15821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Copyright (c) 2012 The Chromium Authors. All rights reserved. 25821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Use of this source code is governed by a BSD-style license that can be 35821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// found in the LICENSE file. 45821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 55821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// This is an implementation of the P224 elliptic curve group. It's written to 65821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// be short and simple rather than fast, although it's still constant-time. 75821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 85821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background. 95821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)#include "crypto/p224.h" 115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)#include <string.h> 135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)#include "base/sys_byteorder.h" 155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)namespace { 175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)using base::HostToNet32; 195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)using base::NetToHost32; 205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Field element functions. 225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1. 245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Field elements are represented by a FieldElement, which is a typedef to an 265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// array of 8 uint32's. The value of a FieldElement, a, is: 275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7] 285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Using 28-bit limbs means that there's only 4 bits of headroom, which is less 305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// than we would really like. But it has the useful feature that we hit 2**224 315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// exactly, making the reflections during a reduce much nicer. 325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)using crypto::p224::FieldElement; 345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// kP is the P224 prime. 365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)const FieldElement kP = { 375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1, 0, 0, 268431360, 385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 268435455, 268435455, 268435455, 268435455, 395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)}; 405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 415821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Contract(FieldElement* inout); 425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// IsZero returns 0xffffffff if a == 0 mod p and 0 otherwise. 445821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)uint32 IsZero(const FieldElement& a) { 455821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement minimal; 465821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) memcpy(&minimal, &a, sizeof(minimal)); 475821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Contract(&minimal); 485821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 495821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 is_zero = 0, is_p = 0; 505821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (unsigned i = 0; i < 8; i++) { 515821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero |= minimal[i]; 525821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_p |= minimal[i] - kP[i]; 535821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 545821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 555821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // If either is_zero or is_p is 0, then we should return 1. 565821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero |= is_zero >> 16; 575821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero |= is_zero >> 8; 585821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero |= is_zero >> 4; 595821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero |= is_zero >> 2; 605821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero |= is_zero >> 1; 615821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 625821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_p |= is_p >> 16; 635821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_p |= is_p >> 8; 645821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_p |= is_p >> 4; 655821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_p |= is_p >> 2; 665821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_p |= is_p >> 1; 675821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 685821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // For is_zero and is_p, the LSB is 0 iff all the bits are zero. 695821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero &= is_p & 1; 705821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero = (~is_zero) << 31; 715821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) is_zero = static_cast<int32>(is_zero) >> 31; 725821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) return is_zero; 735821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 745821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 755821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Add computes *out = a+b 765821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 775821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// a[i] + b[i] < 2**32 785821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Add(FieldElement* out, const FieldElement& a, const FieldElement& b) { 795821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 805821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (*out)[i] = a[i] + b[i]; 815821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 825821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 835821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 845821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint32 kTwo31p3 = (1u<<31) + (1u<<3); 855821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint32 kTwo31m3 = (1u<<31) - (1u<<3); 865821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint32 kTwo31m15m3 = (1u<<31) - (1u<<15) - (1u<<3); 875821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// kZero31ModP is 0 mod p where bit 31 is set in all limbs so that we can 885821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// subtract smaller amounts without underflow. See the section "Subtraction" in 895821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// [1] for why. 905821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const FieldElement kZero31ModP = { 915821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) kTwo31p3, kTwo31m3, kTwo31m3, kTwo31m15m3, 925821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) kTwo31m3, kTwo31m3, kTwo31m3, kTwo31m3 935821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)}; 945821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 955821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Subtract computes *out = a-b 965821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 975821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// a[i], b[i] < 2**30 985821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// out[i] < 2**32 995821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Subtract(FieldElement* out, const FieldElement& a, const FieldElement& b) { 1005821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 1015821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // See the section on "Subtraction" in [1] for details. 