LEFT | RIGHT |
(no file at all) | |
| 1 // Copyright 2013 The Go Authors. All rights reserved. |
| 2 // Use of this source code is governed by a BSD-style |
| 3 // license that can be found in the LICENSE file. |
| 4 |
| 5 package elliptic |
| 6 |
| 7 // This file contains a constant-time, 32-bit implementation of P256. |
| 8 |
| 9 import ( |
| 10 "math/big" |
| 11 ) |
| 12 |
| 13 type p256Curve struct { |
| 14 *CurveParams |
| 15 } |
| 16 |
| 17 var ( |
| 18 p256 p256Curve |
| 19 // RInverse contains 1/R mod p - the inverse of the Montgomery constant |
| 20 // (2**257). |
| 21 p256RInverse *big.Int |
| 22 ) |
| 23 |
| 24 func initP256() { |
| 25 // See FIPS 186-3, section D.2.3 |
| 26 p256.CurveParams = new(CurveParams) |
| 27 p256.P, _ = new(big.Int).SetString("115792089210356248762697446949407573
530086143415290314195533631308867097853951", 10) |
| 28 p256.N, _ = new(big.Int).SetString("115792089210356248762697446949407573
529996955224135760342422259061068512044369", 10) |
| 29 p256.B, _ = new(big.Int).SetString("5ac635d8aa3a93e7b3ebbd55769886bc651d
06b0cc53b0f63bce3c3e27d2604b", 16) |
| 30 p256.Gx, _ = new(big.Int).SetString("6b17d1f2e12c4247f8bce6e563a440f2770
37d812deb33a0f4a13945d898c296", 16) |
| 31 p256.Gy, _ = new(big.Int).SetString("4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bc
e33576b315ececbb6406837bf51f5", 16) |
| 32 p256.BitSize = 256 |
| 33 |
| 34 p256RInverse, _ = new(big.Int).SetString("7fffffff00000001fffffffe800000
0100000000ffffffff0000000180000000", 16) |
| 35 } |
| 36 |
| 37 func (curve p256Curve) Params() *CurveParams { |
| 38 return curve.CurveParams |
| 39 } |
| 40 |
| 41 // p256GetScalar endian-swaps the big-endian scalar value from in and writes it |
| 42 // to out. If the scalar is equal or greater than the order of the group, it's |
| 43 // reduced modulo that order. |
| 44 func p256GetScalar(out *[32]byte, in []byte) { |
| 45 n := new(big.Int).SetBytes(in) |
| 46 var scalarBytes []byte |
| 47 |
| 48 if n.Cmp(p256.N) >= 0 { |
| 49 n.Mod(n, p256.N) |
| 50 scalarBytes = n.Bytes() |
| 51 } else { |
| 52 scalarBytes = in |
| 53 } |
| 54 |
| 55 for i, v := range scalarBytes { |
| 56 out[len(scalarBytes)-(1+i)] = v |
| 57 } |
| 58 } |
| 59 |
| 60 func (p256Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) { |
| 61 var scalarReversed [32]byte |
| 62 p256GetScalar(&scalarReversed, scalar) |
| 63 |
| 64 var x1, y1, z1 [p256Limbs]uint32 |
| 65 p256ScalarBaseMult(&x1, &y1, &z1, &scalarReversed) |
| 66 return p256ToAffine(&x1, &y1, &z1) |
| 67 } |
| 68 |
| 69 func (p256Curve) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int)
{ |
| 70 var scalarReversed [32]byte |
| 71 p256GetScalar(&scalarReversed, scalar) |
| 72 |
| 73 var px, py, x1, y1, z1 [p256Limbs]uint32 |
| 74 p256FromBig(&px, bigX) |
| 75 p256FromBig(&py, bigY) |
| 76 p256ScalarMult(&x1, &y1, &z1, &px, &py, &scalarReversed) |
| 77 return p256ToAffine(&x1, &y1, &z1) |
| 78 } |
| 79 |
| 80 // Field elements are represented as nine, unsigned 32-bit words. |
| 81 // |
| 82 // The value of an field element is: |
| 83 // x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228) |
| 84 // |
| 85 // That is, each limb is alternately 29 or 28-bits wide in little-endian |
| 86 // order. |
| 87 // |
| 88 // This means that a field element hits 2**257, rather than 2**256 as we would |
| 89 // like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes |
