, 0xd03a7480,0xd6a0afd3, 0x9b3cfc27,0xc5ac83d1, 585 0x5d18b99b,0x60b4619a}, 586 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 587 {{0x1ae5aa1c,0xbd6a38e1, 0x49e73658,0xb8b7652b, 0xee5f87ed,0xb130014, 588 0xaeebffcd,0x9d0f27b2}, 589 {0x7a730a55,0xca924631, 0xddbbc83a,0x9c955b2f, 0xac019a71,0x7c1dfe0, 590 0x356ec48d,0x244a566d}, 591 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 592 {{0xf4f8b16a,0x56f8410e, 0xc47b266a,0x97241afe, 0x6d9c87c1,0xa406b8e, 593 0xcd42ab1b,0x803f3e02}, 594 {0x4dbec69,0x7f0309a8, 0x3bbad05f,0xa83b85f7, 0xad8e197f,0xc6097273, 595 0x5067adc1,0xc097440e}, 596 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 597 {{0xc379ab34,0x846a56f2, 0x841df8d1,0xa8ee068b, 0x176c68ef,0x20314459, 598 0x915f1f30,0xf1af32d5}, 599 {0x5d75bd50,0x99c37531, 0xf72f67bc,0x837cffba, 0x48d7723f,0x613a418, 600 0xe2d41c8b,0x23d0f130}, 601 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 602 {{0xd5be5a2b,0xed93e225, 0x5934f3c6,0x6fe79983, 0x22626ffc,0x43140926, 603 0x7990216a,0x50bbb4d9}, 604 {0xe57ec63e,0x378191c6, 0x181dcdb2,0x65422c40, 0x236e0f6,0x41a8099b, 605 0x1fe49c3,0x2b100118}, 606 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 607 {{0x9b391593,0xfc68b5c5, 0x598270fc,0xc385f5a2, 0xd19adcbb,0x7144f3aa, 608 0x83fbae0c,0xdd558999}, 609 {0x74b82ff4,0x93b88b8e, 0x71e734c9,0xd2e03c40, 0x43c0322a,0x9a7a9eaf, 610 0x149d6041,0xe6e4c551}, 611 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 612 {{0x80ec21fe,0x5fe14bfe, 0xc255be82,0xf6ce116a, 0x2f4a5d67,0x98bc5a07, 613 0xdb7e63af,0xfad27148}, 614 {0x29ab05b3,0x90c0b6ac, 0x4e251ae6,0x37a9a83c, 0xc2aade7d,0xa7dc875, 615 0x9f0e1a84,0x77387de3}, 616 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 617 {{0xa56c0dd7,0x1e9ecc49, 0x46086c74,0xa5cffcd8, 0xf505aece,0x8f7a1408, 618 0xbef0c47e,0xb37b85c0}, 619 {0xcc0e6a8f,0x3596b6e4, 0x6b388f23,0xfd6d4bbf, 0xc39cef4e,0xaba453fa, 620 0xf9f628d5,0x9c135ac8}, 621 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 622 {{0x95c8f8be,0xa1c7294, 0x3bf362bf,0x2961c480, 0xdf63d4ac,0x9e418403, 623 0x91ece900,0xc109f9cb}, 624 {0x58945705,0xc2d095d0, 0xddeb85c0,0xb9083d96, 0x7a40449b,0x84692b8d, 625 0x2eee1ee1,0x9bc3344f}, 626 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 627 {{0x42913074,0xd5ae356, 0x48a542b1,0x55491b27, 0xb310732a,0x469ca665, 628 0x5f1a4cc1,0x29591d52}, 629 {0xb84f983f,0xe76f5b6b, 0x9f5f84e1,0xbe7eef41, 0x80baa189,0x1200d496, 630 0x18ef332c,0x6376551f}, 631 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}}, 632 {{{0x0,0x0, 0x0,0x0, 0x0,0x0, 0x0,0x0}, 633 {0x0,0x0, 0x0,0x0, 0x0,0x0, 0x0,0x0}, 634 {0x0,0x0, 0x0,0x0, 0x0,0x0, 0x0,0x0}}, 635 {{0x4147519a,0x20288602, 0x26b372f0,0xd0981eac, 0xa785ebc8,0xa9d4a7ca, 636 0xdbdf58e9,0xd953c50d}, 637 {0xfd590f8f,0x9d6361cc, 0x44e6c917,0x72e9626b, 0x22eb64cf,0x7fd96110, 638 0x9eb288f3,0x863ebb7e}, 639 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 640 {{0xb0e63d34,0x4fe7ee31, 0xa9e54fab,0xf4600572, 0xd5e7b5a4,0xc0493334, 641 0x6d54831,0x8589fb92}, 642 {0x6583553a,0xaa70f5cc, 0xe25649e5,0x879094a, 0x10044652,0xcc904507, 643 0x2541c4f,0xebb0696d}, 644 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 645 {{0x3b89da99,0xabbaa0c0, 0xb8284022,0xa6f2d79e, 0xb81c05e8,0x27847862, 646 0x5e54d63,0x337a4b59}, 647 {0x21f7794a,0x3c67500d, 0x7d6d7f61,0x207005b7, 0x4cfd6e8,0xa5a3781, 648 0xf4c2fbd6,0xd65e0d5}, 649 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 650 {{0x6d3549cf,0xd433e50f, 0xfacd665e,0x6f33696f, 0xce11fcb4,0x695bfdac, 651 0xaf7c9860,0x810ee252}, 652 {0x7159bb2c,0x65450fe1, 0x758b357b,0xf7dfbebe, 0xd69fea72,0x2b057e74, 653 0x92731745,0xd485717a}, 654 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 655 {{0xe83f7669,0xce1f69bb, 0x72877d6b,0x9f8ae82, 0x3244278d,0x9548ae54, 656 0xe3c2c19c,0x207755de}, 657 {0x6fef1945,0x87bd61d9, 0xb12d28c3,0x18813cef, 0x72df64aa,0x9fbcd1d6, 658 0x7154b00d,0x48dc5ee5}, 659 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 660 {{0xf49a3154,0xef0f469e, 0x6e2b2e9a,0x3e85a595, 0xaa924a9c,0x45aaec1e, 661 0xa09e4719,0xaa12dfc8}, 662 {0x4df69f1d,0x26f27227, 0xa2ff5e73,0xe0e4c82c, 0xb7a9dd44,0xb9d8ce73, 663 0xe48ca901,0x6c036e73}, 664 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 665 {{0xa47153f0,0xe1e421e1, 0x920418c9,0xb86c3b79, 0x705d7672,0x93bdce87, 666 0xcab79a77,0xf25ae793}, 667 {0x6d869d0c,0x1f3194a3, 0x4986c264,0x9d55c882, 0x96e945e,0x49fb5ea3, 668 0x13db0a3e,0x39b8e653}, 669 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 670 {{0x35d0b34a,0xe3417bc0, 0x8327c0a7,0x440b386b, 0xac0362d1,0x8fb7262d, 671 0xe0cdf943,0x2c41114c}, 672 {0xad95a0b1,0x2ba5cef1, 0x67d54362,0xc09b37a8, 0x1e486c9,0x26d6cdd2, 673 0x42ff9297,0x20477abf}, 674 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 675 {{0xbc0a67d2,0xf121b41, 0x444d248a,0x62d4760a, 0x659b4737,0xe044f1d, 676 0x250bb4a8,0x8fde365}, 677 {0x848bf287,0xaceec3da, 0xd3369d6e,0xc2a62182, 0x92449482,0x3582dfdc, 678 0x565d6cd7,0x2f7e2fd2}, 679 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 680 {{0x178a876b,0xa0122b5, 0x85104b4,0x51ff96ff, 0x14f29f76,0x50b31ab, 681 0x5f87d4e6,0x84abb28b}, 682 {0x8270790a,0xd5ed439f, 0x85e3f46b,0x2d6cb59d, 0x6c1e2212,0x75f55c1b, 683 0x17655640,0xe5436f67}, 684 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 685 {{0x9aeb596d,0xc2965ecc, 0x23c92b4,0x1ea03e7, 0x2e013961,0x4704b4b6, 686 0x905ea367,0xca8fd3f}, 687 {0x551b2b61,0x92523a42, 0x390fcd06,0x1eb7a89c, 0x392a63e,0xe7f1d2be, 688 0x4ddb0c33,0x96dca264}, 689 