Torne (Richard Coles) | 5821806 | 2012-11-14 11:43:16 +0000 | [diff] [blame] | 1 | // Copyright (c) 2012 The Chromium Authors. All rights reserved. |
| 2 | // Use of this source code is governed by a BSD-style license that can be |
| 3 | // found in the LICENSE file. |
| 4 | |
| 5 | // This is an implementation of the P224 elliptic curve group. It's written to |
| 6 | // be short and simple rather than fast, although it's still constant-time. |
| 7 | // |
| 8 | // See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background. |
| 9 | |
| 10 | #include "crypto/p224.h" |
| 11 | |
| 12 | #include <string.h> |
| 13 | |
| 14 | #include "base/sys_byteorder.h" |
| 15 | |
| 16 | namespace { |
| 17 | |
| 18 | using base::HostToNet32; |
| 19 | using base::NetToHost32; |
| 20 | |
| 21 | // Field element functions. |
| 22 | // |
| 23 | // The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1. |
| 24 | // |
| 25 | // Field elements are represented by a FieldElement, which is a typedef to an |
| 26 | // array of 8 uint32's. The value of a FieldElement, a, is: |
| 27 | // a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7] |
| 28 | // |
| 29 | // Using 28-bit limbs means that there's only 4 bits of headroom, which is less |
| 30 | // than we would really like. But it has the useful feature that we hit 2**224 |
| 31 | // exactly, making the reflections during a reduce much nicer. |
| 32 | |
| 33 | using crypto::p224::FieldElement; |
| 34 | |
| 35 | // kP is the P224 prime. |
| 36 | const FieldElement kP = { |
| 37 | 1, 0, 0, 268431360, |
| 38 | 268435455, 268435455, 268435455, 268435455, |
| 39 | }; |
| 40 | |
| 41 | void Contract(FieldElement* inout); |
| 42 | |
| 43 | // IsZero returns 0xffffffff if a == 0 mod p and 0 otherwise. |
| 44 | uint32 IsZero(const FieldElement& a) { |
| 45 | FieldElement minimal; |
| 46 | memcpy(&minimal, &a, sizeof(minimal)); |
| 47 | Contract(&minimal); |
| 48 | |
| 49 | uint32 is_zero = 0, is_p = 0; |
| 50 | for (unsigned i = 0; i < 8; i++) { |
| 51 | is_zero |= minimal[i]; |
| 52 | is_p |= minimal[i] - kP[i]; |
| 53 | } |
| 54 | |
| 55 | // If either is_zero or is_p is 0, then we should return 1. |
| 56 | is_zero |= is_zero >> 16; |
| 57 | is_zero |= is_zero >> 8; |
| 58 | is_zero |= is_zero >> 4; |
| 59 | is_zero |= is_zero >> 2; |
| 60 | is_zero |= is_zero >> 1; |
| 61 | |
| 62 | is_p |= is_p >> 16; |
| 63 | is_p |= is_p >> 8; |
| 64 | is_p |= is_p >> 4; |
| 65 | is_p |= is_p >> 2; |
| 66 | is_p |= is_p >> 1; |
| 67 | |
| 68 | // For is_zero and is_p, the LSB is 0 iff all the bits are zero. |
| 69 | is_zero &= is_p & 1; |
| 70 | is_zero = (~is_zero) << 31; |
| 71 | is_zero = static_cast<int32>(is_zero) >> 31; |
| 72 | return is_zero; |
| 73 | } |
| 74 | |
| 75 | // Add computes *out = a+b |
| 76 | // |
| 77 | // a[i] + b[i] < 2**32 |
| 78 | void Add(FieldElement* out, const FieldElement& a, const FieldElement& b) { |
| 79 | for (int i = 0; i < 8; i++) { |
| 80 | (*out)[i] = a[i] + b[i]; |
| 81 | } |
| 82 | } |
| 83 | |
| 84 | static const uint32 kTwo31p3 = (1u<<31) + (1u<<3); |
| 85 | static const uint32 kTwo31m3 = (1u<<31) - (1u<<3); |
| 86 | static const uint32 kTwo31m15m3 = (1u<<31) - (1u<<15) - (1u<<3); |
| 87 | // kZero31ModP is 0 mod p where bit 31 is set in all limbs so that we can |
| 88 | // subtract smaller amounts without underflow. See the section "Subtraction" in |
| 89 | // [1] for why. |
| 90 | static const FieldElement kZero31ModP = { |