1025821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (*out)[i] = a[i] + kZero31ModP[i] - b[i]; 1035821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 1045821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 1055821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1065821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint64 kTwo63p35 = (1ull<<63) + (1ull<<35); 1075821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint64 kTwo63m35 = (1ull<<63) - (1ull<<35); 1085821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint64 kTwo63m35m19 = (1ull<<63) - (1ull<<35) - (1ull<<19); 1095821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// kZero63ModP is 0 mod p where bit 63 is set in all limbs. See the section 1105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// "Subtraction" in [1] for why. 1115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint64 kZero63ModP[8] = { 1125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) kTwo63p35, kTwo63m35, kTwo63m35, kTwo63m35, 1135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) kTwo63m35m19, kTwo63m35, kTwo63m35, kTwo63m35, 1145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)}; 1155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const uint32 kBottom28Bits = 0xfffffff; 1175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// LargeFieldElement also represents an element of the field. The limbs are 1195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// still spaced 28-bits apart and in little-endian order. So the limbs are at 1205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 0, 28, 56, ..., 392 bits, each 64-bits wide. 1215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)typedef uint64 LargeFieldElement[15]; 1225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// ReduceLarge converts a LargeFieldElement to a FieldElement. 1245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 1255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// in[i] < 2**62 1265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void ReduceLarge(FieldElement* out, LargeFieldElement* inptr) { 1275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) LargeFieldElement& in(*inptr); 1285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 1305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) in[i] += kZero63ModP[i]; 1315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 1325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Eliminate the coefficients at 2**224 and greater while maintaining the 1345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // same value mod p. 1355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 14; i >= 8; i--) { 1365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) in[i-8] -= in[i]; // reflection off the "+1" term of p. 1375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) in[i-5] += (in[i] & 0xffff) << 12; // part of the "-2**96" reflection. 1385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) in[i-4] += in[i] >> 16; // the rest of the "-2**96" reflection. 1395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 1405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) in[8] = 0; 1415821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // in[0..8] < 2**64 1425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // As the values become small enough, we start to store them in |out| and use 1445821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // 32-bit operations. 1455821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 1; i < 8; i++) { 1465821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) in[i+1] += in[i] >> 28; 1475821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (*out)[i] = static_cast<uint32>(in[i] & kBottom28Bits); 1485821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 1495821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Eliminate the term at 2*224 that we introduced while keeping the same 1505821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // value mod p. 1515821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) in[0] -= in[8]; // reflection off the "+1" term of p. 1525f1c94371a64b3196d4be9466099bb892df9b88eTorne (Richard Coles) (*out)[3] += static_cast<uint32>(in[8] & 0xffff) << 12; // "-2**96" term 1535821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (*out)[4] += static_cast<uint32>(in[8] >> 16); // rest of "-2**96" term 1545821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // in[0] < 2**64 1555821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // out[3] < 2**29 1565821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // out[4] < 2**29 1575821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // out[1,2,5..7] < 2**28 1585821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1595821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (*out)[0] = static_cast<uint32>(in[0] & kBottom28Bits); 1605821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (*out)[1] += static_cast<uint32>((in[0] >> 28) & kBottom28Bits); 1615821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (*out)[2] += static_cast<uint32>(in[0] >> 56); 1625821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // out[0] < 2**28 1635821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // out[1..4] < 2**29 1645821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // out[5..7] < 2**28 1655821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 1665821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1675821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Mul computes *out = a*b 1685821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 1695821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// a[i] < 2**29, b[i] < 2**30 (or vice versa) 1705821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// out[i] < 2**29 1715821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Mul(FieldElement* out, const FieldElement& a, const FieldElement& b) { 1725821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) LargeFieldElement tmp; 1735821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) memset(&tmp, 0, sizeof(tmp)); 1745821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1755821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 1765821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int j = 0; j < 8; j++) { 1775821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) tmp[i+j] += static_cast<uint64>(a[i]) * static_cast<uint64>(b[j]); 1785821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 1795821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 1805821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1815821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) ReduceLarge(out, &tmp); 1825821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 1835821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1845821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Square computes *out = a*a 1855821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 1865821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// a[i] < 