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 690 {{0x15339848,0x231c210e, 0x70778c8d,0xe87a28e8, 0x6956e170,0x9d1de661, 691 0x2bb09c0b,0x4ac3c938}, 692 {0x6998987d,0x19be0551, 0xae09f4d6,0x8b2376c4, 0x1a3f933d,0x1de0b765, 693 0xe39705f4,0x380d94c7}, 694 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 695 {{0x8c31c31d,0x3685954b, 0x5bf21a0c,0x68533d00, 0x75c79ec9,0xbd7626e, 696 0x42c69d54,0xca177547}, 697 {0xf6d2dbb2,0xcc6edaff, 0x174a9d18,0xfd0d8cbd, 0xaa4578e8,0x875e8793, 698 0x9cab2ce6,0xa976a713}, 699 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 700 {{0xb43ea1db,0xce37ab11, 0x5259d292,0xa7ff1a9, 0x8f84f186,0x851b0221, 701 0xdefaad13,0xa7222bea}, 702 {0x2b0a9144,0xa2ac78ec, 0xf2fa59c5,0x5a024051, 0x6147ce38,0x91d1eca5, 703 0xbc2ac690,0xbe94d523}, 704 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}, 705 {{0x79ec1a0f,0x2d8daefd, 0xceb39c97,0x3bbcd6fd, 0x58f61a95,0xf5575ffc, 706 0xadf7b420,0xdbd986c4}, 707 {0x15f39eb7,0x81aa8814, 0xb98d976c,0x6ee2fcf5, 0xcf2f717d,0x5465475d, 708 0x6860bbd0,0x8e24d3c4}, 709 {0x1,0x0, 0x0,0xffffffff, 0xffffffff,0xffffffff, 0xfffffffe,0x0}}}}; 710 #endif 711 712 // select_point selects the |idx|th point from a precomputation table and 713 // copies it to out. 714 static void select_point(const limb_t idx, size_t size, 715 const fe pre_comp[/*size*/][3], 716 fe out[3]) { 717 OPENSSL_memset(out, 0, sizeof(fe) * 3); 718 for (size_t i = 0; i < size; i++) { 719 limb_t mismatch = i ^ idx; 720 fe_cmovznz(out[0], mismatch, pre_comp[i][0], out[0]); 721 fe_cmovznz(out[1], mismatch, pre_comp[i][1], out[1]); 722 fe_cmovznz(out[2], mismatch, pre_comp[i][2], out[2]); 723 } 724 } 725 726 // get_bit returns the |i|th bit in |in| 727 static char get_bit(const uint8_t *in, int i) { 728 if (i < 0 || i >= 256) { 729 return 0; 730 } 731 return (in[i >> 3] >> (i & 7)) & 1; 732 } 733 734 // Interleaved point multiplication using precomputed point multiples: The 735 // small point multiples 0*P, 1*P, ..., 17*P are in p_pre_comp, the scalar 736 // in p_scalar, if non-NULL. If g_scalar is non-NULL, we also add this multiple 737 // of the generator, using certain (large) precomputed multiples in g_pre_comp. 738 // Output point (X, Y, Z) is stored in x_out, y_out, z_out. 739 static void batch_mul(fe x_out, fe y_out, fe z_out, 740 const uint8_t *p_scalar, const uint8_t *g_scalar, 741 const fe p_pre_comp[17][3]) { 742 // set nq to the point at infinity 743 fe nq[3] = {{0},{0},{0}}, ftmp, tmp[3]; 744 uint64_t bits; 745 uint8_t sign, digit; 746 747 // Loop over both scalars msb-to-lsb, interleaving additions of multiples 748 // of the generator (two in each of the last 32 rounds) and additions of p 749 // (every 5th round). 