Alexandre Lision | 7fd5d3d | 2013-12-04 13:06:40 -0500 | [diff] [blame] | 1 | /* Copyright 2008, Google Inc. |
| 2 | * All rights reserved. |
| 3 | * |
| 4 | * Redistribution and use in source and binary forms, with or without |
| 5 | * modification, are permitted provided that the following conditions are |
| 6 | * met: |
| 7 | * |
| 8 | * * Redistributions of source code must retain the above copyright |
| 9 | * notice, this list of conditions and the following disclaimer. |
| 10 | * * Redistributions in binary form must reproduce the above |
| 11 | * copyright notice, this list of conditions and the following disclaimer |
| 12 | * in the documentation and/or other materials provided with the |
| 13 | * distribution. |
| 14 | * * Neither the name of Google Inc. nor the names of its |
| 15 | * contributors may be used to endorse or promote products derived from |
| 16 | * this software without specific prior written permission. |
| 17 | * |
| 18 | * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS |
| 19 | * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT |
| 20 | * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR |
| 21 | * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT |
| 22 | * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
| 23 | * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT |
| 24 | * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, |
| 25 | * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY |
| 26 | * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT |
| 27 | * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE |
| 28 | * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. |
| 29 | * |
| 30 | * curve25519-donna: Curve25519 elliptic curve, public key function |
| 31 | * |
| 32 | * http://code.google.com/p/curve25519-donna/ |
| 33 | * |
| 34 | * Adam Langley <agl@imperialviolet.org> |
| 35 | * |
| 36 | * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to> |
| 37 | * |
| 38 | * More information about curve25519 can be found here |
| 39 | * http://cr.yp.to/ecdh.html |
| 40 | * |
| 41 | * djb's sample implementation of curve25519 is written in a special assembly |
| 42 | * language called qhasm and uses the floating point registers. |
| 43 | * |
| 44 | * This is, almost, a clean room reimplementation from the curve25519 paper. It |
| 45 | * uses many of the tricks described therein. Only the crecip function is taken |
| 46 | * from the sample implementation. |
| 47 | */ |
| 48 | |
| 49 | #include <string.h> |
| 50 | #include <stdint.h> |
| 51 | |
| 52 | #ifdef _MSC_VER |
| 53 | #define inline __inline |
| 54 | #endif |
| 55 | |
| 56 | typedef uint8_t u8; |
| 57 | typedef int32_t s32; |
| 58 | typedef int64_t limb; |
| 59 | |
| 60 | /* Field element representation: |
| 61 | * |
| 62 | * Field elements are written as an array of signed, 64-bit limbs, least |
| 63 | * significant first. The value of the field element is: |
| 64 | * x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ... |
| 65 | * |
| 66 | * i.e. the limbs are 26, 25, 26, 25, ... bits wide. |
| 67 | */ |
| 68 | |
| 69 | /* Sum two numbers: output += in */ |
| 70 | static void fsum(limb *output, const limb *in) { |
| 71 | unsigned i; |
| 72 | for (i = 0; i < 10; i += 2) { |
| 73 | output[0+i] = (output[0+i] + in[0+i]); |
| 74 | output[1+i] = (output[1+i] + in[1+i]); |
| 75 | } |
| 76 | } |
| 77 | |
| 78 | /* Find the difference of two numbers: output = in - output |
| 79 | * (note the order of the arguments!) |
| 80 | */ |
| 81 | static void fdifference(limb *output, const limb *in) { |
| 82 | unsigned i; |
| 83 | for (i = 0; i < 10; ++i) { |
| 84 | output[i] = (in[i] - output[i]); |
| 85 | } |
| 86 | } |
| 87 | |
| 88 | /* Multiply a number by a scalar: output = in * scalar */ |