2**29 1875821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// out[i] < 2**29 1885821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Square(FieldElement* out, const FieldElement& a) { 1895821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) LargeFieldElement tmp; 1905821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) memset(&tmp, 0, sizeof(tmp)); 1915821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 1925821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 1935821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int j = 0; j <= i; j++) { 1945821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint64 r = static_cast<uint64>(a[i]) * static_cast<uint64>(a[j]); 1955821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) if (i == j) { 1965821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) tmp[i+j] += r; 1975821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } else { 1985821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) tmp[i+j] += r << 1; 1995821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2005821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2015821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2025821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2035821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) ReduceLarge(out, &tmp); 2045821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 2055821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2065821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Reduce reduces the coefficients of in_out to smaller bounds. 2075821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 2085821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// On entry: a[i] < 2**31 + 2**30 2095821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// On exit: a[i] < 2**29 2105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Reduce(FieldElement* in_out) { 2115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement& a = *in_out; 2125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 7; i++) { 2145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[i+1] += a[i] >> 28; 2155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[i] &= kBottom28Bits; 2165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 top = a[7] >> 28; 2185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[7] &= kBottom28Bits; 2195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // top < 2**4 2215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Constant-time: mask = (top != 0) ? 0xffffffff : 0 2225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 mask = top; 2235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) mask |= mask >> 2; 2245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) mask |= mask >> 1; 2255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) mask <<= 31; 2265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) mask = static_cast<uint32>(static_cast<int32>(mask) >> 31); 2275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Eliminate top while maintaining the same value mod p. 2295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[0] -= top; 2305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[3] += top << 12; 2315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // We may have just made a[0] negative but, if we did, then we must 2335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // have added something to a[3], thus it's > 2**12. Therefore we can 2345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // carry down to a[0]. 2355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[3] -= 1 & mask; 2365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[2] += mask & ((1<<28) - 1); 2375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[1] += mask & ((1<<28) - 1); 2385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) a[0] += mask & (1<<28); 2395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 2405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2415821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Invert calcuates *out = in**-1 by computing in**(2**224 - 2**96 - 1), i.e. 2425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Fermat's little theorem. 2435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Invert(FieldElement* out, const FieldElement& in) { 2445821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement f1, f2, f3, f4; 2455821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2465821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f1, in); // 2 2475821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f1, f1, in); // 2**2 - 1 2485821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f1, f1); // 2**3 - 2 2495821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f1, f1, in); // 2**3 - 1 2505821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f2, f1); // 2**4 - 2 2515821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f2, f2); // 2**5 - 4 2525821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f2, f2); // 2**6 - 8 2535821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f1, f1, f2); // 2**6 - 1 2545821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f2, f1); // 2**7 - 2 2555821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 5; i++) { // 2**12 - 2**6 2565821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f2, f2); 2575821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2585821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f2, f2, f1); // 2**12 - 1 2595821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f3, f2); // 2**13 - 2 2605821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 11; i++) { // 2**24 - 2**12 2615821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f3, f3); 2625821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2635821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f2, f3, f2); // 2**24 - 1 2645821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f3, f2); // 2**25 - 2 2655821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 23; i++) { // 2**48 - 2**24 2665821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f3, f3); 2675821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2685821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f3, f3, f2); // 2**48 - 1 2695821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f4, f3); // 2**49 - 2 2705821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 47; i++) { // 2**96 - 2**48 2715821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f4, f4); 2725821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2735821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f3, f3, f4); // 2**96 - 1 2745821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f4, f3); // 2**97 - 2 2755821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 23; i++) { // 2**120 - 2**24 2765821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f4, f4); 2775821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2785821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f2, f4, f2); // 2**120 - 1 2795821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 6; i++) { // 2**126 - 2**6 2805821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f2, f2); 2815821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2825821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f1, f1, f2); // 2**126 - 1 