750 751 int skip = 1; // save two point operations in the first round 752 size_t i = p_scalar != NULL ? 255 : 31; 753 for (;;) { 754 // double 755 if (!skip) { 756 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 757 } 758 759 // add multiples of the generator 760 if (g_scalar != NULL && i <= 31) { 761 // first, look 32 bits upwards 762 bits = get_bit(g_scalar, i + 224) << 3; 763 bits |= get_bit(g_scalar, i + 160) << 2; 764 bits |= get_bit(g_scalar, i + 96) << 1; 765 bits |= get_bit(g_scalar, i + 32); 766 // select the point to add, in constant time 767 select_point(bits, 16, g_pre_comp[1], tmp); 768 769 if (!skip) { 770 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 771 tmp[0], tmp[1], tmp[2]); 772 } else { 773 fe_copy(nq[0], tmp[0]); 774 fe_copy(nq[1], tmp[1]); 775 fe_copy(nq[2], tmp[2]); 776 skip = 0; 777 } 778 779 // second, look at the current position 780 bits = get_bit(g_scalar, i + 192) << 3; 781 bits |= get_bit(g_scalar, i + 128) << 2; 782 bits |= get_bit(g_scalar, i + 64) << 1; 783 bits |= get_bit(g_scalar, i); 784 // select the point to add, in constant time 785 select_point(bits, 16, g_pre_comp[0], tmp); 786 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0], 787 tmp[1], tmp[2]); 788 } 789 790 // do other additions every 5 doublings 791 if (p_scalar != NULL && i % 5 == 0) { 792 bits = get_bit(p_scalar, i + 4) << 5; 793 bits |= get_bit(p_scalar, i + 3) << 4; 794 bits |= get_bit(p_scalar, i + 2) << 3; 795 bits |= get_bit(p_scalar, i + 1) << 2; 796 bits |= get_bit(p_scalar, i) << 1; 797 bits |= get_bit(p_scalar, i - 1); 798 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 799 800 // select the point to add or subtract, in constant time. 801 select_point(digit, 17, p_pre_comp, tmp); 802 fe_opp(ftmp, tmp[1]); // (X, -Y, Z) is the negative point. 803 fe_cmovznz(tmp[1], sign, tmp[1], ftmp); 804 805 if (!skip) { 806 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, 807 tmp[0], tmp[1], tmp[2]); 808 } else { 809 fe_copy(nq[0], tmp[0]); 810 fe_copy(nq[1], tmp[1]); 811 fe_copy(nq[2], tmp[2]); 812 skip = 0; 813 } 814 } 815 816 if (i == 0) { 817 break; 818 } 819 --i; 820 } 821 fe_copy(x_out, nq[0]); 822 fe_copy(y_out, nq[1]); 823 fe_copy(z_out, nq[2]); 824 } 825 826 // OPENSSL EC_METHOD FUNCTIONS 827 828 // Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = 829 // (X/Z^2, Y/Z^3). 830 static int ec_GFp_nistp256_point_get_affine_coordinates( 831 const EC_GROUP *group, const EC_RAW_POINT *point, EC_FELEM *x_out, 832 EC_FELEM *y_out) { 833 if (ec_GFp_simple_is_at_infinity(group, point)) { 834 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); 835 return 0; 836 } 837 838 fe z1, z2; 839 fe_from_generic(z1, &point->Z); 840 fe_inv(z2, z1); 841 fe_sqr(z1, z2); 842 843 // Instead of using |fe_from_montgomery| to convert the |x| coordinate and 844 // then calling |fe_from_montgomery| again to convert the |y| coordinate 845 // below, convert the common factor |z1| once now, saving one reduction. 