1/* Copyright 2015 The BoringSSL Authors 2 * 3 * Permission to use, copy, modify, and/or distribute this software for any 4 * purpose with or without fee is hereby granted, provided that the above 5 * copyright notice and this permission notice appear in all copies. 6 * 7 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 8 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 9 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY 10 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 11 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION 12 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN 13 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ 14 15// A 64-bit implementation of the NIST P-224 elliptic curve point multiplication 16// 17// Inspired by Daniel J. Bernstein's public domain nistp224 implementation 18// and Adam Langley's public domain 64-bit C implementation of curve25519. 19 20#include <openssl/base.h> 21 22#include <openssl/bn.h> 23#include <openssl/ec.h> 24#include <openssl/err.h> 25#include <openssl/mem.h> 26 27#include <assert.h> 28#include <string.h> 29 30#include "internal.h" 31#include "../delocate.h" 32#include "../../internal.h" 33 34 35#if defined(BORINGSSL_HAS_UINT128) && !defined(OPENSSL_SMALL) 36 37// Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 38// using 64-bit coefficients called 'limbs', and sometimes (for multiplication 39// results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 40// 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-p224_limb 41// representation is an 'p224_felem'; a 7-p224_widelimb representation is a 42// 'p224_widefelem'. Even within felems, bits of adjacent limbs overlap, and we 43// don't always reduce the representations: we ensure that inputs to each 44// p224_felem multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 45// 4*2^60*2^60, and fit into a 128-bit word without overflow. The coefficients 46// are then again partially reduced to obtain an p224_felem satisfying a_i < 47// 2^57. We only reduce to the unique minimal representation at the end of the 48// computation. 49 50typedef uint64_t p224_limb; 51typedef uint128_t p224_widelimb; 52 53typedef p224_limb p224_felem[4]; 54typedef p224_widelimb p224_widefelem[7]; 55 56// Precomputed multiples of the standard generator 57// Points are given in coordinates (X, Y, Z) where Z normally is 1 58// (0 for the point at infinity). 59// For each field element, slice a_0 is word 0, etc. 60// 61// The table has 2 * 16 elements, starting with the following: 62// index | bits | point 63// ------+---------+------------------------------ 64// 0 | 0 0 0 0 | 0G 65// 1 | 0 0 0 1 | 1G 66// 2 | 0 0 1 0 | 2^56G 67// 3 | 0 0 1 1 | (2^56 + 1)G 68// 4 | 0 1 0 0 | 2^112G 69// 5 | 0 1 0 1 | (2^112 + 1)G 70// 6 | 0 1 1 0 | (2^112 + 2^56)G 71// 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G 72// 8 | 1 0 0 0 | 2^168G 73// 9 | 1 0 0 1 | (2^168 + 1)G 74// 10 | 1 0 1 0 | (2^168 + 2^56)G 75// 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G 76// 12 | 1 1 0 0 | (2^168 + 2^112)G 77// 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G 78// 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G 79// 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G 80// followed by a copy of this with each element multiplied by 2^28. 81// 82// The reason for this is so that we can clock bits into four different 83// locations when doing simple scalar multiplies against the base point, 84// and then another four locations using the second 16 elements. 85static const p224_felem g_p224_pre_comp[2][16][3] = { 86 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 87 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, 88 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, 89 {1, 0, 0, 0}}, 90 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, 91 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, 92 {1, 0, 0, 0}}, 93 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, 94 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, 95 {1, 0, 0, 0}}, 96 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, 97 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, 98 {1, 0, 0, 0}}, 99 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, 100 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, 101 {1, 0, 0, 0}}, 102 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, 103 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, 104 {1, 0, 0, 0}}, 105 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, 106 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, 107 {1, 0, 0, 0}}, 108 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, 109 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, 110 {1, 0, 0, 0}}, 111 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, 112 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, 113 {1, 0, 0, 0}}, 114 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, 115 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, 116 {1, 0, 0, 0}}, 117 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, 118 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, 119 {1, 0, 0, 0}}, 120 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, 121 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, 122 {1, 0, 0, 0}}, 123 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, 124 