| 90 // problems when multiplying as terms end up one bit short of a limb which |
| 91 // would require much bit-shifting to correct. |
| 92 // |
| 93 // Finally, the values stored in a field element are in Montgomery form. So the |
| 94 // value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is |
| 95 // 2**257. |
| 96 |
| 97 const ( |
| 98 p256Limbs = 9 |
| 99 bottom29Bits = 0x1fffffff |
| 100 ) |
| 101 |
| 102 var ( |
| 103 // p256One is the number 1 as a field element. |
| 104 p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff,
0x1fbfffff, 0x1ffffff, 0} |
| 105 p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0} |
| 106 // p256P is the prime modulus as a field element. |
| 107 p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0
, 0x200000, 0xf000000, 0xfffffff} |
| 108 // p2562P is the twice prime modulus as a field element. |
| 109 p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0,
0, 0x400000, 0xe000000, 0x1fffffff} |
| 110 ) |
| 111 |
| 112 // p256Precomputed contains precomputed values to aid the calculation of scalar |
| 113 // multiples of the base point, G. It's actually two, equal length, tables |
| 114 // concatenated. |
| 115 // |
| 116 // The first table contains (x,y) field element pairs for 16 multiples of the |
| 117 // base point, G. |
| 118 // |
| 119 // Index | Index (binary) | Value |
| 120 // 0 | 0000 | 0G (all zeros, omitted) |
| 121 // 1 | 0001 | G |
| 122 // 2 | 0010 | 2**64G |
| 123 // 3 | 0011 | 2**64G + G |
| 124 // 4 | 0100 | 2**128G |
| 125 // 5 | 0101 | 2**128G + G |
| 126 // 6 | 0110 | 2**128G + 2**64G |
| 127 // 7 | 0111 | 2**128G + 2**64G + G |
| 128 // 8 | 1000 | 2**192G |
| 129 // 9 | 1001 | 2**192G + G |
| 130 // 10 | 1010 | 2**192G + 2**64G |
| 131 // 11 | 1011 | 2**192G + 2**64G + G |
| 132 // 12 | 1100 | 2**192G + 2**128G |
| 133 // 13 | 1101 | 2**192G + 2**128G + G |
| 134 // 14 | 1110 | 2**192G + 2**128G + 2**64G |
| 135 // 15 | 1111 | 2**192G + 2**128G + 2**64G + G |
| 136 // |
| 137 // The second table follows the same style, but the terms are 2**32G, |
| 138 // 2**96G, 2**160G, 2**224G. |
| 139 // |
| 140 // This is ~2KB of data. |
| 141 var p256Precomputed = [p256Limbs * 2 * 15 * 2]uint32{ |
| 142 0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x11
9e7edc, 0xd4a6eab, 0x3120bee, |
| 143 0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x1
54ba21, 0x14b10bb, 0xae3fe3, |
| 144 0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe
49073, 0x3fa36cc, 0x5ebcd2c, |
| 145 0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea
12446, 0xe1ade1e, 0xec91f22, |
| 146 0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c
7109, 0xa267a00, 0xb57c050, |
| 147 0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c
5, 0x7d6dee7, 0x2976e4b, |
| 148 0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1
d96a5a9, 0x843a649, 0xc3ab0fa, |
| 149 0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf64
3e11, 0x58c43df, 0xf423fc2, |
| 150 0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x17
2db40f, 0x83e277d, 0xb0dd609, |
| 151 0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449
e3f5, 0xe10c9e, 0x33ab581, |
| 152 0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35
df9f, 0x48764cd, 0x76dbcca, |
| 153 0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f
16b20, 0x4ba3173, 0xc168c33, |
| 154 0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322