846 fe_from_montgomery(z1); 847 848 if (x_out != NULL) { 849 fe x; 850 fe_from_generic(x, &point->X); 851 fe_mul(x, x, z1); 852 fe_to_generic(x_out, x); 853 } 854 855 if (y_out != NULL) { 856 fe y; 857 fe_from_generic(y, &point->Y); 858 fe_mul(z1, z1, z2); 859 fe_mul(y, y, z1); 860 fe_to_generic(y_out, y); 861 } 862 863 return 1; 864 } 865 866 static void ec_GFp_nistp256_add(const EC_GROUP *group, EC_RAW_POINT *r, 867 const EC_RAW_POINT *a, const EC_RAW_POINT *b) { 868 fe x1, y1, z1, x2, y2, z2; 869 fe_from_generic(x1, &a->X); 870 fe_from_generic(y1, &a->Y); 871 fe_from_generic(z1, &a->Z); 872 fe_from_generic(x2, &b->X); 873 fe_from_generic(y2, &b->Y); 874 fe_from_generic(z2, &b->Z); 875 point_add(x1, y1, z1, x1, y1, z1, 0 /* both Jacobian */, x2, y2, z2); 876 fe_to_generic(&r->X, x1); 877 fe_to_generic(&r->Y, y1); 878 fe_to_generic(&r->Z, z1); 879 } 880 881 static void ec_GFp_nistp256_dbl(const EC_GROUP *group, EC_RAW_POINT *r, 882 const EC_RAW_POINT *a) { 883 fe x, y, z; 884 fe_from_generic(x, &a->X); 885 fe_from_generic(y, &a->Y); 886 fe_from_generic(z, &a->Z); 887 point_double(x, y, z, x, y, z); 888 fe_to_generic(&r->X, x); 889 fe_to_generic(&r->Y, y); 890 fe_to_generic(&r->Z, z); 891 } 892 893 static void ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_RAW_POINT *r, 894 const EC_SCALAR *g_scalar, 895 const EC_RAW_POINT *p, 896 const EC_SCALAR *p_scalar) { 897 fe p_pre_comp[17][3]; 898 fe x_out, y_out, z_out; 899 900 if (p != NULL && p_scalar != NULL) { 901 // We treat NULL scalars as 0, and NULL points as points at infinity, i.e., 902 // they contribute nothing to the linear combination. 903 OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp)); 904 // Precompute multiples. 905 fe_from_generic(p_pre_comp[1][0], &p->X); 906 fe_from_generic(p_pre_comp[1][1], &p->Y); 907 fe_from_generic(p_pre_comp[1][2], &p->Z); 908 for (size_t j = 2; j <= 16; ++j) { 909 if (j & 1) { 910 point_add(p_pre_comp[j][0], p_pre_comp[j][1], 911 p_pre_comp[j][2], p_pre_comp[1][0], 912 p_pre_comp[1][1], p_pre_comp[1][2], 913 0, 914 p_pre_comp[j - 1][0], p_pre_comp[j - 1][1], 915 p_pre_comp[j - 1][2]); 916 } else { 917 point_double(p_pre_comp[j][0], p_pre_comp[j][1], 918 p_pre_comp[j][2], p_pre_comp[j / 2][0], 919 p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]); 920 } 921 } 922 } 923 924 batch_mul(x_out, y_out, z_out, 925 (p != NULL && p_scalar != NULL) ? p_scalar->bytes : NULL, 926 g_scalar != NULL ? g_scalar->bytes : NULL, 927 (const fe (*) [3])p_pre_comp); 928 929 fe_to_generic(&r->X, x_out); 930 fe_to_generic(&r->Y, y_out); 931 fe_to_generic(&r->Z, z_out); 932 } 933 934 static void ec_GFp_nistp256_point_mul_public(const EC_GROUP *group, 935 EC_RAW_POINT *r, 936 const EC_SCALAR *g_scalar, 937 const EC_RAW_POINT *p, 938 const EC_SCALAR *p_scalar) { 939 #define P256_WSIZE_PUBLIC 4 940 // Precompute multiples of |p|. p_pre_comp[i] is (2*i+1) * |p|. 