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, 125 {1, 0, 0, 0}}, 126 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, 127 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, 128 {1, 0, 0, 0}}, 129 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, 130 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, 131 {1, 0, 0, 0}}}, 132 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 133 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31}, 134 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d}, 135 {1, 0, 0, 0}}, 136 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3}, 137 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a}, 138 {1, 0, 0, 0}}, 139 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33}, 140 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100}, 141 {1, 0, 0, 0}}, 142 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5}, 143 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea}, 144 {1, 0, 0, 0}}, 145 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be}, 146 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51}, 147 {1, 0, 0, 0}}, 148 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1}, 149 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb}, 150 {1, 0, 0, 0}}, 151 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233}, 152 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def}, 153 {1, 0, 0, 0}}, 154 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae}, 155 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45}, 156 {1, 0, 0, 0}}, 157 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e}, 158 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb}, 159 {1, 0, 0, 0}}, 160 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de}, 161 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3}, 162 {1, 0, 0, 0}}, 163 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05}, 164 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58}, 165 {1, 0, 0, 0}}, 166 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb}, 167 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0}, 168 {1, 0, 0, 0}}, 169 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9}, 170 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea}, 171 {1, 0, 0, 0}}, 172 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba}, 173 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405}, 174 {1, 0, 0, 0}}, 175 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e}, 176 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e}, 177 {1, 0, 0, 0}}}}; 178 179 180// Helper functions to convert field elements to/from internal representation 181 182static void p224_generic_to_felem(p224_felem out, const EC_FELEM *in) { 183 // |p224_felem|'s minimal representation uses four 56-bit words. |EC_FELEM| 184 // uses four 64-bit words. (The top-most word only has 32 bits.) 185 out[0] = in->words[0] & 0x00ffffffffffffff; 186 out[1] = ((in->words[0] >> 56) | (in->words[1] << 8)) & 0x00ffffffffffffff; 187 out[2] = ((in->words[1] >> 48) | (in->words[2] << 16)) & 0x00ffffffffffffff; 188 out[3] = ((in->words[2] >> 40) | (in->words[3] << 24)) & 0x00ffffffffffffff; 189} 190 191// Requires 0 <= in < 2*p (always call p224_felem_reduce first) 192static void p224_felem_to_generic(EC_FELEM *out, const p224_felem in) { 193 // Reduce to unique minimal representation. 194 static const int64_t two56 = ((p224_limb)1) << 56; 195 // 0 <= in < 2*p, p = 2^224 - 2^96 + 1 196 // if in > p , reduce in = in - 2^224 + 2^96 - 1 197 int64_t tmp[4], a; 198 tmp[0] = in[0]; 199 tmp[1] = in[1]; 200 tmp[2] = in[2]; 201 tmp[3] = in[3]; 202 // Case 1: a = 1 iff in >= 2^224 203 a = (in[3] >> 56); 204 tmp[0] -= a; 205 tmp[1] += a << 40; 206 tmp[3] &= 0x00ffffffffffffff; 207 // Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and 208 // the lower part is non-zero 209 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) | 210 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63); 211 a &= 0x00ffffffffffffff; 212 // turn a into an all-one mask (if a = 0) or an all-zero mask 213 a = (a - 1) >> 63; 214 // subtract 2^224 - 2^96 + 1 if a is all-one 215 tmp[3] &= a ^ 0xffffffffffffffff; 216 tmp[2] &= a ^ 0xffffffffffffffff; 217 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; 218 tmp[0] -= 1 & a; 219 220 // eliminate negative coefficients: if tmp[0] is negative, tmp[1] must 221 // be non-zero, so we only need one step 222 a = tmp[0] >> 63; 223 tmp[0] += two56 & a; 224 tmp[1] -= 1 & a; 225 226 // carry 1 -> 2 -> 3 227 tmp[2] += tmp[1] >> 56; 228 tmp[1] &= 0x00ffffffffffffff; 229 230 tmp[3] += tmp[2] >> 56; 231 tmp[2] &= 0x00ffffffffffffff; 232 233 // Now 0 <= tmp < p 234 p224_felem tmp2; 235 tmp2[0] = tmp[0]; 236 tmp2[1] = tmp[1]; 237 tmp2[2] = tmp[2]; 238 tmp2[3] = tmp[3]; 239 240 // |p224_felem|'s minimal representation uses four 56-bit words. |EC_FELEM| 241 // uses four 64-bit words. (The top-most word only has 32 bits.) 242 out->words[0] = tmp2[0] | (tmp2[1] << 56); 243 out->words[1] = (tmp2[1] >> 8) | (tmp2[2] << 48); 244 out->words[2] = (tmp2[2] >> 16) | (tmp2[3] << 40); 245 out->words[3] = tmp2[3] >> 24; 246} 247 248 249// Field operations, using the internal representation of field elements. 250// NB! These operations are specific to our point multiplication and cannot be 251// expected to be correct in general - e.g., multiplication with a large scalar 252// will cause an overflow. 