| 89 | static void fscalar_product(limb *output, const limb *in, const limb scalar) { |
| 90 | unsigned i; |
| 91 | for (i = 0; i < 10; ++i) { |
| 92 | output[i] = in[i] * scalar; |
| 93 | } |
| 94 | } |
| 95 | |
| 96 | /* Multiply two numbers: output = in2 * in |
| 97 | * |
| 98 | * output must be distinct to both inputs. The inputs are reduced coefficient |
| 99 | * form, the output is not. |
| 100 | */ |
| 101 | static void fproduct(limb *output, const limb *in2, const limb *in) { |
| 102 | output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]); |
| 103 | output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) + |
| 104 | ((limb) ((s32) in2[1])) * ((s32) in[0]); |
| 105 | output[2] = 2 * ((limb) ((s32) in2[1])) * ((s32) in[1]) + |
| 106 | ((limb) ((s32) in2[0])) * ((s32) in[2]) + |
| 107 | ((limb) ((s32) in2[2])) * ((s32) in[0]); |
| 108 | output[3] = ((limb) ((s32) in2[1])) * ((s32) in[2]) + |
| 109 | ((limb) ((s32) in2[2])) * ((s32) in[1]) + |
| 110 | ((limb) ((s32) in2[0])) * ((s32) in[3]) + |
| 111 | ((limb) ((s32) in2[3])) * ((s32) in[0]); |
| 112 | output[4] = ((limb) ((s32) in2[2])) * ((s32) in[2]) + |
| 113 | 2 * (((limb) ((s32) in2[1])) * ((s32) in[3]) + |
| 114 | ((limb) ((s32) in2[3])) * ((s32) in[1])) + |
| 115 | ((limb) ((s32) in2[0])) * ((s32) in[4]) + |
| 116 | ((limb) ((s32) in2[4])) * ((s32) in[0]); |
| 117 | output[5] = ((limb) ((s32) in2[2])) * ((s32) in[3]) + |
| 118 | ((limb) ((s32) in2[3])) * ((s32) in[2]) + |
| 119 | ((limb) ((s32) in2[1])) * ((s32) in[4]) + |
| 120 | ((limb) ((s32) in2[4])) * ((s32) in[1]) + |
| 121 | ((limb) ((s32) in2[0])) * ((s32) in[5]) + |
| 122 | ((limb) ((s32) in2[5])) * ((s32) in[0]); |
| 123 | output[6] = 2 * (((limb) ((s32) in2[3])) * ((s32) in[3]) + |
| 124 | ((limb) ((s32) in2[1])) * ((s32) in[5]) + |
| 125 | ((limb) ((s32) in2[5])) * ((s32) in[1])) + |
| 126 | ((limb) ((s32) in2[2])) * ((s32) in[4]) + |
| 127 | ((limb) ((s32) in2[4])) * ((s32) in[2]) + |
| 128 | ((limb) ((s32) in2[0])) * ((s32) in[6]) + |
| 129 | ((limb) ((s32) in2[6])) * ((s32) in[0]); |
| 130 | output[7] = ((limb) ((s32) in2[3])) * ((s32) in[4]) + |
| 131 | ((limb) ((s32) in2[4])) * ((s32) in[3]) + |
| 132 | ((limb) ((s32) in2[2])) * ((s32) in[5]) + |
| 133 | ((limb) ((s32) in2[5])) * ((s32) in[2]) + |
| 134 | ((limb) ((s32) in2[1])) * ((s32) in[6]) + |
| 135 | ((limb) ((s32) in2[6])) * ((s32) in[1]) + |
| 136 | ((limb) ((s32) in2[0])) * ((s32) in[7]) + |
| 137 | ((limb) ((s32) in2[7])) * ((s32) in[0]); |
| 138 | output[8] = ((limb) ((s32) in2[4])) * ((s32) in[4]) + |
| 139 | 2 * (((limb) ((s32) in2[3])) * ((s32) in[5]) + |
| 140 | ((limb) ((s32) in2[5])) * ((s32) in[3]) + |
| 141 | ((limb) ((s32) in2[1])) * ((s32) in[7]) + |
| 142 | ((limb) ((s32) in2[7])) * ((s32) in[1])) + |
| 143 | ((limb) ((s32) in2[2])) * ((s32) in[6]) + |
| 144 | ((limb) ((s32) in2[6])) * ((s32) in[2]) + |
| 145 | ((limb) ((s32) in2[0])) * ((s32) in[8]) + |
| 146 | ((limb) ((s32) in2[8])) * ((s32) in[0]); |
| 147 | output[9] = ((limb) ((s32) in2[4])) * ((s32) in[5]) + |
| 148 | ((limb) ((s32) in2[5])) * ((s32) in[4]) + |
| 149 | ((limb) ((s32) in2[3])) * ((s32) in[6]) + |
| 150 | ((limb) ((s32) in2[6])) * ((s32) in[3]) + |
| 151 | ((limb) ((s32) in2[2])) * ((s32) in[7]) + |
| 152 | ((limb) ((s32) in2[7])) * ((s32) in[2]) + |
| 153 | ((limb) ((s32) in2[1])) * ((s32) in[8]) + |
| 154 | ((limb) ((s32) in2[8])) * ((s32) in[1]) + |
| 155 | ((limb) ((s32) in2[0])) * ((s32) in[9]) + |
| 156 | ((limb) ((s32) in2[9])) * ((s32) in[0]); |
| 157 | output[10] = 2 * (((limb) ((s32) in2[5])) * ((s32) in[5]) + |
| 158 | ((limb) ((s32) in2[3])) * ((s32) in[7]) + |
| 159 | ((limb) ((s32) in2[7])) * ((s32) in[3]) + |
| 160 | ((limb) ((s32) in2[1])) * ((s32) in[9]) + |
| 161 | ((limb) ((s32) in2[9])) * ((s32) in[1])) + |
| 162 | ((limb) ((s32) in2[4])) * ((s32) in[6]) + |