| 91 | kTwo31p3, kTwo31m3, kTwo31m3, kTwo31m15m3, |
| 92 | kTwo31m3, kTwo31m3, kTwo31m3, kTwo31m3 |
| 93 | }; |
| 94 | |
| 95 | // Subtract computes *out = a-b |
| 96 | // |
| 97 | // a[i], b[i] < 2**30 |
| 98 | // out[i] < 2**32 |
| 99 | void Subtract(FieldElement* out, const FieldElement& a, const FieldElement& b) { |
| 100 | for (int i = 0; i < 8; i++) { |
| 101 | // See the section on "Subtraction" in [1] for details. |
| 102 | (*out)[i] = a[i] + kZero31ModP[i] - b[i]; |
| 103 | } |
| 104 | } |
| 105 | |
| 106 | static const uint64 kTwo63p35 = (1ull<<63) + (1ull<<35); |
| 107 | static const uint64 kTwo63m35 = (1ull<<63) - (1ull<<35); |
| 108 | static const uint64 kTwo63m35m19 = (1ull<<63) - (1ull<<35) - (1ull<<19); |
| 109 | // kZero63ModP is 0 mod p where bit 63 is set in all limbs. See the section |
| 110 | // "Subtraction" in [1] for why. |
| 111 | static const uint64 kZero63ModP[8] = { |
| 112 | kTwo63p35, kTwo63m35, kTwo63m35, kTwo63m35, |
| 113 | kTwo63m35m19, kTwo63m35, kTwo63m35, kTwo63m35, |
| 114 | }; |
| 115 | |
| 116 | static const uint32 kBottom28Bits = 0xfffffff; |
| 117 | |
| 118 | // LargeFieldElement also represents an element of the field. The limbs are |
| 119 | // still spaced 28-bits apart and in little-endian order. So the limbs are at |
| 120 | // 0, 28, 56, ..., 392 bits, each 64-bits wide. |
| 121 | typedef uint64 LargeFieldElement[15]; |
| 122 | |
| 123 | // ReduceLarge converts a LargeFieldElement to a FieldElement. |
| 124 | // |
| 125 | // in[i] < 2**62 |
| 126 | void ReduceLarge(FieldElement* out, LargeFieldElement* inptr) { |
| 127 | LargeFieldElement& in(*inptr); |
| 128 | |
| 129 | for (int i = 0; i < 8; i++) { |
| 130 | in[i] += kZero63ModP[i]; |
| 131 | } |
| 132 | |
| 133 | // Eliminate the coefficients at 2**224 and greater while maintaining the |
| 134 | // same value mod p. |
| 135 | for (int i = 14; i >= 8; i--) { |
| 136 | in[i-8] -= in[i]; // reflection off the "+1" term of p. |
| 137 | in[i-5] += (in[i] & 0xffff) << 12; // part of the "-2**96" reflection. |
| 138 | in[i-4] += in[i] >> 16; // the rest of the "-2**96" reflection. |
| 139 | } |
| 140 | in[8] = 0; |
| 141 | // in[0..8] < 2**64 |
| 142 | |
| 143 | // As the values become small enough, we start to store them in |out| and use |
| 144 | // 32-bit operations. |
| 145 | for (int i = 1; i < 8; i++) { |
| 146 | in[i+1] += in[i] >> 28; |
| 147 | (*out)[i] = static_cast<uint32>(in[i] & kBottom28Bits); |
| 148 | } |
| 149 | // Eliminate the term at 2*224 that we introduced while keeping the same |
| 150 | // value mod p. |
| 151 | in[0] -= in[8]; // reflection off the "+1" term of p. |
| 152 | (*out)[3] += static_cast<uint32>(in[8] & 0xffff) << 12; // "-2**96" term |
| 153 | (*out)[4] += static_cast<uint32>(in[8] >> 16); // rest of "-2**96" term |
| 154 | // in[0] < 2**64 |
| 155 | // out[3] < 2**29 |
| 156 | // out[4] < 2**29 |
| 157 | // out[1,2,5..7] < 2**28 |
| 158 | |
| 159 | (*out)[0] = static_cast<uint32>(in[0] & kBottom28Bits); |
| 160 | (*out)[1] += static_cast<uint32>((in[0] >> 28) & kBottom28Bits); |
| 161 | (*out)[2] += static_cast<uint32>(in[0] >> 56); |
| 162 | // out[0] < 2**28 |
| 163 | // out[1..4] < 2**29 |
| 164 | // out[5..7] < 2**28 |
| 165 | } |
| 166 | |
| 167 | // Mul computes *out = a*b |
| 168 | // |
| 169 | // a[i] < 2**29, b[i] < 2**30 (or vice versa) |
| 170 | // out[i] < 2**29 |
| 171 | void Mul(FieldElement* out, const FieldElement& a, const FieldElement& b) { |
| 172 | LargeFieldElement tmp; |