253 254static void p224_felem_assign(p224_felem out, const p224_felem in) { 255 out[0] = in[0]; 256 out[1] = in[1]; 257 out[2] = in[2]; 258 out[3] = in[3]; 259} 260 261// Sum two field elements: out += in 262static void p224_felem_sum(p224_felem out, const p224_felem in) { 263 out[0] += in[0]; 264 out[1] += in[1]; 265 out[2] += in[2]; 266 out[3] += in[3]; 267} 268 269// Subtract field elements: out -= in 270// Assumes in[i] < 2^57 271static void p224_felem_diff(p224_felem out, const p224_felem in) { 272 static const p224_limb two58p2 = 273 (((p224_limb)1) << 58) + (((p224_limb)1) << 2); 274 static const p224_limb two58m2 = 275 (((p224_limb)1) << 58) - (((p224_limb)1) << 2); 276 static const p224_limb two58m42m2 = 277 (((p224_limb)1) << 58) - (((p224_limb)1) << 42) - (((p224_limb)1) << 2); 278 279 // Add 0 mod 2^224-2^96+1 to ensure out > in 280 out[0] += two58p2; 281 out[1] += two58m42m2; 282 out[2] += two58m2; 283 out[3] += two58m2; 284 285 out[0] -= in[0]; 286 out[1] -= in[1]; 287 out[2] -= in[2]; 288 out[3] -= in[3]; 289} 290 291// Subtract in unreduced 128-bit mode: out -= in 292// Assumes in[i] < 2^119 293static void p224_widefelem_diff(p224_widefelem out, const p224_widefelem in) { 294 static const p224_widelimb two120 = ((p224_widelimb)1) << 120; 295 static const p224_widelimb two120m64 = 296 (((p224_widelimb)1) << 120) - (((p224_widelimb)1) << 64); 297 static const p224_widelimb two120m104m64 = (((p224_widelimb)1) << 120) - 298 (((p224_widelimb)1) << 104) - 299 (((p224_widelimb)1) << 64); 300 301 // Add 0 mod 2^224-2^96+1 to ensure out > in 302 out[0] += two120; 303 out[1] += two120m64; 304 out[2] += two120m64; 305 out[3] += two120; 306 out[4] += two120m104m64; 307 out[5] += two120m64; 308 out[6] += two120m64; 309 310 out[0] -= in[0]; 311 out[1] -= in[1]; 312 out[2] -= in[2]; 313 out[3] -= in[3]; 314 out[4] -= in[4]; 315 out[5] -= in[5]; 316 out[6] -= in[6]; 317} 318 319// Subtract in mixed mode: out128 -= in64 320// in[i] < 2^63 321static void p224_felem_diff_128_64(p224_widefelem out, const p224_felem in) { 322 static const p224_widelimb two64p8 = 323 (((p224_widelimb)1) << 64) + (((p224_widelimb)1) << 8); 324 static const p224_widelimb two64m8 = 325 (((p224_widelimb)1) << 64) - (((p224_widelimb)1) << 8); 326 static const p224_widelimb two64m48m8 = (((p224_widelimb)1) << 64) - 327 (((p224_widelimb)1) << 48) - 328 (((p224_widelimb)1) << 8); 329 330 // Add 0 mod 2^224-2^96+1 to ensure out > in 331 out[0] += two64p8; 332 out[1] += two64m48m8; 333 out[2] += two64m8; 334 out[3] += two64m8; 335 336 out[0] -= in[0]; 337 out[1] -= in[1]; 338 out[2] -= in[2]; 339 out[3] -= in[3]; 340} 341 342// Multiply a field element by a scalar: out = out * scalar 343// The scalars we actually use are small, so results fit without overflow 344static void p224_felem_scalar(p224_felem out, const p224_limb scalar) { 345 out[0] *= scalar; 346 out[1] *= scalar; 347 out[2] *= scalar; 348 out[3] *= scalar; 349} 350 351// Multiply an unreduced field element by a scalar: out = out * scalar 352// The scalars we actually use are small, so results fit without overflow 353static void p224_widefelem_scalar(p224_widefelem out, 354 const p224_widelimb scalar) { 355 out[0] *= scalar; 356 out[1] *= scalar; 357 out[2] *= scalar; 358 out[3] *= scalar; 359 out[4] *= scalar; 360 out[5] *= scalar; 361 out[6] *= scalar; 362} 363 364// Square a field element: out = in^2 365static void p224_felem_square(p224_widefelem out, const p224_felem in) { 366 p224_limb tmp0, tmp1, tmp2; 367 tmp0 = 2 * in[0]; 368 tmp1 = 2 * in[1]; 369 tmp2 = 2 * in[2]; 370 out[0] = ((p224_widelimb)in[0]) * in[0]; 371 out[1] = ((p224_widelimb)in[0]) * tmp1; 372 out[2] = ((p224_widelimb)in[0]) * tmp2 + ((p224_widelimb)in[1]) * in[1]; 373 out[3] = ((p224_widelimb)in[3]) * tmp0 + ((p224_widelimb)in[1]) * tmp2; 374 out[4] = ((p224_widelimb)in[3]) * tmp1 + ((p224_widelimb)in[2]) * in[2]; 375 out[5] = ((p224_widelimb)in[3]) * tmp2; 376 out[6] = ((p224_widelimb)in[3]) * in[3]; 377} 378 379// Multiply two field elements: out = in1 * in2 380static void p224_felem_mul(p224_widefelem out, const p224_felem in1, 381 const p224_felem in2) { 382 out[0] = ((p224_widelimb)in1[0]) * in2[0]; 383 out[1] = ((p224_widelimb)in1[0]) * in2[1] + ((p224_widelimb)in1[1]) * in2[0]; 384 out[2] = ((p224_widelimb)in1[0]) * in2[2] + ((p224_widelimb)in1[1]) * in2[1] + 385 ((p224_widelimb)in1[2]) * in2[0]; 386 out[3] = ((p224_widelimb)in1[0]) * in2[3] + ((p224_widelimb)in1[1]) * in2[2] + 387 ((p224_widelimb)in1[2]) * in2[1] + ((p224_widelimb)in1[3]) * in2[0]; 388 out[4] = ((p224_widelimb)in1[1]) * in2[3] + ((p224_widelimb)in1[2]) * in2[2] + 389 ((p224_widelimb)in1[3]) * in2[1]; 390 out[5] = ((p224_widelimb)in1[2]) * in2[3] + ((p224_widelimb)in1[3]) * in2[2]; 391 out[6] = ((p224_widelimb)in1[3]) * in2[3]; 392} 393 394// Reduce seven 128-bit coefficients to four 64-bit coefficients. 