| 163 | ((limb) ((s32) in2[6])) * ((s32) in[4]) + |
| 164 | ((limb) ((s32) in2[2])) * ((s32) in[8]) + |
| 165 | ((limb) ((s32) in2[8])) * ((s32) in[2]); |
| 166 | output[11] = ((limb) ((s32) in2[5])) * ((s32) in[6]) + |
| 167 | ((limb) ((s32) in2[6])) * ((s32) in[5]) + |
| 168 | ((limb) ((s32) in2[4])) * ((s32) in[7]) + |
| 169 | ((limb) ((s32) in2[7])) * ((s32) in[4]) + |
| 170 | ((limb) ((s32) in2[3])) * ((s32) in[8]) + |
| 171 | ((limb) ((s32) in2[8])) * ((s32) in[3]) + |
| 172 | ((limb) ((s32) in2[2])) * ((s32) in[9]) + |
| 173 | ((limb) ((s32) in2[9])) * ((s32) in[2]); |
| 174 | output[12] = ((limb) ((s32) in2[6])) * ((s32) in[6]) + |
| 175 | 2 * (((limb) ((s32) in2[5])) * ((s32) in[7]) + |
| 176 | ((limb) ((s32) in2[7])) * ((s32) in[5]) + |
| 177 | ((limb) ((s32) in2[3])) * ((s32) in[9]) + |
| 178 | ((limb) ((s32) in2[9])) * ((s32) in[3])) + |
| 179 | ((limb) ((s32) in2[4])) * ((s32) in[8]) + |
| 180 | ((limb) ((s32) in2[8])) * ((s32) in[4]); |
| 181 | output[13] = ((limb) ((s32) in2[6])) * ((s32) in[7]) + |
| 182 | ((limb) ((s32) in2[7])) * ((s32) in[6]) + |
| 183 | ((limb) ((s32) in2[5])) * ((s32) in[8]) + |
| 184 | ((limb) ((s32) in2[8])) * ((s32) in[5]) + |
| 185 | ((limb) ((s32) in2[4])) * ((s32) in[9]) + |
| 186 | ((limb) ((s32) in2[9])) * ((s32) in[4]); |
| 187 | output[14] = 2 * (((limb) ((s32) in2[7])) * ((s32) in[7]) + |
| 188 | ((limb) ((s32) in2[5])) * ((s32) in[9]) + |
| 189 | ((limb) ((s32) in2[9])) * ((s32) in[5])) + |
| 190 | ((limb) ((s32) in2[6])) * ((s32) in[8]) + |
| 191 | ((limb) ((s32) in2[8])) * ((s32) in[6]); |
| 192 | output[15] = ((limb) ((s32) in2[7])) * ((s32) in[8]) + |
| 193 | ((limb) ((s32) in2[8])) * ((s32) in[7]) + |
| 194 | ((limb) ((s32) in2[6])) * ((s32) in[9]) + |
| 195 | ((limb) ((s32) in2[9])) * ((s32) in[6]); |
| 196 | output[16] = ((limb) ((s32) in2[8])) * ((s32) in[8]) + |
| 197 | 2 * (((limb) ((s32) in2[7])) * ((s32) in[9]) + |
| 198 | ((limb) ((s32) in2[9])) * ((s32) in[7])); |
| 199 | output[17] = ((limb) ((s32) in2[8])) * ((s32) in[9]) + |
| 200 | ((limb) ((s32) in2[9])) * ((s32) in[8]); |
| 201 | output[18] = 2 * ((limb) ((s32) in2[9])) * ((s32) in[9]); |
| 202 | } |
| 203 | |
| 204 | /* Reduce a long form to a short form by taking the input mod 2^255 - 19. */ |
| 205 | static void freduce_degree(limb *output) { |
| 206 | /* Each of these shifts and adds ends up multiplying the value by 19. */ |
| 207 | output[8] += output[18] << 4; |
| 208 | output[8] += output[18] << 1; |
| 209 | output[8] += output[18]; |
| 210 | output[7] += output[17] << 4; |
| 211 | output[7] += output[17] << 1; |
| 212 | output[7] += output[17]; |
| 213 | output[6] += output[16] << 4; |
| 214 | output[6] += output[16] << 1; |
| 215 | output[6] += output[16]; |
| 216 | output[5] += output[15] << 4; |
| 217 | output[5] += output[15] << 1; |
| 218 | output[5] += output[15]; |
| 219 | output[4] += output[14] << 4; |
| 220 | output[4] += output[14] << 1; |
| 221 | output[4] += output[14]; |
| 222 | output[3] += output[13] << 4; |
| 223 | output[3] += output[13] << 1; |
| 224 | output[3] += output[13]; |
| 225 | output[2] += output[12] << 4; |
| 226 | output[2] += output[12] << 1; |
| 227 | output[2] += output[12]; |
| 228 | output[1] += output[11] << 4; |
| 229 | output[1] += output[11] << 1; |
| 230 | output[1] += output[11]; |
| 231 | output[0] += output[10] << 4; |
| 232 | output[0] += output[10] << 1; |
| 233 | output[0] += output[10]; |
| 234 | } |
| 235 | |
| 236 | #if (-1 & 3) != 3 |
| 237 | #error "This code only works on a two's complement system" |
| 238 | #endif |
| 239 | |
| 240 | /* return v / 2^26, using only shifts and adds. */ |
| 241 | static limb div_by_2_26(const limb v) |
| 242 | { |
| 243 | /* High word of v; no shift needed*/ |
| 244 | const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32); |