c4c0, 0x65dd7ff, 0x3a1e4f6, |
| 155 0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x34
1f077, 0xa6add89, 0x4894acd, |
| 156 0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a29
01a, 0x69a8556, 0x7e7c0, |
| 157 0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051
825c, 0xda0cf5b, 0x812e881, |
| 158 0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe8
0c51, 0xc22be3e, 0xe35e65a, |
| 159 0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c1
84e9, 0x1c5a839, 0x47a1e26, |
| 160 0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a
90c502, 0x2f32042, 0xa17769b, |
| 161 0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10
d06a02, 0x3fc93, 0x5620023, |
| 162 0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc
513c, 0x407f75c, 0xbaab133, |
| 163 0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x104
69ea7, 0x3293ac0, 0xcdc98aa, |
| 164 0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62
f16, 0x2b6fcc7, 0xf5a4e29, |
| 165 0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x102
9f72, 0x73e1c35, 0xee70fbc, |
| 166 0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83
de85, 0x27de188, 0x66f70b8, |
| 167 0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x1
54ae914, 0x2f3ec51, 0x3826b59, |
| 168 0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061
, 0x823d9d2, 0x8213f39, |
| 169 0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x184
11a4a, 0xf5ddc3d, 0x3786689, |
| 170 0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x105
1a729, 0x4be3499, 0x52b23aa, |
| 171 0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb
048035, 0xe31de66, 0xc6ecaa3, |
| 172 0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4
a7529, 0xcb7beb1, 0xb2a78a1, |
| 173 0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1c
bff658, 0xe3d6511, 0xc7d76f, |
| 174 0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc501
24c, 0x50daa90, 0xb13f72, |
| 175 0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d
32411, 0xb04a838, 0xd760d2d, |
| 176 0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c
9e11e, 0x20bca9a, 0x66f496b, |
| 177 0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x15829
68d, 0xbe985f7, 0x1acbc1a, |
| 178 0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17
fa56ff, 0x65ef930, 0x21dc4a, |
| 179 0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112
ac15f, 0x624e62e, 0xa90ae2f, |
| 180 0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x72
5522b, 0xdc78583, 0x40eeabb, |
| 181 0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x36
7ef34, 0xae2a960, 0x91b8bdc, |
| 182 0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316ab
b, 0x2413c8e, 0x5425bf9, |
| 183 0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e
7633, 0x7c91952, 0xd806dce, |
| 184 0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc
4ef73, 0x8956f34, 0xe4b5cf2, |
| 185 0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf92
7ed7, 0x627b614, 0x7371cca, |
| 186 0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e
3edc9, 0x9c19bf2, 0x5882229, |
| 187 0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c
9b5b3, 0xe85ff25, 0x408ef57, |
| 188 0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa03
8113, 0xa4a1769, 0x11fbc6c, |
| 189 0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15c
d60b7, 0x4acbad9, 0x5efc5fa, |
| 190 0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x1961
42cc, 0x7bf0fa9, 0x957651, |