941 fe p_pre_comp[1 << (P256_WSIZE_PUBLIC-1)][3]; 942 fe_from_generic(p_pre_comp[0][0], &p->X); 943 fe_from_generic(p_pre_comp[0][1], &p->Y); 944 fe_from_generic(p_pre_comp[0][2], &p->Z); 945 fe p2[3]; 946 point_double(p2[0], p2[1], p2[2], p_pre_comp[0][0], p_pre_comp[0][1], 947 p_pre_comp[0][2]); 948 for (size_t i = 1; i < OPENSSL_ARRAY_SIZE(p_pre_comp); i++) { 949 point_add(p_pre_comp[i][0], p_pre_comp[i][1], p_pre_comp[i][2], 950 p_pre_comp[i - 1][0], p_pre_comp[i - 1][1], p_pre_comp[i - 1][2], 951 0 /* not mixed */, p2[0], p2[1], p2[2]); 952 } 953 954 // Set up the coefficients for |p_scalar|. 955 int8_t p_wNAF[257]; 956 ec_compute_wNAF(group, p_wNAF, p_scalar, 256, P256_WSIZE_PUBLIC); 957 958 // Set |ret| to the point at infinity. 959 int skip = 1; // Save some point operations. 960 fe ret[3] = {{0},{0},{0}}; 961 for (int i = 256; i >= 0; i--) { 962 if (!skip) { 963 point_double(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2]); 964 } 965 966 // For the |g_scalar|, we use the precomputed table without the 967 // constant-time lookup. 968 if (i <= 31) { 969 // First, look 32 bits upwards. 970 uint64_t bits = get_bit(g_scalar->bytes, i + 224) << 3; 971 bits |= get_bit(g_scalar->bytes, i + 160) << 2; 972 bits |= get_bit(g_scalar->bytes, i + 96) << 1; 973 bits |= get_bit(g_scalar->bytes, i + 32); 974 point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], 1 /* mixed */, 975 g_pre_comp[1][bits][0], g_pre_comp[1][bits][1], 976 g_pre_comp[1][bits][2]); 977 skip = 0; 978 979 // Second, look at the current position. 980 bits = get_bit(g_scalar->bytes, i + 192) << 3; 981 bits |= get_bit(g_scalar->bytes, i + 128) << 2; 982 bits |= get_bit(g_scalar->bytes, i + 64) << 1; 983 bits |= get_bit(g_scalar->bytes, i); 984 point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], 1 /* mixed */, 985 g_pre_comp[0][bits][0], g_pre_comp[0][bits][1], 986 g_pre_comp[0][bits][2]); 987 } 988 989 int digit = p_wNAF[i]; 990 if (digit != 0) { 991 assert(digit & 1); 992 int idx = digit < 0 ? (-digit) >> 1 : digit >> 1; 993 fe *y = &p_pre_comp[idx][1], tmp; 994 if (digit < 0) { 995 fe_opp(tmp, p_pre_comp[idx][1]); 996 y = &tmp; 997 } 998 if (!skip) { 999 point_add(ret[0], ret[1], ret[2], ret[0], ret[1], ret[2], 1000 0 /* not mixed */, p_pre_comp[idx][0], *y, p_pre_comp[idx][2]); 1001 } else { 1002 fe_copy(ret[0], p_pre_comp[idx][0]); 1003 