| 245 | /* Set to all 1s if v was negative; else set to 0s. */ |
| 246 | const int32_t sign = ((int32_t) highword) >> 31; |
| 247 | /* Set to 0x3ffffff if v was negative; else set to 0. */ |
| 248 | const int32_t roundoff = ((uint32_t) sign) >> 6; |
| 249 | /* Should return v / (1<<26) */ |
| 250 | return (v + roundoff) >> 26; |
| 251 | } |
| 252 | |
| 253 | /* return v / (2^25), using only shifts and adds. */ |
| 254 | static limb div_by_2_25(const limb v) |
| 255 | { |
| 256 | /* High word of v; no shift needed*/ |
| 257 | const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32); |
| 258 | /* Set to all 1s if v was negative; else set to 0s. */ |
| 259 | const int32_t sign = ((int32_t) highword) >> 31; |
| 260 | /* Set to 0x1ffffff if v was negative; else set to 0. */ |
| 261 | const int32_t roundoff = ((uint32_t) sign) >> 7; |
| 262 | /* Should return v / (1<<25) */ |
| 263 | return (v + roundoff) >> 25; |
| 264 | } |
| 265 | |
| 266 | static s32 div_s32_by_2_25(const s32 v) |
| 267 | { |
| 268 | const s32 roundoff = ((uint32_t)(v >> 31)) >> 7; |
| 269 | return (v + roundoff) >> 25; |
| 270 | } |
| 271 | |
| 272 | /* Reduce all coefficients of the short form input so that |x| < 2^26. |
| 273 | * |
| 274 | * On entry: |output[i]| < 2^62 |
| 275 | */ |
| 276 | static void freduce_coefficients(limb *output) { |
| 277 | unsigned i; |
| 278 | |
| 279 | output[10] = 0; |
| 280 | |
| 281 | for (i = 0; i < 10; i += 2) { |
| 282 | limb over = div_by_2_26(output[i]); |
| 283 | output[i] -= over << 26; |
| 284 | output[i+1] += over; |
| 285 | |
| 286 | over = div_by_2_25(output[i+1]); |
| 287 | output[i+1] -= over << 25; |
| 288 | output[i+2] += over; |
| 289 | } |
| 290 | /* Now |output[10]| < 2 ^ 38 and all other coefficients are reduced. */ |
| 291 | output[0] += output[10] << 4; |
| 292 | output[0] += output[10] << 1; |
| 293 | output[0] += output[10]; |
| 294 | |
| 295 | output[10] = 0; |
| 296 | |
| 297 | /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19 * 2^38 |
| 298 | * So |over| will be no more than 77825 */ |
| 299 | { |
| 300 | limb over = div_by_2_26(output[0]); |
| 301 | output[0] -= over << 26; |
| 302 | output[1] += over; |
| 303 | } |
| 304 | |
| 305 | /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 77825 |
| 306 | * So |over| will be no more than 1. */ |
| 307 | { |
| 308 | /* output[1] fits in 32 bits, so we can use div_s32_by_2_25 here. */ |
| 309 | s32 over32 = div_s32_by_2_25((s32) output[1]); |
| 310 | output[1] -= over32 << 25; |
| 311 | output[2] += over32; |
| 312 | } |
| 313 | |
| 314 | /* Finally, output[0,1,3..9] are reduced, and output[2] is "nearly reduced": |
| 315 | * we have |output[2]| <= 2^26. This is good enough for all of our math, |
| 316 | * but it will require an extra freduce_coefficients before fcontract. */ |
| 317 | } |
| 318 | |
| 319 | /* A helpful wrapper around fproduct: output = in * in2. |
| 320 | * |
| 321 | * output must be distinct to both inputs. The output is reduced degree and |
| 322 | * reduced coefficient. |
| 323 | */ |
| 324 | static void |
| 325 | fmul(limb *output, const limb *in, const limb *in2) { |
| 326 | limb t[19]; |
| 327 | fproduct(t, in, in2); |
| 328 | freduce_degree(t); |
| 329 | freduce_coefficients(t); |
| 330 | memcpy(output, t, sizeof(limb) * 10); |
| 331 | } |
| 332 | |
| 333 | static void fsquare_inner(limb *output, const limb *in) { |
| 334 | output[0] = ((limb) ((s32) in[0])) * ((s32) in[0]); |
| 335 | output[1] = 2 * ((limb) ((s32) in[0])) * ((s32) in[1]); |
| 336 | output[2] = 2 * (((limb) ((s32) in[1])) * ((s32) in[1]) + |
| 337 | ((limb) ((s32) in[0])) * ((s32) in[2])); |
| 338 | output[3] = 2 * (((limb) ((s32) in[1])) * ((s32) in[2]) + |
| 339 | ((limb) ((s32) in[0])) * ((s32) in[3])); |