| 191 0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7
d57, 0xf2ecaac, 0xca86dec, |
| 192 0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19
d1c12d, 0xf20bd46, 0x1951fa7, |
| 193 0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9
fc74, 0x99bb618, 0x2db944c, |
| 194 0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e
74779, 0x576138, 0x9587927, |
| 195 0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d0
7782d, 0xfc72e0b, 0x701b298, |
| 196 0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14
e2f5d8, 0xf858d3a, 0x942eea8, |
| 197 0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x198
1d7a1, 0x8395659, 0x52ed4e2, |
| 198 0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c
146c0, 0x6bdf55a, 0x4e4457d, |
| 199 0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x40
4747b, 0x878558d, 0x7d29aa4, |
| 200 0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81
d55d7, 0xa5bef68, 0xb7b30d8, |
| 201 0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f
51951, 0x9d0c177, 0x1c49a78, |
| 202 } |
| 203 |
| 204 // Field element operations: |
| 205 |
| 206 // nonZeroToAllOnes returns: |
| 207 // 0xffffffff for 0 < x <= 2**31 |
| 208 // 0 for x == 0 or x > 2**31. |
| 209 func nonZeroToAllOnes(x uint32) uint32 { |
| 210 return ((x - 1) >> 31) - 1 |
| 211 } |
| 212 |
| 213 // p256ReduceCarry adds a multiple of p in order to cancel |carry|, |
| 214 // which is a term at 2**257. |
| 215 // |
| 216 // On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28. |
| 217 // On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29. |
| 218 func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) { |
| 219 carry_mask := nonZeroToAllOnes(carry) |
| 220 |
| 221 inout[0] += carry << 1 |
| 222 inout[3] += 0x10000000 & carry_mask |
| 223 // carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the |
| 224 // previous line therefore this doesn't underflow. |
| 225 inout[3] -= carry << 11 |
| 226 inout[4] += (0x20000000 - 1) & carry_mask |
| 227 inout[5] += (0x10000000 - 1) & carry_mask |
| 228 inout[6] += (0x20000000 - 1) & carry_mask |
| 229 inout[6] -= carry << 22 |
| 230 // This may underflow if carry is non-zero but, if so, we'll fix it in t
he |
| 231 // next line. |
| 232 inout[7] -= 1 & carry_mask |
| 233 inout[7] += carry << 25 |
| 234 } |
| 235 |
| 236 // p256Sum sets out = in+in2. |
| 237 // |
| 238 // On entry, in[i]+in2[i] must not overflow a 32-bit word. |
| 239 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 |
| 240 func p256Sum(out, in, in2 *[p256Limbs]uint32) { |
| 241 carry := uint32(0) |
| 242 for i := 0; ; i++ { |
| 243 out[i] = in[i] + in2[i] |
| 244 out[i] += carry |
| 245 carry = out[i] >> 29 |
| 246 out[i] &= bottom29Bits |
| 247 |
| 248 i++ |
| 249 if i == p256Limbs { |
| 250 break |
| 251 } |
| 252 |
| 253 out[i] = in[i] + in2[i] |
| 254 out[i] += carry |
| 255 carry = out[i] >> 28 |
| 256 out[i] &= bottom28Bits |
| 257 } |
| 258 |
| 259 p256ReduceCarry(out, carry) |
| 260 } |
| 261 |
| 262 const ( |
| 263 two30m2 = 1<<30 - 1<<2 |
| 264 two30p13m2 = 1<<30 + 1<<13 - 1<<2 |
| 265 two31m2 = 1<<31 - 1<<2 |
| 266 two31p24m2 = 1<<31 + 1<<24 - 1<<2 |
| 267 two30m27m2 = 1<<30 - 1<<27 - 1<<2 |
| 268 ) |
| 269 |
| 270 // p256Zero31 is 0 mod p. |
| 271 var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m
2, two30m2, two31p24m2, two30m27m2, two31m2} |
| 272 |
| 273 // p256Diff sets out = in-in2. |
| 274 // |