395// Requires in[i] < 2^126, 396// ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 397static void p224_felem_reduce(p224_felem out, const p224_widefelem in) { 398 static const p224_widelimb two127p15 = 399 (((p224_widelimb)1) << 127) + (((p224_widelimb)1) << 15); 400 static const p224_widelimb two127m71 = 401 (((p224_widelimb)1) << 127) - (((p224_widelimb)1) << 71); 402 static const p224_widelimb two127m71m55 = (((p224_widelimb)1) << 127) - 403 (((p224_widelimb)1) << 71) - 404 (((p224_widelimb)1) << 55); 405 p224_widelimb output[5]; 406 407 // Add 0 mod 2^224-2^96+1 to ensure all differences are positive 408 output[0] = in[0] + two127p15; 409 output[1] = in[1] + two127m71m55; 410 output[2] = in[2] + two127m71; 411 output[3] = in[3]; 412 output[4] = in[4]; 413 414 // Eliminate in[4], in[5], in[6] 415 output[4] += in[6] >> 16; 416 output[3] += (in[6] & 0xffff) << 40; 417 output[2] -= in[6]; 418 419 output[3] += in[5] >> 16; 420 output[2] += (in[5] & 0xffff) << 40; 421 output[1] -= in[5]; 422 423 output[2] += output[4] >> 16; 424 output[1] += (output[4] & 0xffff) << 40; 425 output[0] -= output[4]; 426 427 // Carry 2 -> 3 -> 4 428 output[3] += output[2] >> 56; 429 output[2] &= 0x00ffffffffffffff; 430 431 output[4] = output[3] >> 56; 432 output[3] &= 0x00ffffffffffffff; 433 434 // Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 435 436 // Eliminate output[4] 437 output[2] += output[4] >> 16; 438 // output[2] < 2^56 + 2^56 = 2^57 439 output[1] += (output[4] & 0xffff) << 40; 440 output[0] -= output[4]; 441 442 // Carry 0 -> 1 -> 2 -> 3 443 output[1] += output[0] >> 56; 444 out[0] = output[0] & 0x00ffffffffffffff; 445 446 output[2] += output[1] >> 56; 447 // output[2] < 2^57 + 2^72 448 out[1] = output[1] & 0x00ffffffffffffff; 449 output[3] += output[2] >> 56; 450 // output[3] <= 2^56 + 2^16 451 out[2] = output[2] & 0x00ffffffffffffff; 452 453 // out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, 454 // out[3] <= 2^56 + 2^16 (due to final carry), 455 // so out < 2*p 456 out[3] = output[3]; 457} 458 459// Get negative value: out = -in 460// Requires in[i] < 2^63, 461// ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 462static void p224_felem_neg(p224_felem out, const p224_felem in) { 463 p224_widefelem tmp = {0}; 464 p224_felem_diff_128_64(tmp, in); 465 p224_felem_reduce(out, tmp); 466} 467 468// Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field 469// elements are reduced to in < 2^225, so we only need to check three cases: 0, 470// 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 471static p224_limb p224_felem_is_zero(const p224_felem in) { 472 p224_limb zero = in[0] | in[1] | in[2] | in[3]; 473 zero = (((int64_t)(zero)-1) >> 63) & 1; 474 475 p224_limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) | 476 (in[2] ^ 0x00ffffffffffffff) | 477 (in[3] ^ 0x00ffffffffffffff); 478 two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1; 479 p224_limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) | 480 (in[2] ^ 0x00ffffffffffffff) | 481 (in[3] ^ 0x01ffffffffffffff); 482 two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1; 483 return (zero | two224m96p1 | two225m97p2); 484} 485 486// Invert a field element 487// Computation chain copied from djb's code 488static void p224_felem_inv(p224_felem out, const p224_felem in) { 489 p224_felem ftmp, ftmp2, ftmp3, ftmp4; 490 p224_widefelem tmp; 491 492 p224_felem_square(tmp, in); 493 p224_felem_reduce(ftmp, tmp); // 2 494 p224_felem_mul(tmp, in, ftmp); 495 p224_felem_reduce(ftmp, tmp); // 2^2 - 1 496 p224_felem_square(tmp, ftmp); 497 p224_felem_reduce(ftmp, tmp); // 2^3 - 2 498 p224_felem_mul(tmp, in, ftmp); 499 p224_felem_reduce(ftmp, tmp); // 2^3 - 1 500 p224_felem_square(tmp, ftmp); 501 p224_felem_reduce(ftmp2, tmp); // 2^4 - 2 502 p224_felem_square(tmp, ftmp2); 503 p224_felem_reduce(ftmp2, tmp); // 2^5 - 4 504 p224_felem_square(tmp, ftmp2); 505 p224_felem_reduce(ftmp2, tmp); // 2^6 - 8 506 p224_felem_mul(tmp, ftmp2, ftmp); 507 p224_felem_reduce(ftmp, tmp); // 2^6 - 1 508 p224_felem_square(tmp, ftmp); 509 p224_felem_reduce(ftmp2, tmp); // 2^7 - 2 510 for (size_t i = 0; i < 5; ++i) { // 2^12 - 2^6 511 p224_felem_square(tmp, ftmp2); 512 p224_felem_reduce(ftmp2, tmp); 513 } 514 p224_felem_mul(tmp, ftmp2, ftmp); 515 p224_felem_reduce(ftmp2, tmp); // 2^12 - 1 516 p224_felem_square(tmp, ftmp2); 517 p224_felem_reduce(ftmp3, tmp); // 2^13 - 2 518 for (size_t i = 0; i < 11; ++i) { // 2^24 - 2^12 519 p224_felem_square(tmp, ftmp3); 520 p224_felem_reduce(ftmp3, tmp); 521 } 522 p224_felem_mul(tmp, ftmp3, ftmp2); 523 p224_felem_reduce(ftmp2, tmp); // 2^24 - 1 524 p224_felem_square(tmp, ftmp2); 525 p224_felem_reduce(ftmp3, tmp); // 2^25 - 2 526 for (size_t i = 0; i < 23; ++i) { // 2^48 - 2^24 527 p224_felem_square(tmp, ftmp3); 528 p224_felem_reduce(ftmp3, tmp); 529 } 530 p224_felem_mul(tmp, ftmp3, ftmp2); 531 p224_felem_reduce(ftmp3, tmp); // 2^48 - 1 532 p224_felem_square(tmp, ftmp3); 533 p224_felem_reduce(ftmp4, tmp); // 2^49 - 2 534 for (size_t i = 0; i < 47; ++i) { // 2^96 - 2^48 535 p224_felem_square(tmp, ftmp4); 536 p224_felem_reduce(ftmp4, tmp); 537 } 538 p224_felem_mul(tmp, ftmp3, ftmp4); 539 p224_felem_reduce(ftmp3, tmp); // 2^96 - 1 540 p224_felem_square(tmp, ftmp3); 541 p224_felem_reduce(ftmp4, tmp); // 2^97 - 2 542 for (size_t i = 0; i < 23; ++i) { // 2^120 - 2^24 543 p224_felem_square(tmp, ftmp4); 544 p224_felem_reduce(ftmp4, tmp); 545 } 546 p224_felem_mul(tmp, ftmp2, ftmp4); 547 p224_felem_reduce(ftmp2, tmp); // 2^120 - 1 548 for (size_t i = 0; i < 6; ++i) { // 2^126 - 2^6 549 p224_felem_square(tmp, ftmp2); 550 p224_felem_reduce(ftmp2, tmp); 551 } 552 p224_felem_mul(tmp, ftmp2, ftmp); 553 p224_felem_reduce(ftmp, tmp); // 2^126 - 1 554 p224_felem_square(tmp, ftmp); 555 p224_felem_reduce(ftmp, tmp); // 2^127 - 2 556 p224_felem_mul(tmp, ftmp, in); 557 p224_felem_reduce(ftmp, tmp); // 2^127 - 1 558 for (size_t i = 0; i < 97; ++i) { // 2^224 - 2^97 559 p224_felem_square(tmp, ftmp); 560 p224_felem_reduce(ftmp, tmp); 561 } 562 p224_felem_mul(tmp, ftmp, ftmp3); 563 p224_felem_reduce(out, tmp); // 2^224 - 2^96 - 1 564} 565 566// Copy in constant time: 567// if icopy == 1, copy in to out, 568// if icopy == 0, copy out to itself. 