| 340 | output[4] = ((limb) ((s32) in[2])) * ((s32) in[2]) + |
| 341 | 4 * ((limb) ((s32) in[1])) * ((s32) in[3]) + |
| 342 | 2 * ((limb) ((s32) in[0])) * ((s32) in[4]); |
| 343 | output[5] = 2 * (((limb) ((s32) in[2])) * ((s32) in[3]) + |
| 344 | ((limb) ((s32) in[1])) * ((s32) in[4]) + |
| 345 | ((limb) ((s32) in[0])) * ((s32) in[5])); |
| 346 | output[6] = 2 * (((limb) ((s32) in[3])) * ((s32) in[3]) + |
| 347 | ((limb) ((s32) in[2])) * ((s32) in[4]) + |
| 348 | ((limb) ((s32) in[0])) * ((s32) in[6]) + |
| 349 | 2 * ((limb) ((s32) in[1])) * ((s32) in[5])); |
| 350 | output[7] = 2 * (((limb) ((s32) in[3])) * ((s32) in[4]) + |
| 351 | ((limb) ((s32) in[2])) * ((s32) in[5]) + |
| 352 | ((limb) ((s32) in[1])) * ((s32) in[6]) + |
| 353 | ((limb) ((s32) in[0])) * ((s32) in[7])); |
| 354 | output[8] = ((limb) ((s32) in[4])) * ((s32) in[4]) + |
| 355 | 2 * (((limb) ((s32) in[2])) * ((s32) in[6]) + |
| 356 | ((limb) ((s32) in[0])) * ((s32) in[8]) + |
| 357 | 2 * (((limb) ((s32) in[1])) * ((s32) in[7]) + |
| 358 | ((limb) ((s32) in[3])) * ((s32) in[5]))); |
| 359 | output[9] = 2 * (((limb) ((s32) in[4])) * ((s32) in[5]) + |
| 360 | ((limb) ((s32) in[3])) * ((s32) in[6]) + |
| 361 | ((limb) ((s32) in[2])) * ((s32) in[7]) + |
| 362 | ((limb) ((s32) in[1])) * ((s32) in[8]) + |
| 363 | ((limb) ((s32) in[0])) * ((s32) in[9])); |
| 364 | output[10] = 2 * (((limb) ((s32) in[5])) * ((s32) in[5]) + |
| 365 | ((limb) ((s32) in[4])) * ((s32) in[6]) + |
| 366 | ((limb) ((s32) in[2])) * ((s32) in[8]) + |
| 367 | 2 * (((limb) ((s32) in[3])) * ((s32) in[7]) + |
| 368 | ((limb) ((s32) in[1])) * ((s32) in[9]))); |
| 369 | output[11] = 2 * (((limb) ((s32) in[5])) * ((s32) in[6]) + |
| 370 | ((limb) ((s32) in[4])) * ((s32) in[7]) + |
| 371 | ((limb) ((s32) in[3])) * ((s32) in[8]) + |
| 372 | ((limb) ((s32) in[2])) * ((s32) in[9])); |
| 373 | output[12] = ((limb) ((s32) in[6])) * ((s32) in[6]) + |
| 374 | 2 * (((limb) ((s32) in[4])) * ((s32) in[8]) + |
| 375 | 2 * (((limb) ((s32) in[5])) * ((s32) in[7]) + |
| 376 | ((limb) ((s32) in[3])) * ((s32) in[9]))); |
| 377 | output[13] = 2 * (((limb) ((s32) in[6])) * ((s32) in[7]) + |
| 378 | ((limb) ((s32) in[5])) * ((s32) in[8]) + |
| 379 | ((limb) ((s32) in[4])) * ((s32) in[9])); |
| 380 | output[14] = 2 * (((limb) ((s32) in[7])) * ((s32) in[7]) + |
| 381 | ((limb) ((s32) in[6])) * ((s32) in[8]) + |
| 382 | 2 * ((limb) ((s32) in[5])) * ((s32) in[9])); |
| 383 | output[15] = 2 * (((limb) ((s32) in[7])) * ((s32) in[8]) + |
| 384 | ((limb) ((s32) in[6])) * ((s32) in[9])); |
| 385 | output[16] = ((limb) ((s32) in[8])) * ((s32) in[8]) + |
| 386 | 4 * ((limb) ((s32) in[7])) * ((s32) in[9]); |
| 387 | output[17] = 2 * ((limb) ((s32) in[8])) * ((s32) in[9]); |
| 388 | output[18] = 2 * ((limb) ((s32) in[9])) * ((s32) in[9]); |
| 389 | } |
| 390 | |
| 391 | static void |
| 392 | fsquare(limb *output, const limb *in) { |
| 393 | limb t[19]; |
| 394 | fsquare_inner(t, in); |
| 395 | freduce_degree(t); |
| 396 | freduce_coefficients(t); |
| 397 | memcpy(output, t, sizeof(limb) * 10); |
| 398 | } |
| 399 | |
| 400 | /* Take a little-endian, 32-byte number and expand it into polynomial form */ |
| 401 | static void |
| 402 | fexpand(limb *output, const u8 *input) { |
| 403 | #define F(n,start,shift,mask) \ |
| 404 | output[n] = ((((limb) input[start + 0]) | \ |
| 405 | ((limb) input[start + 1]) << 8 | \ |
| 406 | ((limb) input[start + 2]) << 16 | \ |
| 407 | ((limb) input[start + 3]) << 24) >> shift) & mask; |
| 408 | F(0, 0, 0, 0x3ffffff); |
| 409 | F(1, 3, 2, 0x1ffffff); |
| 410 | F(2, 6, 3, 0x3ffffff); |
| 411 | F(3, 9, 5, 0x1ffffff); |
| 412 | F(4, 12, 6, 0x3ffffff); |
| 413 | F(5, 16, 0, 0x1ffffff); |
| 414 | F(6, 19, 1, 0x3ffffff); |
| 415 | F(7, 22, 3, 0x1ffffff); |
| 416 | F(8, 25, 4, 0x3ffffff); |
| 417 | F(9, 28, 6, 0x1ffffff); |