| 275 // On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and |
| 276 // in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. |
| 277 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 278 func p256Diff(out, in, in2 *[p256Limbs]uint32) { |
| 279 var carry uint32 |
| 280 |
| 281 for i := 0; ; i++ { |
| 282 out[i] = in[i] - in2[i] |
| 283 out[i] += p256Zero31[i] |
| 284 out[i] += carry |
| 285 carry = out[i] >> 29 |
| 286 out[i] &= bottom29Bits |
| 287 |
| 288 i++ |
| 289 if i == p256Limbs { |
| 290 break |
| 291 } |
| 292 |
| 293 out[i] = in[i] - in2[i] |
| 294 out[i] += p256Zero31[i] |
| 295 out[i] += carry |
| 296 carry = out[i] >> 28 |
| 297 out[i] &= bottom28Bits |
| 298 } |
| 299 |
| 300 p256ReduceCarry(out, carry) |
| 301 } |
| 302 |
| 303 // p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with |
| 304 // the same 29,28,... bit positions as an field element. |
| 305 // |
| 306 // The values in field elements are in Montgomery form: x*R mod p where R = |
| 307 // 2**257. Since we just multiplied two Montgomery values together, the result |
| 308 // is x*y*R*R mod p. We wish to divide by R in order for the result also to be |
| 309 // in Montgomery form. |
| 310 // |
| 311 // On entry: tmp[i] < 2**64 |
| 312 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 |
| 313 func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) { |
| 314 // The following table may be helpful when reading this code: |
| 315 // |
| 316 // Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10... |
| 317 // Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29 |
| 318 // Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285 |
| 319 // (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285 |
| 320 var tmp2 [18]uint32 |
| 321 var carry, x, xMask uint32 |
| 322 |
| 323 // tmp contains 64-bit words with the same 29,28,29-bit positions as an |
| 324 // field element. So the top of an element of tmp might overlap with |
| 325 // another element two positions down. The following loop eliminates |
| 326 // this overlap. |
| 327 tmp2[0] = uint32(tmp[0]) & bottom29Bits |
| 328 |
| 329 tmp2[1] = uint32(tmp[0]) >> 29 |
| 330 tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits |
| 331 tmp2[1] += uint32(tmp[1]) & bottom28Bits |
| 332 carry = tmp2[1] >> 28 |
| 333 tmp2[1] &= bottom28Bits |
| 334 |
| 335 for i := 2; i < 17; i++ { |
| 336 tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25 |
| 337 tmp2[i] += (uint32(tmp[i-1])) >> 28 |
| 338 tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits |
| 339 tmp2[i] += uint32(tmp[i]) & bottom29Bits |
| 340 tmp2[i] += carry |
| 341 carry = tmp2[i] >> 29 |
| 342 tmp2[i] &= bottom29Bits |
| 343 |
| 344 i++ |
| 345 if i == 17 { |
| 346 break |
| 347 } |
| 348 tmp2[i] = uint32(tmp[i-2]>>32) >> 25 |
| 349 tmp2[i] += uint32(tmp[i-1]) >> 29 |
| 350 tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits |
| 351 tmp2[i] += uint32(tmp[i]) & bottom28Bits |
| 352 tmp2[i] += carry |
| 353 carry = tmp2[i] >> 28 |
| 354 tmp2[i] &= bottom28Bits |
| 355 } |
| 356 |
| 357 tmp2[17] = uint32(tmp[15]>>32) >> 25 |
| 358 tmp2[17] += uint32(tmp[16]) >> 29 |
| 359 tmp2[17] += uint32(tmp[16]>>32) << 3 |
| 360 tmp2[17] += carry |
| 361 |
| 362 // Montgomery elimination of terms: |
| 363 // |
| 364 // Since R is 2**257, we can divide by R with a bitwise shift if we can |
| 365 // ensure that the right-most 257 bits are all zero. We can make that tr