2835821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f1, f1); // 2**127 - 2 2845821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&f1, f1, in); // 2**127 - 1 2855821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 97; i++) { // 2**224 - 2**97 2865821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&f1, f1); 2875821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 2885821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(out, f1, f3); // 2**224 - 2**96 - 1 2895821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 2905821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2915821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Contract converts a FieldElement to its minimal, distinguished form. 2925821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 2935821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// On entry, in[i] < 2**29 2945821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// On exit, in[i] < 2**28 2955821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Contract(FieldElement* inout) { 2965821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement& out = *inout; 2975821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 2985821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Reduce the coefficients to < 2**28. 2995821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 7; i++) { 3005821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i+1] += out[i] >> 28; 3015821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i] &= kBottom28Bits; 3025821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 3035821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 top = out[7] >> 28; 3045821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[7] &= kBottom28Bits; 3055821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3065821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Eliminate top while maintaining the same value mod p. 3075821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[0] -= top; 3085821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[3] += top << 12; 3095821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // We may just have made out[0] negative. So we carry down. If we made 3115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // out[0] negative then we know that out[3] is sufficiently positive 3125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // because we just added to it. 3135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 3; i++) { 3145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 mask = static_cast<uint32>(static_cast<int32>(out[i]) >> 31); 3155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i] += (1 << 28) & mask; 3165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i+1] -= 1 & mask; 3175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 3185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // We might have pushed out[3] over 2**28 so we perform another, partial 3205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // carry chain. 3215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 3; i < 7; i++) { 3225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i+1] += out[i] >> 28; 3235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i] &= kBottom28Bits; 3245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 3255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top = out[7] >> 28; 3265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[7] &= kBottom28Bits; 3275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Eliminate top while maintaining the same value mod p. 3295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[0] -= top; 3305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[3] += top << 12; 3315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // There are two cases to consider for out[3]: 3335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // 1) The first time that we eliminated top, we didn't push out[3] over 3345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // 2**28. In this case, the partial carry chain didn't change any values 3355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // and top is zero. 3365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // 2) We did push out[3] over 2**28 the first time that we eliminated top. 3375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // The first value of top was in [0..16), therefore, prior to eliminating 3385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after 3395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // overflowing and being reduced by the second carry chain, out[3] <= 3405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // 0xf000. Thus it cannot have overflowed when we eliminated top for the 3415821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // second time. 3425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Again, we may just have made out[0] negative, so do the same carry down. 3445821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // As before, if we made out[0] negative then we know that out[3] is 3455821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // sufficiently positive. 3465821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 3; i++) { 3475821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 mask = static_cast<uint32>(static_cast<int32>(out[i]) >> 31); 3485821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i] += (1 << 28) & mask; 3495821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[i+1] -= 1 & mask; 3505821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 3515821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3525821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // The value is < 2**224, but maybe greater than p. In order to reduce to a 3535821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // unique, minimal value we see if the value is >= p and, if so, subtract p. 3545821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3555821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // First we build a mask from the top four limbs, which must all be 3565821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // equal to bottom28Bits if the whole value is >= p. If top_4_all_ones 3575821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // ends up with any zero bits in the bottom 28 bits, then this wasn't 3585821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // true. 3595821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 top_4_all_ones = 0xffffffffu; 3605821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 4; i < 8; i++) { 3615821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones &= out[i]; 3625821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 3635821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones |= 0xf0000000; 3645821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Now we replicate any zero bits to all the bits in top_4_all_ones. 3655821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones &= top_4_all_ones >> 16; 3665821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones &= top_4_all_ones >> 8; 3675821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones &= top_4_all_ones >> 4; 3685821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones &= top_4_all_ones >> 2; 3695821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones &= top_4_all_ones >> 1; 