| 173 | memset(&tmp, 0, sizeof(tmp)); |
| 174 | |
| 175 | for (int i = 0; i < 8; i++) { |
| 176 | for (int j = 0; j < 8; j++) { |
| 177 | tmp[i+j] += static_cast<uint64>(a[i]) * static_cast<uint64>(b[j]); |
| 178 | } |
| 179 | } |
| 180 | |
| 181 | ReduceLarge(out, &tmp); |
| 182 | } |
| 183 | |
| 184 | // Square computes *out = a*a |
| 185 | // |
| 186 | // a[i] < 2**29 |
| 187 | // out[i] < 2**29 |
| 188 | void Square(FieldElement* out, const FieldElement& a) { |
| 189 | LargeFieldElement tmp; |
| 190 | memset(&tmp, 0, sizeof(tmp)); |
| 191 | |
| 192 | for (int i = 0; i < 8; i++) { |
| 193 | for (int j = 0; j <= i; j++) { |
| 194 | uint64 r = static_cast<uint64>(a[i]) * static_cast<uint64>(a[j]); |
| 195 | if (i == j) { |
| 196 | tmp[i+j] += r; |
| 197 | } else { |
| 198 | tmp[i+j] += r << 1; |
| 199 | } |
| 200 | } |
| 201 | } |
| 202 | |
| 203 | ReduceLarge(out, &tmp); |
| 204 | } |
| 205 | |
| 206 | // Reduce reduces the coefficients of in_out to smaller bounds. |
| 207 | // |
| 208 | // On entry: a[i] < 2**31 + 2**30 |
| 209 | // On exit: a[i] < 2**29 |
| 210 | void Reduce(FieldElement* in_out) { |
| 211 | FieldElement& a = *in_out; |
| 212 | |
| 213 | for (int i = 0; i < 7; i++) { |
| 214 | a[i+1] += a[i] >> 28; |
| 215 | a[i] &= kBottom28Bits; |
| 216 | } |
| 217 | uint32 top = a[7] >> 28; |
| 218 | a[7] &= kBottom28Bits; |
| 219 | |
| 220 | // top < 2**4 |
| 221 | // Constant-time: mask = (top != 0) ? 0xffffffff : 0 |
| 222 | uint32 mask = top; |
| 223 | mask |= mask >> 2; |
| 224 | mask |= mask >> 1; |
| 225 | mask <<= 31; |
| 226 | mask = static_cast<uint32>(static_cast<int32>(mask) >> 31); |
| 227 | |
| 228 | // Eliminate top while maintaining the same value mod p. |
| 229 | a[0] -= top; |
| 230 | a[3] += top << 12; |
| 231 | |
| 232 | // We may have just made a[0] negative but, if we did, then we must |
| 233 | // have added something to a[3], thus it's > 2**12. Therefore we can |
| 234 | // carry down to a[0]. |
| 235 | a[3] -= 1 & mask; |
| 236 | a[2] += mask & ((1<<28) - 1); |
| 237 | a[1] += mask & ((1<<28) - 1); |
| 238 | a[0] += mask & (1<<28); |
| 239 | } |
| 240 | |
| 241 | // Invert calcuates *out = in**-1 by computing in**(2**224 - 2**96 - 1), i.e. |
| 242 | // Fermat's little theorem. |
| 243 | void Invert(FieldElement* out, const FieldElement& in) { |
| 244 | FieldElement f1, f2, f3, f4; |
| 245 | |
| 246 | Square(&f1, in); // 2 |
| 247 | Mul(&f1, f1, in); // 2**2 - 1 |
| 248 | Square(&f1, f1); // 2**3 - 2 |
| 249 | Mul(&f1, f1, in); // 2**3 - 1 |
| 250 | Square(&f2, f1); // 2**4 - 2 |
| 251 | Square(&f2, f2); // 2**5 - 4 |
| 252 | Square(&f2, f2); // 2**6 - 8 |
| 253 | Mul(&f1, f1, f2); // 2**6 - 1 |
| 254 | Square(&f2, f1); // 2**7 - 2 |
| 255 | for (int i = 0; i < 5; i++) { // 2**12 - 2**6 |
| 256 | Square(&f2, f2); |
| 257 | } |
| 258 | Mul(&f2, f2, f1); // 2**12 - 1 |
| 259 | Square(&f3, f2); // 2**13 - 2 |
| 260 | for (int i = 0; i < 11; i++) { // 2**24 - 2**12 |
| 261 | Square(&f3, f3); |
| 262 | } |
| 263 | Mul(&f2, f3, f2); // 2**24 - 1 |
| 264 | Square(&f3, f2); // 2**25 - 2 |
| 265 | for (int i = 0; i < 23; i++) { // 2**48 - 2**24 |
| 266 | Square(&f3, f3); |
| 267 | } |
| 268 | Mul(&f3, f3, f2); // 2**48 - 1 |
| 269 | Square(&f4, f3); // 2**49 - 2 |
| 270 | for (int i = 0; i < 47; i++) { // 2**96 - 2**48 |
| 271 | Square(&f4, f4); |
| 272 | } |
| 273 | Mul(&f3, f3, f4); // 2**96 - 1 |
| 274 | Square(&f4, f3); // 2**97 - 2 |