| 418 | #undef F |
| 419 | } |
| 420 | |
| 421 | #if (-32 >> 1) != -16 |
| 422 | #error "This code only works when >> does sign-extension on negative numbers" |
| 423 | #endif |
| 424 | |
| 425 | /* Take a fully reduced polynomial form number and contract it into a |
| 426 | * little-endian, 32-byte array |
| 427 | */ |
| 428 | static void |
| 429 | fcontract(u8 *output, limb *input) { |
| 430 | int i; |
| 431 | int j; |
| 432 | |
| 433 | for (j = 0; j < 2; ++j) { |
| 434 | for (i = 0; i < 9; ++i) { |
| 435 | if ((i & 1) == 1) { |
| 436 | /* This calculation is a time-invariant way to make input[i] positive |
| 437 | by borrowing from the next-larger limb. |
| 438 | */ |
| 439 | const s32 mask = (s32)(input[i]) >> 31; |
| 440 | const s32 carry = -(((s32)(input[i]) & mask) >> 25); |
| 441 | input[i] = (s32)(input[i]) + (carry << 25); |
| 442 | input[i+1] = (s32)(input[i+1]) - carry; |
| 443 | } else { |
| 444 | const s32 mask = (s32)(input[i]) >> 31; |
| 445 | const s32 carry = -(((s32)(input[i]) & mask) >> 26); |
| 446 | input[i] = (s32)(input[i]) + (carry << 26); |
| 447 | input[i+1] = (s32)(input[i+1]) - carry; |
| 448 | } |
| 449 | } |
| 450 | { |
| 451 | const s32 mask = (s32)(input[9]) >> 31; |
| 452 | const s32 carry = -(((s32)(input[9]) & mask) >> 25); |
| 453 | input[9] = (s32)(input[9]) + (carry << 25); |
| 454 | input[0] = (s32)(input[0]) - (carry * 19); |
| 455 | } |
| 456 | } |
| 457 | |
| 458 | /* The first borrow-propagation pass above ended with every limb |
| 459 | except (possibly) input[0] non-negative. |
| 460 | |
| 461 | Since each input limb except input[0] is decreased by at most 1 |
| 462 | by a borrow-propagation pass, the second borrow-propagation pass |
| 463 | could only have wrapped around to decrease input[0] again if the |
| 464 | first pass left input[0] negative *and* input[1] through input[9] |
| 465 | were all zero. In that case, input[1] is now 2^25 - 1, and this |
| 466 | last borrow-propagation step will leave input[1] non-negative. |
| 467 | */ |
| 468 | { |
| 469 | const s32 mask = (s32)(input[0]) >> 31; |
| 470 | const s32 carry = -(((s32)(input[0]) & mask) >> 26); |
| 471 | input[0] = (s32)(input[0]) + (carry << 26); |
| 472 | input[1] = (s32)(input[1]) - carry; |
| 473 | } |
| 474 | |
| 475 | /* Both passes through the above loop, plus the last 0-to-1 step, are |
| 476 | necessary: if input[9] is -1 and input[0] through input[8] are 0, |
| 477 | negative values will remain in the array until the end. |
| 478 | */ |
| 479 | |
| 480 | input[1] <<= 2; |
| 481 | input[2] <<= 3; |
| 482 | input[3] <<= 5; |
| 483 | input[4] <<= 6; |
| 484 | input[6] <<= 1; |
| 485 | input[7] <<= 3; |
| 486 | input[8] <<= 4; |
| 487 | input[9] <<= 6; |
| 488 | #define F(i, s) \ |
| 489 | output[s+0] |= input[i] & 0xff; \ |
| 490 | output[s+1] = (input[i] >> 8) & 0xff; \ |
| 491 | output[s+2] = (input[i] >> 16) & 0xff; \ |
| 492 | output[s+3] = (input[i] >> 24) & 0xff; |
| 493 | output[0] = 0; |
| 494 | output[16] = 0; |
| 495 | F(0,0); |
| 496 | F(1,3); |
| 497 | F(2,6); |
| 498 | F(3,9); |
| 499 | F(4,12); |
| 500 | F(5,16); |
| 501 | F(6,19); |
| 502 | F(7,22); |
| 503 | F(8,25); |
| 504 | F(9,28); |
| 505 | #undef F |
| 506 | } |
| 507 | |
| 508 | /* Input: Q, Q', Q-Q' |
| 509 | * Output: 2Q, Q+Q' |
| 510 | * |
| 511 | * x2 z3: long form |
| 512 | * x3 z3: long form |
| 513 | * x z: short form, destroyed |
| 514 | * xprime zprime: short form, destroyed |
| 515 | * qmqp: short form, preserved |
| 516 | */ |
| 517 | static void fmonty(limb *x2, limb *z2, /* output 2Q */ |
| 518 | limb *x3, limb *z3, /* output Q + Q' */ |
| 519 | limb *x, limb *z, /* input Q */ |
| 520 | limb *xprime, limb *zprime, /* input Q' */ |
| 521 | const limb *qmqp /* input Q - Q' */) { |
| 522 | limb origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19], |