3705821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) top_4_all_ones = 3715821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) static_cast<uint32>(static_cast<int32>(top_4_all_ones << 31) >> 31); 3725821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3735821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Now we test whether the bottom three limbs are non-zero. 3745821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 bottom_3_non_zero = out[0] | out[1] | out[2]; 3755821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) bottom_3_non_zero |= bottom_3_non_zero >> 16; 3765821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) bottom_3_non_zero |= bottom_3_non_zero >> 8; 3775821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) bottom_3_non_zero |= bottom_3_non_zero >> 4; 3785821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) bottom_3_non_zero |= bottom_3_non_zero >> 2; 3795821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) bottom_3_non_zero |= bottom_3_non_zero >> 1; 3805821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) bottom_3_non_zero = 3815821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) static_cast<uint32>(static_cast<int32>(bottom_3_non_zero) >> 31); 3825821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3835821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Everything depends on the value of out[3]. 3845821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // If it's > 0xffff000 and top_4_all_ones != 0 then the whole value is >= p 3855821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // If it's = 0xffff000 and top_4_all_ones != 0 and bottom_3_non_zero != 0, 3865821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // then the whole value is >= p 3875821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // If it's < 0xffff000, then the whole value is < p 3885821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 n = out[3] - 0xffff000; 3895821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 out_3_equal = n; 3905821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out_3_equal |= out_3_equal >> 16; 3915821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out_3_equal |= out_3_equal >> 8; 3925821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out_3_equal |= out_3_equal >> 4; 3935821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out_3_equal |= out_3_equal >> 2; 3945821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out_3_equal |= out_3_equal >> 1; 3955821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out_3_equal = 3965821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) ~static_cast<uint32>(static_cast<int32>(out_3_equal << 31) >> 31); 3975821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 3985821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // If out[3] > 0xffff000 then n's MSB will be zero. 3995821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 out_3_gt = ~static_cast<uint32>(static_cast<int32>(n << 31) >> 31); 4005821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4015821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 mask = top_4_all_ones & ((out_3_equal & bottom_3_non_zero) | out_3_gt); 4025821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[0] -= 1 & mask; 4035821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[3] -= 0xffff000 & mask; 4045821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[4] -= 0xfffffff & mask; 4055821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[5] -= 0xfffffff & mask; 4065821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[6] -= 0xfffffff & mask; 4075821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[7] -= 0xfffffff & mask; 4085821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 4095821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Group element functions. 4125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 4135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// These functions deal with group elements. The group is an elliptic curve 4145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// group with a = -3 defined in FIPS 186-3, section D.2.2. 4155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)using crypto::p224::Point; 4175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// kB is parameter of the elliptic curve. 4195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)const FieldElement kB = { 4205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 55967668, 11768882, 265861671, 185302395, 4215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 39211076, 180311059, 84673715, 188764328, 4225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)}; 4235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void CopyConditional(Point* out, const Point& a, uint32 mask); 4255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void DoubleJacobian(Point* out, const Point& a); 4265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// AddJacobian computes *out = a+b where a != b. 4285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void AddJacobian(Point *out, 4295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) const Point& a, 4305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) const Point& b) { 4315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl 4325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v; 4335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 z1_is_zero = IsZero(a.z); 4355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 z2_is_zero = IsZero(b.z); 4365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Z1Z1 = Z1² 4385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&z1z1, a.z); 4395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Z2Z2 = Z2² 4415821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&z2z2, b.z); 4425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // U1 = X1*Z2Z2 4445821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&u1, a.x, z2z2); 4455821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4465821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // U2 = X2*Z1Z1 4475821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&u2, b.x, z1z1); 4485821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4495821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // S1 = Y1*Z2*Z2Z2 4505821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&s1, b.z, z2z2); 4515821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&s1, a.y, s1); 4525821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4535821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // S2 = Y2*Z1*Z1Z1 4545821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&s2, a.z, z1z1); 4555821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&s2, b.y, s2); 4565821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4575821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // H = U2-U1 4585821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&h, u2, u1); 4595821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&h); 4605821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 x_equal = IsZero(h); 4615821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4625821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // I = (2*H)² 4635821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int j = 0; j < 8; j++) { 4645821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) i[j] = h[j] << 1; 4655821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 4665821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&i); 4675821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&i, i); 4685821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4695821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // J = H*I 4705821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&j, h, i); 4715821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // r = 2*(S2-S1) 4725821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&r, s2, s1); 4735821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&r); 4745821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 y_equal = IsZero(r); 4755821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4765821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { 4775821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // The two input points are the same therefore we must use the dedicated 4785821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // doubling function as the slope of the line is undefined. 