ue |
| 366 // by adding multiplies of p without affecting the value. |
| 367 // |
| 368 // So we eliminate limbs from right to left. Since the bottom 29 bits of
p |
| 369 // are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] ==
0. |
| 370 // We can do that for 8 further limbs and then right shift to eliminate
the |
| 371 // extra factor of R. |
| 372 for i := 0; ; i += 2 { |
| 373 tmp2[i+1] += tmp2[i] >> 29 |
| 374 x = tmp2[i] & bottom29Bits |
| 375 xMask = nonZeroToAllOnes(x) |
| 376 tmp2[i] = 0 |
| 377 |
| 378 // The bounds calculations for this loop are tricky. Each iterat
ion of |
| 379 // the loop eliminates two words by adding values to words to th
eir |
| 380 // right. |
| 381 // |
| 382 // The following table contains the amounts added to each word (
as an |
| 383 // offset from the value of i at the top of the loop). The amoun
ts are |
| 384 // accounted for from the first and second half of the loop sepa
rately |
| 385 // and are written as, for example, 28 to mean a value <2**28. |
| 386 // |
| 387 // Word: 3 4 5 6 7 8 9 10 |
| 388 // Added in top half: 28 11 29 21 29 28 |
| 389 // 28 29 |
| 390 // 29 |
| 391 // Added in bottom half: 29 10 28 21 28 28 |
| 392 // 29 |
| 393 // |
| 394 // The value that is currently offset 7 will be offset 5 for the
next |
| 395 // iteration and then offset 3 for the iteration after that. The
refore |
| 396 // the total value added will be the values added at 7, 5 and 3. |
| 397 // |
| 398 // The following table accumulates these values. The sums at the
bottom |
| 399 // are written as, for example, 29+28, to mean a value < 2**29+2
**28. |
| 400 // |
| 401 // Word: 3 4 5 6 7 8 9 10 11 12
13 |
| 402 // 28 11 10 29 21 29 28 28 28 28
28 |
| 403 // 29 28 11 28 29 28 29 28 29
28 |
| 404 // 29 28 21 21 29 21 29
21 |
| 405 // 10 29 28 21 28 21
28 |
| 406 // 28 29 28 29 28 29
28 |
| 407 // 11 10 29 10 29
10 |
| 408 // 29 28 11 28 11 |
| 409 // 29 29 |
| 410 // --------------------------------------
------ |
| 411 // 30+ 31+ 30+ 31
+ 30+ |
| 412 // 28+ 29+ 28+ 29
+ 21+ |
| 413 // 21+ 28+ 21+ 28
+ 10 |
| 414 // 10 21+ 10 21
+ |
| 415 // 11 11 |
| 416 // |
| 417 // So the greatest amount is added to tmp2[10] and tmp2[12]. If |
| 418 // tmp2[10/12] has an initial value of <2**29, then the maximum
value |
| 419 // will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2
**32, |
| 420 // as required. |
| 421 tmp2[i+3] += (x << 10) & bottom28Bits |
| 422 tmp2[i+4] += (x >> 18) |
| 423 |
| 424 tmp2[i+6] += (x << 21) & bottom29Bits |
| 425 tmp2[i+7] += x >> 8 |
| 426 |
| 427 // At position 200, which is the starting bit position for word
7, we |
| 428 // have a factor of 0xf000000 = 2**28 - 2**24. |
| 429 tmp2[i+7] += 0x10000000 & xMask |
| 430 tmp2[i+8] += (x - 1) & xMask |
| 431 tmp2[i+7] -= (x << 24) & bottom28Bits |
| 432 tmp2[i+8] -= x >> 4 |
| 433 |
| 434 tmp2[i+8] += 0x20000000 & xMask |
| 435 tmp2[i+8] -= x |
| 436 tmp2[i+8] += (x << 28) & bottom29Bits |
| 437 tmp2[i+9] += ((x >> 1) - 1) & xMask |
| 438 |
| 439 if i+1 == p256Limbs { |
| 440 break |
| 441 } |
| 442 tmp2[i+2] += tmp2[i+1] >> 28 |
| 443 x = tmp2[i+1] & bottom28Bits |
| 444 xMask = nonZeroToAllOnes(x) |
| 445 tmp2[i+1] = 0 |
| 446 |
| 447 tmp2[i+4] += (x << 11) & bottom29Bits |
| 448 tmp2[i+5] += (x >> 18) |
| 449 |
| 450 tmp2[i+7] += (x << 21) & bottom28Bits |
| 451 tmp2[i+8] += x >> 7 |
| 452 |