| 275 | for (int i = 0; i < 23; i++) { // 2**120 - 2**24 |
| 276 | Square(&f4, f4); |
| 277 | } |
| 278 | Mul(&f2, f4, f2); // 2**120 - 1 |
| 279 | for (int i = 0; i < 6; i++) { // 2**126 - 2**6 |
| 280 | Square(&f2, f2); |
| 281 | } |
| 282 | Mul(&f1, f1, f2); // 2**126 - 1 |
| 283 | Square(&f1, f1); // 2**127 - 2 |
| 284 | Mul(&f1, f1, in); // 2**127 - 1 |
| 285 | for (int i = 0; i < 97; i++) { // 2**224 - 2**97 |
| 286 | Square(&f1, f1); |
| 287 | } |
| 288 | Mul(out, f1, f3); // 2**224 - 2**96 - 1 |
| 289 | } |
| 290 | |
| 291 | // Contract converts a FieldElement to its minimal, distinguished form. |
| 292 | // |
| 293 | // On entry, in[i] < 2**29 |
| 294 | // On exit, in[i] < 2**28 |
| 295 | void Contract(FieldElement* inout) { |
| 296 | FieldElement& out = *inout; |
| 297 | |
| 298 | // Reduce the coefficients to < 2**28. |
| 299 | for (int i = 0; i < 7; i++) { |
| 300 | out[i+1] += out[i] >> 28; |
| 301 | out[i] &= kBottom28Bits; |
| 302 | } |
| 303 | uint32 top = out[7] >> 28; |
| 304 | out[7] &= kBottom28Bits; |
| 305 | |
| 306 | // Eliminate top while maintaining the same value mod p. |
| 307 | out[0] -= top; |
| 308 | out[3] += top << 12; |
| 309 | |
| 310 | // We may just have made out[0] negative. So we carry down. If we made |
| 311 | // out[0] negative then we know that out[3] is sufficiently positive |
| 312 | // because we just added to it. |
| 313 | for (int i = 0; i < 3; i++) { |
| 314 | uint32 mask = static_cast<uint32>(static_cast<int32>(out[i]) >> 31); |
| 315 | out[i] += (1 << 28) & mask; |
| 316 | out[i+1] -= 1 & mask; |
| 317 | } |
| 318 | |
| 319 | // We might have pushed out[3] over 2**28 so we perform another, partial |
| 320 | // carry chain. |
| 321 | for (int i = 3; i < 7; i++) { |
| 322 | out[i+1] += out[i] >> 28; |
| 323 | out[i] &= kBottom28Bits; |
| 324 | } |
| 325 | top = out[7] >> 28; |
| 326 | out[7] &= kBottom28Bits; |
| 327 | |
| 328 | // Eliminate top while maintaining the same value mod p. |
| 329 | out[0] -= top; |
| 330 | out[3] += top << 12; |
| 331 | |
| 332 | // There are two cases to consider for out[3]: |
| 333 | // 1) The first time that we eliminated top, we didn't push out[3] over |
| 334 | // 2**28. In this case, the partial carry chain didn't change any values |
| 335 | // and top is zero. |
| 336 | // 2) We did push out[3] over 2**28 the first time that we eliminated top. |
| 337 | // The first value of top was in [0..16), therefore, prior to eliminating |
| 338 | // the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after |
| 339 | // overflowing and being reduced by the second carry chain, out[3] <= |
| 340 | // 0xf000. Thus it cannot have overflowed when we eliminated top for the |
| 341 | // second time. |
| 342 | |
| 343 | // Again, we may just have made out[0] negative, so do the same carry down. |
| 344 | // As before, if we made out[0] negative then we know that out[3] is |
| 345 | // sufficiently positive. |
| 346 | for (int i = 0; i < 3; i++) { |
| 347 | uint32 mask = static_cast<uint32>(static_cast<int32>(out[i]) >> 31); |
| 348 | out[i] += (1 << 28) & mask; |
| 349 | out[i+1] -= 1 & mask; |
| 350 | } |
| 351 | |
| 352 | // The value is < 2**224, but maybe greater than p. In order to reduce to a |
| 353 | // unique, minimal value we see if the value is >= p and, if so, subtract p. |
| 354 | |
| 355 | // First we build a mask from the top four limbs, which must all be |
| 356 | // equal to bottom28Bits if the whole value is >= p. If top_4_all_ones |