4795821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) DoubleJacobian(out, a); 4805821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) return; 4815821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 4825821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4835821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 4845821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) r[i] <<= 1; 4855821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 4865821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&r); 4875821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4885821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // V = U1*I 4895821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&v, u1, i); 4905821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 4915821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H 4925821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Add(&z1z1, z1z1, z2z2); 4935821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Add(&z2z2, a.z, b.z); 4945821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&z2z2); 4955821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&z2z2, z2z2); 4965821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->z, z2z2, z1z1); 4975821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->z); 4985821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&out->z, out->z, h); 4995821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5005821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // X3 = r²-J-2*V 5015821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 5025821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) z1z1[i] = v[i] << 1; 5035821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 5045821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Add(&z1z1, j, z1z1); 5055821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&z1z1); 5065821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&out->x, r); 5075821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->x, out->x, z1z1); 5085821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->x); 5095821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Y3 = r*(V-X3)-2*S1*J 5115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 5125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) s1[i] <<= 1; 5135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 5145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&s1, s1, j); 5155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&z1z1, v, out->x); 5165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&z1z1); 5175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&z1z1, z1z1, r); 5185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->y, z1z1, s1); 5195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->y); 5205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) CopyConditional(out, a, z2_is_zero); 5225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) CopyConditional(out, b, z1_is_zero); 5235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 5245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// DoubleJacobian computes *out = a+a. 5265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void DoubleJacobian(Point* out, const Point& a) { 5275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b 5285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement delta, gamma, beta, alpha, t; 5295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&delta, a.z); 5315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&gamma, a.y); 5325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&beta, a.x, gamma); 5335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // alpha = 3*(X1-delta)*(X1+delta) 5355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Add(&t, a.x, delta); 5365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 5375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) t[i] += t[i] << 1; 5385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 5395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&t); 5405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&alpha, a.x, delta); 5415821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&alpha); 5425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&alpha, alpha, t); 5435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5445821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Z3 = (Y1+Z1)²-gamma-delta 5455821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Add(&out->z, a.y, a.z); 5465821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->z); 5475821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&out->z, out->z); 5485821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->z, out->z, gamma); 5495821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->z); 5505821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->z, out->z, delta); 5515821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->z); 5525821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5535821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // X3 = alpha²-8*beta 5545821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 5555821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) delta[i] = beta[i] << 3; 5565821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 5575821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&delta); 5585821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&out->x, alpha); 5595821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->x, out->x, delta); 5605821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->x); 5615821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5625821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Y3 = alpha*(4*beta-X3)-8*gamma² 5635821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 5645821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) beta[i] <<= 2; 5655821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 5665821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&beta); 5675821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&beta, beta, out->x); 5685821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&beta); 5695821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&gamma, gamma); 5705821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 5715821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) gamma[i] <<= 3; 5725821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 5735821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&gamma); 5745821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&out->y, alpha, beta); 5755821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->y, out->y, gamma); 5765821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->y); 5775821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 5785821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5795821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// CopyConditional sets *out=a if mask is 0xffffffff. mask must be either 0 of 5805821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// 0xffffffff. 