| 453 // At position 199, which is the starting bit of the 8th word wh
en |
| 454 // dealing with a context starting on an odd word, we have a fac
tor of |
| 455 // 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the
8th |
| 456 // word from i+1 is i+8. |
| 457 tmp2[i+8] += 0x20000000 & xMask |
| 458 tmp2[i+9] += (x - 1) & xMask |
| 459 tmp2[i+8] -= (x << 25) & bottom29Bits |
| 460 tmp2[i+9] -= x >> 4 |
| 461 |
| 462 tmp2[i+9] += 0x10000000 & xMask |
| 463 tmp2[i+9] -= x |
| 464 tmp2[i+10] += (x - 1) & xMask |
| 465 } |
| 466 |
| 467 // We merge the right shift with a carry chain. The words above 2**257 h
ave |
| 468 // widths of 28,29,... which we need to correct when copying them down. |
| 469 carry = 0 |
| 470 for i := 0; i < 8; i++ { |
| 471 // The maximum value of tmp2[i + 9] occurs on the first iteratio
n and |
| 472 // is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is |
| 473 // therefore safe. |
| 474 out[i] = tmp2[i+9] |
| 475 out[i] += carry |
| 476 out[i] += (tmp2[i+10] << 28) & bottom29Bits |
| 477 carry = out[i] >> 29 |
| 478 out[i] &= bottom29Bits |
| 479 |
| 480 i++ |
| 481 out[i] = tmp2[i+9] >> 1 |
| 482 out[i] += carry |
| 483 carry = out[i] >> 28 |
| 484 out[i] &= bottom28Bits |
| 485 } |
| 486 |
| 487 out[8] = tmp2[17] |
| 488 out[8] += carry |
| 489 carry = out[8] >> 29 |
| 490 out[8] &= bottom29Bits |
| 491 |
| 492 p256ReduceCarry(out, carry) |
| 493 } |
| 494 |
| 495 // p256Square sets out=in*in. |
| 496 // |
| 497 // On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29. |
| 498 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 499 func p256Square(out, in *[p256Limbs]uint32) { |
| 500 var tmp [17]uint64 |
| 501 |
| 502 tmp[0] = uint64(in[0]) * uint64(in[0]) |
| 503 tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1) |
| 504 tmp[2] = uint64(in[0])*(uint64(in[2])<<1) + |
| 505 uint64(in[1])*(uint64(in[1])<<1) |
| 506 tmp[3] = uint64(in[0])*(uint64(in[3])<<1) + |
| 507 uint64(in[1])*(uint64(in[2])<<1) |
| 508 tmp[4] = uint64(in[0])*(uint64(in[4])<<1) + |
| 509 uint64(in[1])*(uint64(in[3])<<2) + |
| 510 uint64(in[2])*uint64(in[2]) |
| 511 tmp[5] = uint64(in[0])*(uint64(in[5])<<1) + |
| 512 uint64(in[1])*(uint64(in[4])<<1) + |
| 513 uint64(in[2])*(uint64(in[3])<<1) |
| 514 tmp[6] = uint64(in[0])*(uint64(in[6])<<1) + |
| 515 uint64(in[1])*(uint64(in[5])<<2) + |
| 516 uint64(in[2])*(uint64(in[4])<<1) + |
| 517 uint64(in[3])*(uint64(in[3])<<1) |
| 518 tmp[7] = uint64(in[0])*(uint64(in[7])<<1) + |
| 519 uint64(in[1])*(uint64(in[6])<<1) + |
| 520 uint64(in[2])*(uint64(in[5])<<1) + |
| 521 uint64(in[3])*(uint64(in[4])<<1) |
| 522 // tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**6
0, |
| 523 // which is < 2**64 as required. |
| 524 tmp[8] = uint64(in[0])*(uint64(in[8])<<1) + |
| 525 uint64(in[1])*(uint64(in[7])<<2) + |
| 526 uint64(in[2])*(uint64(in[6])<<1) + |
| 527 uint64(in[3])*(uint64(in[5])<<2) + |
| 528 uint64(in[4])*uint64(in[4]) |
| 529 tmp[9] = uint64(in[1])*(uint64(in[8])<<1) + |
| 530 uint64(in[2])*(uint64(in[7])<<1) + |
| 531 uint64(in[3])*(uint64(in[6])<<1) + |
| 532 uint64(in[4])*(uint64(in[5])<<1) |
| 533 tmp[10] = uint64(in[2])*(uint64(in[8])<<1) + |
| 534 uint64(in[3])*(uint64(in[7])<<2) + |
| 535 uint64(in[4])*(uint64(in[6])<<1) + |
| 536 uint64(in[5])*(uint64(in[5])<<1) |
| 537 tmp[11] = uint64(in[3])*(uint64(in[8])<<1) + |
| 538 uint64(in[4])*(uint64(in[7])<<1) + |
| 539 uint64(in[5])*(uint64(in[6])<<1) |