| 357 | // ends up with any zero bits in the bottom 28 bits, then this wasn't |
| 358 | // true. |
| 359 | uint32 top_4_all_ones = 0xffffffffu; |
| 360 | for (int i = 4; i < 8; i++) { |
| 361 | top_4_all_ones &= out[i]; |
| 362 | } |
| 363 | top_4_all_ones |= 0xf0000000; |
| 364 | // Now we replicate any zero bits to all the bits in top_4_all_ones. |
| 365 | top_4_all_ones &= top_4_all_ones >> 16; |
| 366 | top_4_all_ones &= top_4_all_ones >> 8; |
| 367 | top_4_all_ones &= top_4_all_ones >> 4; |
| 368 | top_4_all_ones &= top_4_all_ones >> 2; |
| 369 | top_4_all_ones &= top_4_all_ones >> 1; |
| 370 | top_4_all_ones = |
| 371 | static_cast<uint32>(static_cast<int32>(top_4_all_ones << 31) >> 31); |
| 372 | |
| 373 | // Now we test whether the bottom three limbs are non-zero. |
| 374 | uint32 bottom_3_non_zero = out[0] | out[1] | out[2]; |
| 375 | bottom_3_non_zero |= bottom_3_non_zero >> 16; |
| 376 | bottom_3_non_zero |= bottom_3_non_zero >> 8; |
| 377 | bottom_3_non_zero |= bottom_3_non_zero >> 4; |
| 378 | bottom_3_non_zero |= bottom_3_non_zero >> 2; |
| 379 | bottom_3_non_zero |= bottom_3_non_zero >> 1; |
| 380 | bottom_3_non_zero = |
| 381 | static_cast<uint32>(static_cast<int32>(bottom_3_non_zero) >> 31); |
| 382 | |
| 383 | // Everything depends on the value of out[3]. |
| 384 | // If it's > 0xffff000 and top_4_all_ones != 0 then the whole value is >= p |
| 385 | // If it's = 0xffff000 and top_4_all_ones != 0 and bottom_3_non_zero != 0, |
| 386 | // then the whole value is >= p |
| 387 | // If it's < 0xffff000, then the whole value is < p |
| 388 | uint32 n = out[3] - 0xffff000; |
| 389 | uint32 out_3_equal = n; |
| 390 | out_3_equal |= out_3_equal >> 16; |
| 391 | out_3_equal |= out_3_equal >> 8; |
| 392 | out_3_equal |= out_3_equal >> 4; |
| 393 | out_3_equal |= out_3_equal >> 2; |
| 394 | out_3_equal |= out_3_equal >> 1; |
| 395 | out_3_equal = |
| 396 | ~static_cast<uint32>(static_cast<int32>(out_3_equal << 31) >> 31); |
| 397 | |
| 398 | // If out[3] > 0xffff000 then n's MSB will be zero. |
| 399 | uint32 out_3_gt = ~static_cast<uint32>(static_cast<int32>(n << 31) >> 31); |
| 400 | |
| 401 | uint32 mask = top_4_all_ones & ((out_3_equal & bottom_3_non_zero) | out_3_gt); |
| 402 | out[0] -= 1 & mask; |
| 403 | out[3] -= 0xffff000 & mask; |
| 404 | out[4] -= 0xfffffff & mask; |
| 405 | out[5] -= 0xfffffff & mask; |
| 406 | out[6] -= 0xfffffff & mask; |
| 407 | out[7] -= 0xfffffff & mask; |
| 408 | } |
| 409 | |
| 410 | |
| 411 | // Group element functions. |
| 412 | // |
| 413 | // These functions deal with group elements. The group is an elliptic curve |
| 414 | // group with a = -3 defined in FIPS 186-3, section D.2.2. |
| 415 | |
| 416 | using crypto::p224::Point; |
| 417 | |
| 418 | // kB is parameter of the elliptic curve. |
| 419 | const FieldElement kB = { |
| 420 | 55967668, 11768882, 265861671, 185302395, |
| 421 | 39211076, 180311059, 84673715, 188764328, |
| 422 | }; |
| 423 | |
| 424 | void CopyConditional(Point* out, const Point& a, uint32 mask); |
| 425 | void DoubleJacobian(Point* out, const Point& a); |
| 426 | |
| 427 | // AddJacobian computes *out = a+b where a != b. |
| 428 | void AddJacobian(Point *out, |
| 429 | const Point& a, |
| 430 | const Point& b) { |
| 431 | // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl |
| 432 | FieldElement z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v; |
| 433 | |
| 434 | uint32 z1_is_zero = IsZero(a.z); |