569static void p224_copy_conditional(p224_felem out, const p224_felem in, 570 p224_limb icopy) { 571 // icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one 572 const p224_limb copy = -icopy; 573 for (size_t i = 0; i < 4; ++i) { 574 const p224_limb tmp = copy & (in[i] ^ out[i]); 575 out[i] ^= tmp; 576 } 577} 578 579// ELLIPTIC CURVE POINT OPERATIONS 580// 581// Points are represented in Jacobian projective coordinates: 582// (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3), 583// or to the point at infinity if Z == 0. 584 585// Double an elliptic curve point: 586// (X', Y', Z') = 2 * (X, Y, Z), where 587// X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2 588// Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2 589// Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z 590// Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed, 591// while x_out == y_in is not (maybe this works, but it's not tested). 592static void p224_point_double(p224_felem x_out, p224_felem y_out, 593 p224_felem z_out, const p224_felem x_in, 594 const p224_felem y_in, const p224_felem z_in) { 595 p224_widefelem tmp, tmp2; 596 p224_felem delta, gamma, beta, alpha, ftmp, ftmp2; 597 598 p224_felem_assign(ftmp, x_in); 599 p224_felem_assign(ftmp2, x_in); 600 601 // delta = z^2 602 p224_felem_square(tmp, z_in); 603 p224_felem_reduce(delta, tmp); 604 605 // gamma = y^2 606 p224_felem_square(tmp, y_in); 607 p224_felem_reduce(gamma, tmp); 608 609 // beta = x*gamma 610 p224_felem_mul(tmp, x_in, gamma); 611 p224_felem_reduce(beta, tmp); 612 613 // alpha = 3*(x-delta)*(x+delta) 614 p224_felem_diff(ftmp, delta); 615 // ftmp[i] < 2^57 + 2^58 + 2 < 2^59 616 p224_felem_sum(ftmp2, delta); 617 // ftmp2[i] < 2^57 + 2^57 = 2^58 618 p224_felem_scalar(ftmp2, 3); 619 // ftmp2[i] < 3 * 2^58 < 2^60 620 p224_felem_mul(tmp, ftmp, ftmp2); 621 // tmp[i] < 2^60 * 2^59 * 4 = 2^121 622 p224_felem_reduce(alpha, tmp); 623 624 // x' = alpha^2 - 8*beta 625 p224_felem_square(tmp, alpha); 626 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 627 p224_felem_assign(ftmp, beta); 628 p224_felem_scalar(ftmp, 8); 629 // ftmp[i] < 8 * 2^57 = 2^60 630 p224_felem_diff_128_64(tmp, ftmp); 631 // tmp[i] < 2^116 + 2^64 + 8 < 2^117 632 p224_felem_reduce(x_out, tmp); 633 634 // z' = (y + z)^2 - gamma - delta 635 p224_felem_sum(delta, gamma); 636 // delta[i] < 2^57 + 2^57 = 2^58 637 p224_felem_assign(ftmp, y_in); 638 p224_felem_sum(ftmp, z_in); 639 // ftmp[i] < 2^57 + 2^57 = 2^58 640 p224_felem_square(tmp, ftmp); 641 // tmp[i] < 4 * 2^58 * 2^58 = 2^118 642 p224_felem_diff_128_64(tmp, delta); 643 // tmp[i] < 2^118 + 2^64 + 8 < 2^119 644 p224_felem_reduce(z_out, tmp); 645 646 // y' = alpha*(4*beta - x') - 8*gamma^2 647 p224_felem_scalar(beta, 4); 648 // beta[i] < 4 * 2^57 = 2^59 649 p224_felem_diff(beta, x_out); 650 // beta[i] < 2^59 + 2^58 + 2 < 2^60 651 p224_felem_mul(tmp, alpha, beta); 652 // tmp[i] < 4 * 2^57 * 2^60 = 2^119 653 p224_felem_square(tmp2, gamma); 654 // tmp2[i] < 4 * 2^57 * 2^57 = 2^116 655 p224_widefelem_scalar(tmp2, 8); 656 // tmp2[i] < 8 * 2^116 = 2^119 657 p224_widefelem_diff(tmp, tmp2); 658 // tmp[i] < 2^119 + 2^120 < 2^121 659 p224_felem_reduce(y_out, tmp); 660} 661 662// Add two elliptic curve points: 663// (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where 664// X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 - 665// 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 666// Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * 667// X_1)^2 - X_3) - 668// Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3 669// Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) 670// 671// This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. 672 673// This function is not entirely constant-time: it includes a branch for 674// checking whether the two input points are equal, (while not equal to the 675// point at infinity). This case never happens during single point 676// multiplication, so there is no timing leak for ECDH or ECDSA signing. 677static void p224_point_add(p224_felem x3, p224_felem y3, p224_felem z3, 678 const p224_felem x1, const p224_felem y1, 679 const p224_felem z1, const int mixed, 680 const p224_felem x2, const p224_felem y2, 681 const p224_felem z2) { 682 p224_felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out; 683 p224_widefelem tmp, tmp2; 684 p224_limb z1_is_zero, z2_is_zero, x_equal, y_equal; 685 686 if (!mixed) { 687 // ftmp2 = z2^2 688 p224_felem_square(tmp, z2); 689 p224_felem_reduce(ftmp2, tmp); 690 691 // ftmp4 = z2^3 692 p224_felem_mul(tmp, ftmp2, z2); 693 p224_felem_reduce(ftmp4, tmp); 694 695 // ftmp4 = z2^3*y1 696 p224_felem_mul(tmp2, ftmp4, y1); 697 p224_felem_reduce(ftmp4, tmp2); 698 699 // ftmp2 = z2^2*x1 700 p224_felem_mul(tmp2, ftmp2, x1); 701 p224_felem_reduce(ftmp2, tmp2); 702 } else { 703 // We'll assume z2 = 1 (special case z2 = 0 is handled later) 704 705 // ftmp4 = z2^3*y1 706 p224_felem_assign(ftmp4, y1); 