5815821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void CopyConditional(Point* out, 5825821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) const Point& a, 5835821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 mask) { 5845821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 5855821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out->x[i] ^= mask & (a.x[i] ^ out->x[i]); 5865821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out->y[i] ^= mask & (a.y[i] ^ out->y[i]); 5875821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out->z[i] ^= mask & (a.z[i] ^ out->z[i]); 5885821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 5895821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 5905821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5915821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// ScalarMult calculates *out = a*scalar where scalar is a big-endian number of 5925821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// length scalar_len and != 0. 5935821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void ScalarMult(Point* out, const Point& a, 5945821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) const uint8* scalar, size_t scalar_len) { 5955821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) memset(out, 0, sizeof(*out)); 5965821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Point tmp; 5975821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 5985821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (size_t i = 0; i < scalar_len; i++) { 5995821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (unsigned int bit_num = 0; bit_num < 8; bit_num++) { 6005821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) DoubleJacobian(out, *out); 6015821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 bit = static_cast<uint32>(static_cast<int32>( 6025821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (((scalar[i] >> (7 - bit_num)) & 1) << 31) >> 31)); 6035821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) AddJacobian(&tmp, a, *out); 6045821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) CopyConditional(out, tmp, bit); 6055821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 6065821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 6075821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 6085821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6095821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Get224Bits reads 7 words from in and scatters their contents in 6105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// little-endian form into 8 words at out, 28 bits per output word. 6115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Get224Bits(uint32* out, const uint32* in) { 6125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[0] = NetToHost32(in[6]) & kBottom28Bits; 6135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[1] = ((NetToHost32(in[5]) << 4) | 6145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (NetToHost32(in[6]) >> 28)) & kBottom28Bits; 6155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[2] = ((NetToHost32(in[4]) << 8) | 6165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (NetToHost32(in[5]) >> 24)) & kBottom28Bits; 6175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[3] = ((NetToHost32(in[3]) << 12) | 6185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (NetToHost32(in[4]) >> 20)) & kBottom28Bits; 6195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[4] = ((NetToHost32(in[2]) << 16) | 6205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (NetToHost32(in[3]) >> 16)) & kBottom28Bits; 6215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[5] = ((NetToHost32(in[1]) << 20) | 6225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (NetToHost32(in[2]) >> 12)) & kBottom28Bits; 6235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[6] = ((NetToHost32(in[0]) << 24) | 6245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) (NetToHost32(in[1]) >> 8)) & kBottom28Bits; 6255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[7] = (NetToHost32(in[0]) >> 4) & kBottom28Bits; 6265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 6275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// Put224Bits performs the inverse operation to Get224Bits: taking 28 bits from 6295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// each of 8 input words and writing them in big-endian order to 7 words at 6305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// out. 6315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Put224Bits(uint32* out, const uint32* in) { 6325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[6] = HostToNet32((in[0] >> 0) | (in[1] << 28)); 6335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[5] = HostToNet32((in[1] >> 4) | (in[2] << 24)); 6345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[4] = HostToNet32((in[2] >> 8) | (in[3] << 20)); 6355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[3] = HostToNet32((in[3] >> 12) | (in[4] << 16)); 6365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[2] = HostToNet32((in[4] >> 16) | (in[5] << 12)); 6375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[1] = HostToNet32((in[5] >> 20) | (in[6] << 8)); 6385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out[0] = HostToNet32((in[6] >> 24) | (in[7] << 4)); 6395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 6405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6415f1c94371a64b3196d4be9466099bb892df9b88eTorne (Richard Coles)} // anonymous namespace 6425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)namespace crypto { 6445821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6455821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)namespace p224 { 6465821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6475821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)bool Point::SetFromString(const base::StringPiece& in) { 6485821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) if (in.size() != 2*28) 6495821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) return false; 6505821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) const uint32* inwords = reinterpret_cast<const uint32*>(in.data()); 6515821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Get224Bits(x, inwords); 6525821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Get224Bits(y, inwords + 7); 6535821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) memset(&z, 0, sizeof(z)); 6545821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) z[0] = 1; 