fe_copy(ret[1], *y); 1004 fe_copy(ret[2], p_pre_comp[idx][2]); 1005 skip = 0; 1006 } 1007 } 1008 } 1009 1010 fe_to_generic(&r->X, ret[0]); 1011 fe_to_generic(&r->Y, ret[1]); 1012 fe_to_generic(&r->Z, ret[2]); 1013 } 1014 1015 static int ec_GFp_nistp256_cmp_x_coordinate(const EC_GROUP *group, 1016 const EC_RAW_POINT *p, 1017 const EC_SCALAR *r) { 1018 if (ec_GFp_simple_is_at_infinity(group, p)) { 1019 return 0; 1020 } 1021 1022 // We wish to compare X/Z^2 with r. This is equivalent to comparing X with 1023 // r*Z^2. Note that X and Z are represented in Montgomery form, while r is 1024 // not. 1025 fe Z2_mont; 1026 fe_from_generic(Z2_mont, &p->Z); 1027 fe_mul(Z2_mont, Z2_mont, Z2_mont); 1028 1029 fe r_Z2; 1030 fe_frombytes(r_Z2, r->bytes); // r < order < p, so this is valid. 1031 fe_mul(r_Z2, r_Z2, Z2_mont); 1032 1033 fe X; 1034 fe_from_generic(X, &p->X); 1035 fe_from_montgomery(X); 1036 1037 if (OPENSSL_memcmp(&r_Z2, &X, sizeof(r_Z2)) == 0) { 1038 return 1; 1039 } 1040 1041 // During signing the x coefficient is reduced modulo the group order. 1042 // Therefore there is a small possibility, less than 1/2^128, that group_order 1043 // < p.x < P. in that case we need not only to compare against |r| but also to 1044 // compare against r+group_order. 1045 assert(group->field.width == group->order.width); 1046 if (bn_less_than_words(r->words, group->field_minus_order.words, 1047 group->field.width)) { 1048 // We can ignore the carry because: r + group_order < p < 2^256. 1049 EC_FELEM tmp; 1050 bn_add_words(tmp.words, r->words, group->order.d, group->order.width); 1051 fe_from_generic(r_Z2, &tmp); 1052 fe_mul(r_Z2, r_Z2, Z2_mont); 1053 if (OPENSSL_memcmp(&r_Z2, &X, sizeof(r_Z2)) == 0) { 1054 return 1; 1055 } 1056 } 1057 1058 return 0; 1059 } 1060 1061 DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp256_method) { 1062 out->group_init = ec_GFp_mont_group_init; 1063 out->group_finish = ec_GFp_mont_group_finish; 1064 out->group_set_curve = ec_GFp_mont_group_set_curve; 1065 out->point_get_affine_coordinates = 1066 ec_GFp_nistp256_point_get_affine_coordinates; 1067 out->add = ec_GFp_nistp256_add; 1068 out->dbl = ec_GFp_nistp256_dbl; 1069 out->mul = ec_GFp_nistp256_points_mul; 1070 out->mul_public = ec_GFp_nistp256_point_mul_public; 1071 out->felem_mul = ec_GFp_mont_felem_mul; 1072 out->felem_sqr = ec_GFp_mont_felem_sqr; 1073 out->bignum_to_felem = ec_GFp_mont_bignum_to_felem; 1074 out->felem_to_bignum = ec_GFp_mont_felem_to_bignum; 1075 out->scalar_inv_montgomery = ec_simple_scalar_inv_montgomery; 1076 out->scalar_inv_montgomery_vartime = ec_GFp_simple_mont_inv_mod_ord_vartime; 1077 out->cmp_x_coordinate = ec_GFp_nistp256_cmp_x_coordinate; 1078 } 1079 1080 #undef BORINGSSL_NISTP256_64BIT 1081