707 708 // ftmp2 = z2^2*x1 709 p224_felem_assign(ftmp2, x1); 710 } 711 712 // ftmp = z1^2 713 p224_felem_square(tmp, z1); 714 p224_felem_reduce(ftmp, tmp); 715 716 // ftmp3 = z1^3 717 p224_felem_mul(tmp, ftmp, z1); 718 p224_felem_reduce(ftmp3, tmp); 719 720 // tmp = z1^3*y2 721 p224_felem_mul(tmp, ftmp3, y2); 722 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 723 724 // ftmp3 = z1^3*y2 - z2^3*y1 725 p224_felem_diff_128_64(tmp, ftmp4); 726 // tmp[i] < 2^116 + 2^64 + 8 < 2^117 727 p224_felem_reduce(ftmp3, tmp); 728 729 // tmp = z1^2*x2 730 p224_felem_mul(tmp, ftmp, x2); 731 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 732 733 // ftmp = z1^2*x2 - z2^2*x1 734 p224_felem_diff_128_64(tmp, ftmp2); 735 // tmp[i] < 2^116 + 2^64 + 8 < 2^117 736 p224_felem_reduce(ftmp, tmp); 737 738 // The formulae are incorrect if the points are equal, so we check for this 739 // and do doubling if this happens. 740 x_equal = p224_felem_is_zero(ftmp); 741 y_equal = p224_felem_is_zero(ftmp3); 742 z1_is_zero = p224_felem_is_zero(z1); 743 z2_is_zero = p224_felem_is_zero(z2); 744 // In affine coordinates, (X_1, Y_1) == (X_2, Y_2) 745 p224_limb is_nontrivial_double = 746 x_equal & y_equal & (1 - z1_is_zero) & (1 - z2_is_zero); 747 if (constant_time_declassify_w(is_nontrivial_double)) { 748 p224_point_double(x3, y3, z3, x1, y1, z1); 749 return; 750 } 751 752 // ftmp5 = z1*z2 753 if (!mixed) { 754 p224_felem_mul(tmp, z1, z2); 755 p224_felem_reduce(ftmp5, tmp); 756 } else { 757 // special case z2 = 0 is handled later 758 p224_felem_assign(ftmp5, z1); 759 } 760 761 // z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) 762 p224_felem_mul(tmp, ftmp, ftmp5); 763 p224_felem_reduce(z_out, tmp); 764 765 // ftmp = (z1^2*x2 - z2^2*x1)^2 766 p224_felem_assign(ftmp5, ftmp); 767 p224_felem_square(tmp, ftmp); 768 p224_felem_reduce(ftmp, tmp); 769 770 // ftmp5 = (z1^2*x2 - z2^2*x1)^3 771 p224_felem_mul(tmp, ftmp, ftmp5); 772 p224_felem_reduce(ftmp5, tmp); 773 774 // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 775 p224_felem_mul(tmp, ftmp2, ftmp); 776 p224_felem_reduce(ftmp2, tmp); 777 778 // tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 779 p224_felem_mul(tmp, ftmp4, ftmp5); 780 // tmp[i] < 4 * 2^57 * 2^57 = 2^116 781 782 // tmp2 = (z1^3*y2 - z2^3*y1)^2 783 p224_felem_square(tmp2, ftmp3); 784 // tmp2[i] < 4 * 2^57 * 2^57 < 2^116 785 786 // tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 787 p224_felem_diff_128_64(tmp2, ftmp5); 788 // tmp2[i] < 2^116 + 2^64 + 8 < 2^117 789 790 // ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 791 p224_felem_assign(ftmp5, ftmp2); 792 p224_felem_scalar(ftmp5, 2); 793 // ftmp5[i] < 2 * 2^57 = 2^58 794 795 /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - 796 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ 797 p224_felem_diff_128_64(tmp2, ftmp5); 798 // tmp2[i] < 2^117 + 2^64 + 8 < 2^118 799 p224_felem_reduce(x_out, tmp2); 800 801 // ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out 802 p224_felem_diff(ftmp2, x_out); 803 // ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 804 805 // tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) 806 p224_felem_mul(tmp2, ftmp3, ftmp2); 807 // tmp2[i] < 4 * 2^57 * 2^59 = 2^118 808 809 /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) - 810 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ 811 p224_widefelem_diff(tmp2, tmp); 812 // tmp2[i] < 2^118 + 2^120 < 2^121 813 p224_felem_reduce(y_out, tmp2); 814 815 // the result (x_out, y_out, z_out) is incorrect if one of the inputs is 816 // the point at infinity, so we need to check for this separately 817 818 // if point 1 is at infinity, copy point 2 to output, and vice versa 819 p224_copy_conditional(x_out, x2, z1_is_zero); 820 p224_copy_conditional(x_out, x1, z2_is_zero); 821 p224_copy_conditional(y_out, y2, z1_is_zero); 822 p224_copy_conditional(y_out, y1, z2_is_zero); 823 p224_copy_conditional(z_out, z2, z1_is_zero); 824 p224_copy_conditional(z_out, z1, z2_is_zero); 825 p224_felem_assign(x3, x_out); 826 p224_felem_assign(y3, y_out); 827 p224_felem_assign(z3, z_out); 828} 829 830// p224_select_point selects the |idx|th point from a precomputation table and 831// copies it to out. 832static void p224_select_point(const uint64_t idx, size_t size, 833 const p224_felem pre_comp[/*size*/][3], 834 p224_felem out[3]) { 835 p224_limb *outlimbs = &out[0][0]; 836 OPENSSL_memset(outlimbs, 0, 3 * sizeof(p224_felem)); 837 838 for (size_t i = 0; i < size; i++) { 839 const p224_limb *inlimbs = &pre_comp[i][0][0]; 840 static_assert(sizeof(uint64_t) <= sizeof(crypto_word_t), 841 "crypto_word_t too small"); 842 static_assert(sizeof(size_t) <= sizeof(crypto_word_t), 843 "crypto_word_t too small"); 844 // Without a value barrier, Clang adds a branch here. 845 uint64_t mask = value_barrier_w(constant_time_eq_w(i, idx)); 846 for (size_t j = 0; j < 4 * 3; j++) { 847 outlimbs[j] |= inlimbs[j] & mask; 848 } 849 } 850} 851 852// p224_get_bit returns the |i|th bit in |in|. 