| 523 | zzprime[19], zzzprime[19], xxxprime[19]; |
| 524 | |
| 525 | memcpy(origx, x, 10 * sizeof(limb)); |
| 526 | fsum(x, z); |
| 527 | fdifference(z, origx); /* does x - z */ |
| 528 | |
| 529 | memcpy(origxprime, xprime, sizeof(limb) * 10); |
| 530 | fsum(xprime, zprime); |
| 531 | fdifference(zprime, origxprime); |
| 532 | fproduct(xxprime, xprime, z); |
| 533 | fproduct(zzprime, x, zprime); |
| 534 | freduce_degree(xxprime); |
| 535 | freduce_coefficients(xxprime); |
| 536 | freduce_degree(zzprime); |
| 537 | freduce_coefficients(zzprime); |
| 538 | memcpy(origxprime, xxprime, sizeof(limb) * 10); |
| 539 | fsum(xxprime, zzprime); |
| 540 | fdifference(zzprime, origxprime); |
| 541 | fsquare(xxxprime, xxprime); |
| 542 | fsquare(zzzprime, zzprime); |
| 543 | fproduct(zzprime, zzzprime, qmqp); |
| 544 | freduce_degree(zzprime); |
| 545 | freduce_coefficients(zzprime); |
| 546 | memcpy(x3, xxxprime, sizeof(limb) * 10); |
| 547 | memcpy(z3, zzprime, sizeof(limb) * 10); |
| 548 | |
| 549 | fsquare(xx, x); |
| 550 | fsquare(zz, z); |
| 551 | fproduct(x2, xx, zz); |
| 552 | freduce_degree(x2); |
| 553 | freduce_coefficients(x2); |
| 554 | fdifference(zz, xx); /* does zz = xx - zz */ |
| 555 | memset(zzz + 10, 0, sizeof(limb) * 9); |
| 556 | fscalar_product(zzz, zz, 121665); |
| 557 | /* No need to call freduce_degree here: |
| 558 | fscalar_product doesn't increase the degree of its input. |
| 559 | */ |
| 560 | freduce_coefficients(zzz); |
| 561 | fsum(zzz, xx); |
| 562 | fproduct(z2, zz, zzz); |
| 563 | freduce_degree(z2); |
| 564 | freduce_coefficients(z2); |
| 565 | } |
| 566 | |
| 567 | /* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave |
| 568 | * them unchanged if 'iswap' is 0. Runs in data-invariant time to avoid |
| 569 | * side-channel attacks. |
| 570 | * |
| 571 | * NOTE that this function requires that 'iswap' be 1 or 0; other values give |
| 572 | * wrong results. Also, the two limb arrays must be in reduced-coefficient, |
| 573 | * reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped, |
| 574 | * and all all values in a[0..9],b[0..9] must have magnitude less than |
| 575 | * INT32_MAX. |
| 576 | */ |
| 577 | static void |
| 578 | swap_conditional(limb a[19], limb b[19], limb iswap) { |
| 579 | unsigned i; |
| 580 | const s32 swap = (s32) -iswap; |
| 581 | |
| 582 | for (i = 0; i < 10; ++i) { |
| 583 | const s32 x = swap & ( ((s32)a[i]) ^ ((s32)b[i]) ); |
| 584 | a[i] = ((s32)a[i]) ^ x; |
| 585 | b[i] = ((s32)b[i]) ^ x; |
| 586 | } |
| 587 | } |
| 588 | |
| 589 | /* Calculates nQ where Q is the x-coordinate of a point on the curve |
| 590 | * |
| 591 | * resultx/resultz: the x coordinate of the resulting curve point (short form) |
| 592 | * n: a little endian, 32-byte number |
| 593 | * q: a point of the curve (short form) |
| 594 | */ |
| 595 | static void |
| 596 | cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) { |
| 597 | limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0}; |
| 598 | limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t; |
| 599 | limb e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1}; |
| 600 | limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h; |
| 601 | |
| 602 | unsigned i, j; |
| 603 | |
| 604 | memcpy(nqpqx, q, sizeof(limb) * 10); |
| 605 | |
| 606 | for (i = 0; i < 32; ++i) { |
| 607 | u8 byte = n[31 - i]; |
| 608 | for (j = 0; j < 8; ++j) { |
| 609 | const limb bit = byte >> 7; |
| 610 | |
| 611 | swap_conditional(nqx, nqpqx, bit); |
| 612 | swap_conditional(nqz, nqpqz, bit); |
| 613 | fmonty(nqx2, nqz2, |
| 614 | nqpqx2, nqpqz2, |
| 615 | nqx, nqz, |
| 616 | nqpqx, nqpqz, |
| 617 | q); |
| 618 | swap_conditional(nqx2, nqpqx2, bit); |
| 619 | swap_conditional(nqz2, nqpqz2, bit); |
| 620 | |
| 621 | t = nqx; |