| 435 | uint32 z2_is_zero = IsZero(b.z); |
| 436 | |
| 437 | // Z1Z1 = Z1² |
| 438 | Square(&z1z1, a.z); |
| 439 | |
| 440 | // Z2Z2 = Z2² |
| 441 | Square(&z2z2, b.z); |
| 442 | |
| 443 | // U1 = X1*Z2Z2 |
| 444 | Mul(&u1, a.x, z2z2); |
| 445 | |
| 446 | // U2 = X2*Z1Z1 |
| 447 | Mul(&u2, b.x, z1z1); |
| 448 | |
| 449 | // S1 = Y1*Z2*Z2Z2 |
| 450 | Mul(&s1, b.z, z2z2); |
| 451 | Mul(&s1, a.y, s1); |
| 452 | |
| 453 | // S2 = Y2*Z1*Z1Z1 |
| 454 | Mul(&s2, a.z, z1z1); |
| 455 | Mul(&s2, b.y, s2); |
| 456 | |
| 457 | // H = U2-U1 |
| 458 | Subtract(&h, u2, u1); |
| 459 | Reduce(&h); |
| 460 | uint32 x_equal = IsZero(h); |
| 461 | |
| 462 | // I = (2*H)² |
| 463 | for (int j = 0; j < 8; j++) { |
| 464 | i[j] = h[j] << 1; |
| 465 | } |
| 466 | Reduce(&i); |
| 467 | Square(&i, i); |
| 468 | |
| 469 | // J = H*I |
| 470 | Mul(&j, h, i); |
| 471 | // r = 2*(S2-S1) |
| 472 | Subtract(&r, s2, s1); |
| 473 | Reduce(&r); |
| 474 | uint32 y_equal = IsZero(r); |
| 475 | |
| 476 | if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { |
| 477 | // The two input points are the same therefore we must use the dedicated |
| 478 | // doubling function as the slope of the line is undefined. |
| 479 | DoubleJacobian(out, a); |
| 480 | return; |
| 481 | } |
| 482 | |
| 483 | for (int i = 0; i < 8; i++) { |
| 484 | r[i] <<= 1; |
| 485 | } |
| 486 | Reduce(&r); |
| 487 | |
| 488 | // V = U1*I |
| 489 | Mul(&v, u1, i); |
| 490 | |
| 491 | // Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H |
| 492 | Add(&z1z1, z1z1, z2z2); |
| 493 | Add(&z2z2, a.z, b.z); |
| 494 | Reduce(&z2z2); |
| 495 | Square(&z2z2, z2z2); |
| 496 | Subtract(&out->z, z2z2, z1z1); |
| 497 | Reduce(&out->z); |
| 498 | Mul(&out->z, out->z, h); |
| 499 | |
| 500 | // X3 = r²-J-2*V |
| 501 | for (int i = 0; i < 8; i++) { |
| 502 | z1z1[i] = v[i] << 1; |
| 503 | } |
| 504 | Add(&z1z1, j, z1z1); |
| 505 | Reduce(&z1z1); |
| 506 | Square(&out->x, r); |
| 507 | Subtract(&out->x, out->x, z1z1); |
| 508 | Reduce(&out->x); |
| 509 | |
| 510 | // Y3 = r*(V-X3)-2*S1*J |
| 511 | for (int i = 0; i < 8; i++) { |
| 512 | s1[i] <<= 1; |
| 513 | } |
| 514 | Mul(&s1, s1, j); |
| 515 | Subtract(&z1z1, v, out->x); |
| 516 | Reduce(&z1z1); |
| 517 | Mul(&z1z1, z1z1, r); |
| 518 | Subtract(&out->y, z1z1, s1); |
| 519 | Reduce(&out->y); |
| 520 | |
| 521 | CopyConditional(out, a, z2_is_zero); |
| 522 | CopyConditional(out, b, z1_is_zero); |
| 523 | } |
| 524 | |
| 525 | // DoubleJacobian computes *out = a+a. |
| 526 | void DoubleJacobian(Point* out, const Point& a) { |
| 527 | // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b |
| 528 | FieldElement delta, gamma, beta, alpha, t; |
| 529 | |
| 530 | Square(&delta, a.z); |
| 531 | Square(&gamma, a.y); |
| 532 | Mul(&beta, a.x, gamma); |
| 533 | |
| 534 | // alpha = 3*(X1-delta)*(X1+delta) |
| 535 | Add(&t, a.x, delta); |
| 536 | for (int i = 0; i < 8; i++) { |
| 537 | t[i] += t[i] << 1; |
| 538 | } |
| 539 | Reduce(&t); |
| 540 | Subtract(&alpha, a.x, delta); |
| 541 | Reduce(&alpha); |
| 542 | Mul(&alpha, alpha, t); |
| 543 | |
| 544 | // Z3 = (Y1+Z1)²-gamma-delta |
| 545 | Add(&out->z, a.y, a.z); |
| 546 | Reduce(&out->z); |
| 547 | Square(&out->z, out->z); |
| 548 | Subtract(&out->z, out->z, gamma); |
| 549 | Reduce(&out->z); |
| 550 | Subtract(&out->z, out->z, delta); |
| 551 | Reduce(&out->z); |
| 552 | |
| 553 | // X3 = alpha²-8*beta |
| 554 | for (int i = 0; i < 8; i++) { |
| 555 | delta[i] = beta[i] << 3; |
| 556 | } |