853static crypto_word_t p224_get_bit(const EC_SCALAR *in, size_t i) { 854 if (i >= 224) { 855 return 0; 856 } 857 static_assert(sizeof(in->words[0]) == 8, "BN_ULONG is not 64-bit"); 858 return (in->words[i >> 6] >> (i & 63)) & 1; 859} 860 861// Takes the Jacobian coordinates (X, Y, Z) of a point and returns 862// (X', Y') = (X/Z^2, Y/Z^3) 863static int ec_GFp_nistp224_point_get_affine_coordinates( 864 const EC_GROUP *group, const EC_JACOBIAN *point, EC_FELEM *x, 865 EC_FELEM *y) { 866 if (constant_time_declassify_int( 867 ec_GFp_simple_is_at_infinity(group, point))) { 868 OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); 869 return 0; 870 } 871 872 p224_felem z1, z2; 873 p224_widefelem tmp; 874 p224_generic_to_felem(z1, &point->Z); 875 p224_felem_inv(z2, z1); 876 p224_felem_square(tmp, z2); 877 p224_felem_reduce(z1, tmp); 878 879 if (x != NULL) { 880 p224_felem x_in, x_out; 881 p224_generic_to_felem(x_in, &point->X); 882 p224_felem_mul(tmp, x_in, z1); 883 p224_felem_reduce(x_out, tmp); 884 p224_felem_to_generic(x, x_out); 885 } 886 887 if (y != NULL) { 888 p224_felem y_in, y_out; 889 p224_generic_to_felem(y_in, &point->Y); 890 p224_felem_mul(tmp, z1, z2); 891 p224_felem_reduce(z1, tmp); 892 p224_felem_mul(tmp, y_in, z1); 893 p224_felem_reduce(y_out, tmp); 894 p224_felem_to_generic(y, y_out); 895 } 896 897 return 1; 898} 899 900static void ec_GFp_nistp224_add(const EC_GROUP *group, EC_JACOBIAN *r, 901 const EC_JACOBIAN *a, const EC_JACOBIAN *b) { 902 p224_felem x1, y1, z1, x2, y2, z2; 903 p224_generic_to_felem(x1, &a->X); 904 p224_generic_to_felem(y1, &a->Y); 905 p224_generic_to_felem(z1, &a->Z); 906 p224_generic_to_felem(x2, &b->X); 907 p224_generic_to_felem(y2, &b->Y); 908 p224_generic_to_felem(z2, &b->Z); 909 p224_point_add(x1, y1, z1, x1, y1, z1, 0 /* both Jacobian */, x2, y2, z2); 910 // The outputs are already reduced, but still need to be contracted. 911 p224_felem_to_generic(&r->X, x1); 912 p224_felem_to_generic(&r->Y, y1); 913 p224_felem_to_generic(&r->Z, z1); 914} 915 916static void ec_GFp_nistp224_dbl(const EC_GROUP *group, EC_JACOBIAN *r, 917 const EC_JACOBIAN *a) { 918 p224_felem x, y, z; 919 p224_generic_to_felem(x, &a->X); 920 p224_generic_to_felem(y, &a->Y); 921 p224_generic_to_felem(z, &a->Z); 922 p224_point_double(x, y, z, x, y, z); 923 // The outputs are already reduced, but still need to be contracted. 924 p224_felem_to_generic(&r->X, x); 925 p224_felem_to_generic(&r->Y, y); 926 p224_felem_to_generic(&r->Z, z); 927} 928 929static void ec_GFp_nistp224_make_precomp(p224_felem out[17][3], 930 const EC_JACOBIAN *p) { 931 OPENSSL_memset(out[0], 0, sizeof(p224_felem) * 3); 932 933 p224_generic_to_felem(out[1][0], &p->X); 934 p224_generic_to_felem(out[1][1], &p->Y); 935 p224_generic_to_felem(out[1][2], &p->Z); 936 937 for (size_t j = 2; j <= 16; ++j) { 938 if (j & 1) { 939 p224_point_add(out[j][0], out[j][1], out[j][2], out[1][0], out[1][1], 940 out[1][2], 0, out[j - 1][0], out[j - 1][1], out[j - 1][2]); 941 } else { 942 p224_point_double(out[j][0], out[j][1], out[j][2], out[j / 2][0], 943 out[j / 2][1], out[j / 2][2]); 944 } 945 } 946} 947 948static void ec_GFp_nistp224_point_mul(const EC_GROUP *group, EC_JACOBIAN *r, 949 const EC_JACOBIAN *p, 950 const EC_SCALAR *scalar) { 951 p224_felem p_pre_comp[17][3]; 952 ec_GFp_nistp224_make_precomp(p_pre_comp, p); 953 954 // Set nq to the point at infinity. 955 p224_felem nq[3], tmp[4]; 956 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem)); 957 958 int skip = 1; // Save two point operations in the first round. 959 for (size_t i = 220; i < 221; i--) { 960 if (!skip) { 961 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 962 } 963 964 // Add every 5 doublings. 965 if (i % 5 == 0) { 966 crypto_word_t bits = p224_get_bit(scalar, i + 4) << 5; 967 bits |= p224_get_bit(scalar, i + 3) << 4; 968 bits |= p224_get_bit(scalar, i + 2) << 3; 969 bits |= p224_get_bit(scalar, i + 1) << 2; 970 bits |= p224_get_bit(scalar, i) << 1; 971 bits |= p224_get_bit(scalar, i - 1); 972 crypto_word_t sign, digit; 973 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 974 975 // Select the point to add or subtract. 976 p224_select_point(digit, 17, (const p224_felem(*)[3])p_pre_comp, tmp); 977 p224_felem_neg(tmp[3], tmp[1]); // (X, -Y, Z) is the negative point 978 p224_copy_conditional(tmp[1], tmp[3], sign); 979 980 if (!skip) { 981 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, 982 tmp[0], tmp[1], tmp[2]); 983 } else { 984 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem)); 985 skip = 0; 986 } 987 } 988 } 989 990 // Reduce the output to its unique minimal representation. 991 p224_felem_to_generic(&r->X, nq[0]); 992 p224_felem_to_generic(&r->Y, nq[1]); 993 p224_felem_to_generic(&r->Z, nq[2]); 994} 995 996static void ec_GFp_nistp224_point_mul_base(const EC_GROUP *group, 997 EC_JACOBIAN *r, 998 const EC_SCALAR *scalar) { 999 // Set nq to the point at infinity. 1000 p224_felem nq[3], tmp[3]; 1001 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem)); 1002 1003 int skip = 1; // Save two point operations in the first round. 1004 for (size_t i = 27; i < 28; i--) { 1005 // double 1006 if (!skip) { 1007 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 1008 } 1009 1010 // First, look 28 bits upwards. 