6555821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6565821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Check that the point is on the curve, i.e. that y² = x³ - 3x + b. 6575821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement lhs; 6585821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&lhs, y); 6595821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Contract(&lhs); 6605821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6615821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement rhs; 6625821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&rhs, x); 6635821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&rhs, x, rhs); 6645821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6655821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement three_x; 6665821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) for (int i = 0; i < 8; i++) { 6675821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) three_x[i] = x[i] * 3; 6685821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 6695821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&three_x); 6705821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&rhs, rhs, three_x); 6715821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&rhs); 6725821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6735821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) ::Add(&rhs, rhs, kB); 6745821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Contract(&rhs); 6755821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) return memcmp(&lhs, &rhs, sizeof(lhs)) == 0; 6765821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 6775821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6785821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)std::string Point::ToString() const { 6795821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement zinv, zinv_sq, x, y; 6805821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6815821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // If this is the point at infinity we return a string of all zeros. 6825821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) if (IsZero(this->z)) { 6835821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) static const char zeros[56] = {0}; 6845821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) return std::string(zeros, sizeof(zeros)); 6855821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) } 6865821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6875821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Invert(&zinv, this->z); 6885821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&zinv_sq, zinv); 6895821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&x, this->x, zinv_sq); 6905821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&zinv_sq, zinv_sq, zinv); 6915821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&y, this->y, zinv_sq); 6925821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6935821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Contract(&x); 6945821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Contract(&y); 6955821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 6965821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) uint32 outwords[14]; 6975821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Put224Bits(outwords, x); 6985821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Put224Bits(outwords + 7, y); 6995821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) return std::string(reinterpret_cast<const char*>(outwords), sizeof(outwords)); 7005821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 7015821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7025821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void ScalarMult(const Point& in, const uint8* scalar, Point* out) { 7035821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) ::ScalarMult(out, in, scalar, 28); 7045821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 7055821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7065821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)// kBasePoint is the base point (generator) of the elliptic curve group. 7075821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)static const Point kBasePoint = { 7085821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) {22813985, 52956513, 34677300, 203240812, 7095821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 12143107, 133374265, 225162431, 191946955}, 7105821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) {83918388, 223877528, 122119236, 123340192, 7115821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 266784067, 263504429, 146143011, 198407736}, 7125821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) {1, 0, 0, 0, 0, 0, 0, 0}, 7135821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)}; 7145821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7155821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void ScalarBaseMult(const uint8* scalar, Point* out) { 7165821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) ::ScalarMult(out, kBasePoint, scalar, 28); 7175821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 7185821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7195821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Add(const Point& a, const Point& b, Point* out) { 7205821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) AddJacobian(out, a, b); 7215821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 7225821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7235821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)void Negate(const Point& in, Point* out) { 7245821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // Guide to elliptic curve cryptography, page 89 suggests that (X : X+Y : Z) 7255821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // is the negative in Jacobian coordinates, but it doesn't actually appear to 7265821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) // be true in testing so this performs the negation in affine coordinates. 7275821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) FieldElement zinv, zinv_sq, y; 7285821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Invert(&zinv, in.z); 7295821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Square(&zinv_sq, zinv); 7305821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&out->x, in.x, zinv_sq); 7315821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&zinv_sq, zinv_sq, zinv); 7325821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Mul(&y, in.y, zinv_sq); 7335821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7345821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Subtract(&out->y, kP, y); 7355821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) Reduce(&out->y); 7365821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7375821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) memset(&out->z, 0, sizeof(out->z)); 7385821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) out->z[0] = 1; 7395821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} 7405821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7415821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} // namespace p224 7425821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles) 7435821806d5e7f356e8fa4b058a389a808ea183019Torne (Richard Coles)} // namespace crypto 744