| 557 | Reduce(&delta); |
| 558 | Square(&out->x, alpha); |
| 559 | Subtract(&out->x, out->x, delta); |
| 560 | Reduce(&out->x); |
| 561 | |
| 562 | // Y3 = alpha*(4*beta-X3)-8*gamma² |
| 563 | for (int i = 0; i < 8; i++) { |
| 564 | beta[i] <<= 2; |
| 565 | } |
| 566 | Reduce(&beta); |
| 567 | Subtract(&beta, beta, out->x); |
| 568 | Reduce(&beta); |
| 569 | Square(&gamma, gamma); |
| 570 | for (int i = 0; i < 8; i++) { |
| 571 | gamma[i] <<= 3; |
| 572 | } |
| 573 | Reduce(&gamma); |
| 574 | Mul(&out->y, alpha, beta); |
| 575 | Subtract(&out->y, out->y, gamma); |
| 576 | Reduce(&out->y); |
| 577 | } |
| 578 | |
| 579 | // CopyConditional sets *out=a if mask is 0xffffffff. mask must be either 0 of |
| 580 | // 0xffffffff. |
| 581 | void CopyConditional(Point* out, |
| 582 | const Point& a, |
| 583 | uint32 mask) { |
| 584 | for (int i = 0; i < 8; i++) { |
| 585 | out->x[i] ^= mask & (a.x[i] ^ out->x[i]); |
| 586 | out->y[i] ^= mask & (a.y[i] ^ out->y[i]); |
| 587 | out->z[i] ^= mask & (a.z[i] ^ out->z[i]); |
| 588 | } |
| 589 | } |
| 590 | |
| 591 | // ScalarMult calculates *out = a*scalar where scalar is a big-endian number of |
| 592 | // length scalar_len and != 0. |
| 593 | void ScalarMult(Point* out, const Point& a, |
| 594 | const uint8* scalar, size_t scalar_len) { |
| 595 | memset(out, 0, sizeof(*out)); |
| 596 | Point tmp; |
| 597 | |
| 598 | for (size_t i = 0; i < scalar_len; i++) { |
| 599 | for (unsigned int bit_num = 0; bit_num < 8; bit_num++) { |
| 600 | DoubleJacobian(out, *out); |
| 601 | uint32 bit = static_cast<uint32>(static_cast<int32>( |
| 602 | (((scalar[i] >> (7 - bit_num)) & 1) << 31) >> 31)); |
| 603 | AddJacobian(&tmp, a, *out); |
| 604 | CopyConditional(out, tmp, bit); |
| 605 | } |
| 606 | } |
| 607 | } |
| 608 | |
| 609 | // Get224Bits reads 7 words from in and scatters their contents in |
| 610 | // little-endian form into 8 words at out, 28 bits per output word. |
| 611 | void Get224Bits(uint32* out, const uint32* in) { |
| 612 | out[0] = NetToHost32(in[6]) & kBottom28Bits; |
| 613 | out[1] = ((NetToHost32(in[5]) << 4) | |
| 614 | (NetToHost32(in[6]) >> 28)) & kBottom28Bits; |
| 615 | out[2] = ((NetToHost32(in[4]) << 8) | |
| 616 | (NetToHost32(in[5]) >> 24)) & kBottom28Bits; |
| 617 | out[3] = ((NetToHost32(in[3]) << 12) | |
| 618 | (NetToHost32(in[4]) >> 20)) & kBottom28Bits; |
| 619 | out[4] = ((NetToHost32(in[2]) << 16) | |
| 620 | (NetToHost32(in[3]) >> 16)) & kBottom28Bits; |
| 621 | out[5] = ((NetToHost32(in[1]) << 20) | |
| 622 | (NetToHost32(in[2]) >> 12)) & kBottom28Bits; |
| 623 | out[6] = ((NetToHost32(in[0]) << 24) | |
| 624 | (NetToHost32(in[1]) >> 8)) & kBottom28Bits; |
| 625 | out[7] = (NetToHost32(in[0]) >> 4) & kBottom28Bits; |
| 626 | } |
| 627 | |
| 628 | // Put224Bits performs the inverse operation to Get224Bits: taking 28 bits from |
| 629 | // each of 8 input words and writing them in big-endian order to 7 words at |
| 630 | // out. |
| 631 | void Put224Bits(uint32* out, const uint32* in) { |
| 632 | out[6] = HostToNet32((in[0] >> 0) | (in[1] << 28)); |
| 633 | out[5] = HostToNet32((in[1] >> 4) | (in[2] << 24)); |
| 634 | out[4] = HostToNet32((in[2] >> 8) | (in[3] << 20)); |
| 635 | out[3] = HostToNet32((in[3] >> 12) | (in[4] << 16)); |
| 636 | out[2] = HostToNet32((in[4] >> 16) | (in[5] << 12)); |
| 637 | out[1] = HostToNet32((in[5] >> 20) | (in[6] << 8)); |
| 638 | out[0] = HostToNet32((in[6] >> 24) | (in[7] << 4)); |
| 639 | } |
| 640 | |