1011 crypto_word_t bits = p224_get_bit(scalar, i + 196) << 3; 1012 bits |= p224_get_bit(scalar, i + 140) << 2; 1013 bits |= p224_get_bit(scalar, i + 84) << 1; 1014 bits |= p224_get_bit(scalar, i + 28); 1015 // Select the point to add, in constant time. 1016 p224_select_point(bits, 16, g_p224_pre_comp[1], tmp); 1017 1018 if (!skip) { 1019 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 1020 tmp[0], tmp[1], tmp[2]); 1021 } else { 1022 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem)); 1023 skip = 0; 1024 } 1025 1026 // Second, look at the current position/ 1027 bits = p224_get_bit(scalar, i + 168) << 3; 1028 bits |= p224_get_bit(scalar, i + 112) << 2; 1029 bits |= p224_get_bit(scalar, i + 56) << 1; 1030 bits |= p224_get_bit(scalar, i); 1031 // Select the point to add, in constant time. 1032 p224_select_point(bits, 16, g_p224_pre_comp[0], tmp); 1033 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 1034 tmp[0], tmp[1], tmp[2]); 1035 } 1036 1037 // Reduce the output to its unique minimal representation. 1038 p224_felem_to_generic(&r->X, nq[0]); 1039 p224_felem_to_generic(&r->Y, nq[1]); 1040 p224_felem_to_generic(&r->Z, nq[2]); 1041} 1042 1043static void ec_GFp_nistp224_point_mul_public(const EC_GROUP *group, 1044 EC_JACOBIAN *r, 1045 const EC_SCALAR *g_scalar, 1046 const EC_JACOBIAN *p, 1047 const EC_SCALAR *p_scalar) { 1048 // TODO(davidben): If P-224 ECDSA verify performance ever matters, using 1049 // |ec_compute_wNAF| for |p_scalar| would likely be an easy improvement. 1050 p224_felem p_pre_comp[17][3]; 1051 ec_GFp_nistp224_make_precomp(p_pre_comp, p); 1052 1053 // Set nq to the point at infinity. 1054 p224_felem nq[3], tmp[3]; 1055 OPENSSL_memset(nq, 0, 3 * sizeof(p224_felem)); 1056 1057 // Loop over both scalars msb-to-lsb, interleaving additions of multiples of 1058 // the generator (two in each of the last 28 rounds) and additions of p (every 1059 // 5th round). 1060 int skip = 1; // Save two point operations in the first round. 1061 for (size_t i = 220; i < 221; i--) { 1062 if (!skip) { 1063 p224_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 1064 } 1065 1066 // Add multiples of the generator. 1067 if (i <= 27) { 1068 // First, look 28 bits upwards. 1069 crypto_word_t bits = p224_get_bit(g_scalar, i + 196) << 3; 1070 bits |= p224_get_bit(g_scalar, i + 140) << 2; 1071 bits |= p224_get_bit(g_scalar, i + 84) << 1; 1072 bits |= p224_get_bit(g_scalar, i + 28); 1073 1074 size_t index = (size_t)bits; 1075 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 1076 g_p224_pre_comp[1][index][0], g_p224_pre_comp[1][index][1], 1077 g_p224_pre_comp[1][index][2]); 1078 assert(!skip); 1079 1080 // Second, look at the current position. 1081 bits = p224_get_bit(g_scalar, i + 168) << 3; 1082 bits |= p224_get_bit(g_scalar, i + 112) << 2; 1083 bits |= p224_get_bit(g_scalar, i + 56) << 1; 1084 bits |= p224_get_bit(g_scalar, i); 1085 index = (size_t)bits; 1086 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, 1087 g_p224_pre_comp[0][index][0], g_p224_pre_comp[0][index][1], 1088 g_p224_pre_comp[0][index][2]); 1089 } 1090 1091 // Incorporate |p_scalar| every 5 doublings. 1092 if (i % 5 == 0) { 1093 crypto_word_t bits = p224_get_bit(p_scalar, i + 4) << 5; 1094 bits |= p224_get_bit(p_scalar, i + 3) << 4; 1095 bits |= p224_get_bit(p_scalar, i + 2) << 3; 1096 bits |= p224_get_bit(p_scalar, i + 1) << 2; 1097 bits |= p224_get_bit(p_scalar, i) << 1; 1098 bits |= p224_get_bit(p_scalar, i - 1); 1099 crypto_word_t sign, digit; 1100 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 1101 1102 // Select the point to add or subtract. 1103 OPENSSL_memcpy(tmp, p_pre_comp[digit], 3 * sizeof(p224_felem)); 1104 if (sign) { 1105 p224_felem_neg(tmp[1], tmp[1]); // (X, -Y, Z) is the negative point 1106 } 1107 1108 if (!skip) { 1109 p224_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, 1110 tmp[0], tmp[1], tmp[2]); 1111 } else { 1112 OPENSSL_memcpy(nq, tmp, 3 * sizeof(p224_felem)); 1113 skip = 0; 1114 } 1115 } 1116 } 1117 1118 // Reduce the output to its unique minimal representation. 1119 p224_felem_to_generic(&r->X, nq[0]); 1120 p224_felem_to_generic(&r->Y, nq[1]); 1121 p224_felem_to_generic(&r->Z, nq[2]); 1122} 1123 1124static void ec_GFp_nistp224_felem_mul(const EC_GROUP *group, EC_FELEM *r, 1125 const EC_FELEM *a, const EC_FELEM *b) { 1126 p224_felem felem1, felem2; 1127 p224_widefelem wide; 1128 p224_generic_to_felem(felem1, a); 1129 p224_generic_to_felem(felem2, b); 1130 p224_felem_mul(wide, felem1, felem2); 1131 p224_felem_reduce(felem1, wide); 1132 p224_felem_to_generic(r, felem1); 1133} 1134 1135static void ec_GFp_nistp224_felem_sqr(const EC_GROUP *group, EC_FELEM *r, 1136 const EC_FELEM *a) { 1137 p224_felem felem; 1138 p224_generic_to_felem(felem, a); 1139 p224_widefelem wide; 1140 p224_felem_square(wide, felem); 1141 p224_felem_reduce(felem, wide); 1142 p224_felem_to_generic(r, felem); 1143} 1144 1145DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp224_method) { 1146 out->point_get_affine_coordinates = 1147 ec_GFp_nistp224_point_get_affine_coordinates; 1148 out->add = ec_GFp_nistp224_add; 1149 out->dbl = ec_GFp_nistp224_dbl; 1150 out->mul = ec_GFp_nistp224_point_mul; 1151 out->mul_base = ec_GFp_nistp224_point_mul_base; 1152 out->mul_public = ec_GFp_nistp224_point_mul_public; 1153 out->felem_mul = ec_GFp_nistp224_felem_mul; 1154 out->felem_sqr = ec_GFp_nistp224_felem_sqr; 1155 out->felem_to_bytes = ec_GFp_simple_felem_to_bytes; 1156 out->felem_from_bytes = ec_GFp_simple_felem_from_bytes; 1157 out->scalar_inv0_montgomery = ec_simple_scalar_inv0_montgomery; 1158 out->scalar_to_montgomery_inv_vartime = 1159 ec_simple_scalar_to_montgomery_inv_vartime; 1160 out->cmp_x_coordinate = ec_GFp_simple_cmp_x_coordinate; 1161} 1162 1163#endif // BORINGSSL_HAS_UINT128 && !SMALL 1164