| 622 | nqx = nqx2; |
| 623 | nqx2 = t; |
| 624 | t = nqz; |
| 625 | nqz = nqz2; |
| 626 | nqz2 = t; |
| 627 | t = nqpqx; |
| 628 | nqpqx = nqpqx2; |
| 629 | nqpqx2 = t; |
| 630 | t = nqpqz; |
| 631 | nqpqz = nqpqz2; |
| 632 | nqpqz2 = t; |
| 633 | |
| 634 | byte <<= 1; |
| 635 | } |
| 636 | } |
| 637 | |
| 638 | memcpy(resultx, nqx, sizeof(limb) * 10); |
| 639 | memcpy(resultz, nqz, sizeof(limb) * 10); |
| 640 | } |
| 641 | |
| 642 | /* ----------------------------------------------------------------------------- |
| 643 | * Shamelessly copied from djb's code |
| 644 | * ----------------------------------------------------------------------------- */ |
| 645 | static void |
| 646 | crecip(limb *out, const limb *z) { |
| 647 | limb z2[10]; |
| 648 | limb z9[10]; |
| 649 | limb z11[10]; |
| 650 | limb z2_5_0[10]; |
| 651 | limb z2_10_0[10]; |
| 652 | limb z2_20_0[10]; |
| 653 | limb z2_50_0[10]; |
| 654 | limb z2_100_0[10]; |
| 655 | limb t0[10]; |
| 656 | limb t1[10]; |
| 657 | int i; |
| 658 | |
| 659 | /* 2 */ fsquare(z2,z); |
| 660 | /* 4 */ fsquare(t1,z2); |
| 661 | /* 8 */ fsquare(t0,t1); |
| 662 | /* 9 */ fmul(z9,t0,z); |
| 663 | /* 11 */ fmul(z11,z9,z2); |
| 664 | /* 22 */ fsquare(t0,z11); |
| 665 | /* 2^5 - 2^0 = 31 */ fmul(z2_5_0,t0,z9); |
| 666 | |
| 667 | /* 2^6 - 2^1 */ fsquare(t0,z2_5_0); |
| 668 | /* 2^7 - 2^2 */ fsquare(t1,t0); |
| 669 | /* 2^8 - 2^3 */ fsquare(t0,t1); |
| 670 | /* 2^9 - 2^4 */ fsquare(t1,t0); |
| 671 | /* 2^10 - 2^5 */ fsquare(t0,t1); |
| 672 | /* 2^10 - 2^0 */ fmul(z2_10_0,t0,z2_5_0); |
| 673 | |
| 674 | /* 2^11 - 2^1 */ fsquare(t0,z2_10_0); |
| 675 | /* 2^12 - 2^2 */ fsquare(t1,t0); |
| 676 | /* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
| 677 | /* 2^20 - 2^0 */ fmul(z2_20_0,t1,z2_10_0); |
| 678 | |
| 679 | /* 2^21 - 2^1 */ fsquare(t0,z2_20_0); |
| 680 | /* 2^22 - 2^2 */ fsquare(t1,t0); |
| 681 | /* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
| 682 | /* 2^40 - 2^0 */ fmul(t0,t1,z2_20_0); |
| 683 | |
| 684 | /* 2^41 - 2^1 */ fsquare(t1,t0); |
| 685 | /* 2^42 - 2^2 */ fsquare(t0,t1); |
| 686 | /* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t1,t0); fsquare(t0,t1); } |
| 687 | /* 2^50 - 2^0 */ fmul(z2_50_0,t0,z2_10_0); |
| 688 | |
| 689 | /* 2^51 - 2^1 */ fsquare(t0,z2_50_0); |
| 690 | /* 2^52 - 2^2 */ fsquare(t1,t0); |
| 691 | /* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
| 692 | /* 2^100 - 2^0 */ fmul(z2_100_0,t1,z2_50_0); |
| 693 | |
| 694 | /* 2^101 - 2^1 */ fsquare(t1,z2_100_0); |
| 695 | /* 2^102 - 2^2 */ fsquare(t0,t1); |
| 696 | /* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fsquare(t1,t0); fsquare(t0,t1); } |
| 697 | /* 2^200 - 2^0 */ fmul(t1,t0,z2_100_0); |
| 698 | |
| 699 | /* 2^201 - 2^1 */ fsquare(t0,t1); |
| 700 | /* 2^202 - 2^2 */ fsquare(t1,t0); |
| 701 | /* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
| 702 | /* 2^250 - 2^0 */ fmul(t0,t1,z2_50_0); |
| 703 | |
| 704 | /* 2^251 - 2^1 */ fsquare(t1,t0); |
| 705 | /* 2^252 - 2^2 */ fsquare(t0,t1); |
| 706 | /* 2^253 - 2^3 */ fsquare(t1,t0); |
| 707 | /* 2^254 - 2^4 */ fsquare(t0,t1); |
| 708 | /* 2^255 - 2^5 */ fsquare(t1,t0); |
| 709 | /* 2^255 - 21 */ fmul(out,t1,z11); |
| 710 | } |
| 711 | |
| 712 | int curve25519_donna(u8 *, const u8 *, const u8 *); |
| 713 | |
| 714 | int curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) { |
| 715 | limb bp[10], x[10], z[11], zmone[10]; |
| 716 | uint8_t e[32]; |
| 717 | int i; |
| 718 | |
| 719 | for (i = 0; i < 32; ++i) e[i] = secret[i]; |
| 720 | e[0] &= 248; |
| 721 | e[31] &= 127; |
| 722 | e[31] |= 64; |
| 723 | |
| 724 | fexpand(bp, basepoint); |
| 725 | cmult(x, z, e, bp); |
| 726 | crecip(zmone, z); |
| 727 | fmul(z, x, zmone); |
| 728 | freduce_coefficients(z); |
| 729 | fcontract(mypublic, z); |
| 730 | return 0; |
| 731 | } |