| 641 | } // anonymous namespace |
| 642 | |
| 643 | namespace crypto { |
| 644 | |
| 645 | namespace p224 { |
| 646 | |
| 647 | bool Point::SetFromString(const base::StringPiece& in) { |
| 648 | if (in.size() != 2*28) |
| 649 | return false; |
| 650 | const uint32* inwords = reinterpret_cast<const uint32*>(in.data()); |
| 651 | Get224Bits(x, inwords); |
| 652 | Get224Bits(y, inwords + 7); |
| 653 | memset(&z, 0, sizeof(z)); |
| 654 | z[0] = 1; |
| 655 | |
| 656 | // Check that the point is on the curve, i.e. that y² = x³ - 3x + b. |
| 657 | FieldElement lhs; |
| 658 | Square(&lhs, y); |
| 659 | Contract(&lhs); |
| 660 | |
| 661 | FieldElement rhs; |
| 662 | Square(&rhs, x); |
| 663 | Mul(&rhs, x, rhs); |
| 664 | |
| 665 | FieldElement three_x; |
| 666 | for (int i = 0; i < 8; i++) { |
| 667 | three_x[i] = x[i] * 3; |
| 668 | } |
| 669 | Reduce(&three_x); |
| 670 | Subtract(&rhs, rhs, three_x); |
| 671 | Reduce(&rhs); |
| 672 | |
| 673 | ::Add(&rhs, rhs, kB); |
| 674 | Contract(&rhs); |
| 675 | return memcmp(&lhs, &rhs, sizeof(lhs)) == 0; |
| 676 | } |
| 677 | |
| 678 | std::string Point::ToString() const { |
| 679 | FieldElement zinv, zinv_sq, x, y; |
| 680 | |
| 681 | // If this is the point at infinity we return a string of all zeros. |
| 682 | if (IsZero(this->z)) { |
| 683 | static const char zeros[56] = {0}; |
| 684 | return std::string(zeros, sizeof(zeros)); |
| 685 | } |
| 686 | |
| 687 | Invert(&zinv, this->z); |
| 688 | Square(&zinv_sq, zinv); |
| 689 | Mul(&x, this->x, zinv_sq); |
| 690 | Mul(&zinv_sq, zinv_sq, zinv); |
| 691 | Mul(&y, this->y, zinv_sq); |
| 692 | |
| 693 | Contract(&x); |
| 694 | Contract(&y); |
| 695 | |
| 696 | uint32 outwords[14]; |
| 697 | Put224Bits(outwords, x); |
| 698 | Put224Bits(outwords + 7, y); |
| 699 | return std::string(reinterpret_cast<const char*>(outwords), sizeof(outwords)); |
| 700 | } |
| 701 | |
| 702 | void ScalarMult(const Point& in, const uint8* scalar, Point* out) { |
| 703 | ::ScalarMult(out, in, scalar, 28); |
| 704 | } |
| 705 | |
| 706 | // kBasePoint is the base point (generator) of the elliptic curve group. |
| 707 | static const Point kBasePoint = { |
| 708 | {22813985, 52956513, 34677300, 203240812, |
| 709 | 12143107, 133374265, 225162431, 191946955}, |
| 710 | {83918388, 223877528, 122119236, 123340192, |
| 711 | 266784067, 263504429, 146143011, 198407736}, |
| 712 | {1, 0, 0, 0, 0, 0, 0, 0}, |
| 713 | }; |
| 714 | |
| 715 | void ScalarBaseMult(const uint8* scalar, Point* out) { |
| 716 | ::ScalarMult(out, kBasePoint, scalar, 28); |
| 717 | } |
| 718 | |
| 719 | void Add(const Point& a, const Point& b, Point* out) { |
| 720 | AddJacobian(out, a, b); |
| 721 | } |
| 722 | |
| 723 | void Negate(const Point& in, Point* out) { |
| 724 | // Guide to elliptic curve cryptography, page 89 suggests that (X : X+Y : Z) |
| 725 | // is the negative in Jacobian coordinates, but it doesn't actually appear to |
| 726 | // be true in testing so this performs the negation in affine coordinates. |
| 727 | FieldElement zinv, zinv_sq, y; |
| 728 | Invert(&zinv, in.z); |
| 729 | Square(&zinv_sq, zinv); |
| 730 | Mul(&out->x, in.x, zinv_sq); |
| 731 | Mul(&zinv_sq, zinv_sq, zinv); |
| 732 | Mul(&y, in.y, zinv_sq); |
| 733 | |
| 734 | Subtract(&out->y, kP, y); |
| 735 | Reduce(&out->y); |
| 736 | |
| 737 | memset(&out->z, 0, sizeof(out->z)); |
| 738 | out->z[0] = 1; |
| 739 | } |
| 740 | |
| 741 | } // namespace p224 |
| 742 | |
| 743 | } // namespace crypto |