| 540 tmp[12] = uint64(in[4])*(uint64(in[8])<<1) + |
| 541 uint64(in[5])*(uint64(in[7])<<2) + |
| 542 uint64(in[6])*uint64(in[6]) |
| 543 tmp[13] = uint64(in[5])*(uint64(in[8])<<1) + |
| 544 uint64(in[6])*(uint64(in[7])<<1) |
| 545 tmp[14] = uint64(in[6])*(uint64(in[8])<<1) + |
| 546 uint64(in[7])*(uint64(in[7])<<1) |
| 547 tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1) |
| 548 tmp[16] = uint64(in[8]) * uint64(in[8]) |
| 549 |
| 550 p256ReduceDegree(out, tmp) |
| 551 } |
| 552 |
| 553 // p256Mul sets out=in*in2. |
| 554 // |
| 555 // On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and |
| 556 // in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. |
| 557 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 558 func p256Mul(out, in, in2 *[p256Limbs]uint32) { |
| 559 var tmp [17]uint64 |
| 560 |
| 561 tmp[0] = uint64(in[0]) * uint64(in2[0]) |
| 562 tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) + |
| 563 uint64(in[1])*(uint64(in2[0])<<0) |
| 564 tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) + |
| 565 uint64(in[1])*(uint64(in2[1])<<1) + |
| 566 uint64(in[2])*(uint64(in2[0])<<0) |
| 567 tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) + |
| 568 uint64(in[1])*(uint64(in2[2])<<0) + |
| 569 uint64(in[2])*(uint64(in2[1])<<0) + |
| 570 uint64(in[3])*(uint64(in2[0])<<0) |
| 571 tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) + |
| 572 uint64(in[1])*(uint64(in2[3])<<1) + |
| 573 uint64(in[2])*(uint64(in2[2])<<0) + |
| 574 uint64(in[3])*(uint64(in2[1])<<1) + |
| 575 uint64(in[4])*(uint64(in2[0])<<0) |
| 576 tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) + |
| 577 uint64(in[1])*(uint64(in2[4])<<0) + |
| 578 uint64(in[2])*(uint64(in2[3])<<0) + |
| 579 uint64(in[3])*(uint64(in2[2])<<0) + |
| 580 uint64(in[4])*(uint64(in2[1])<<0) + |
| 581 uint64(in[5])*(uint64(in2[0])<<0) |
| 582 tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) + |
| 583 uint64(in[1])*(uint64(in2[5])<<1) + |
| 584 uint64(in[2])*(uint64(in2[4])<<0) + |
| 585 uint64(in[3])*(uint64(in2[3])<<1) + |
| 586 uint64(in[4])*(uint64(in2[2])<<0) + |
| 587 uint64(in[5])*(uint64(in2[1])<<1) + |
| 588 uint64(in[6])*(uint64(in2[0])<<0) |
| 589 tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) + |
| 590 uint64(in[1])*(uint64(in2[6])<<0) + |
| 591 uint64(in[2])*(uint64(in2[5])<<0) + |
| 592 uint64(in[3])*(uint64(in2[4])<<0) + |
| 593 uint64(in[4])*(uint64(in2[3])<<0) + |
| 594 uint64(in[5])*(uint64(in2[2])<<0) + |
| 595 uint64(in[6])*(uint64(in2[1])<<0) + |
| 596 uint64(in[7])*(uint64(in2[0])<<0) |
| 597 // tmp[8] has the greatest value but doesn't overflow. See logic in |
| 598 // p256Square. |
| 599 tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) + |
| 600 uint64(in[1])*(uint64(in2[7])<<1) + |
| 601 uint64(in[2])*(uint64(in2[6])<<0) + |
| 602 uint64(in[3])*(uint64(in2[5])<<1) + |
| 603 uint64(in[4])*(uint64(in2[4])<<0) + |
| 604 uint64(in[5])*(uint64(in2[3])<<1) + |
| 605 uint64(in[6])*(uint64(in2[2])<<0) + |
| 606 uint64(in[7])*(uint64(in2[1])<<1) + |
| 607 uint64(in[8])*(uint64(in2[0])<<0) |
| 608 tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) + |
| 609 uint64(in[2])*(uint64(in2[7])<<0) + |
| 610 uint64(in[3])*(uint64(in2[6])<<0) + |
| 611 uint64(in[4])*(uint64(in2[5])<<0) + |
| 612 uint64(in[5])*(uint64(in2[4])<<0) + |
| 613 uint64(in[6])*(uint64(in2[3])<<0) + |
| 614 uint64(in[7])*(uint64(in2[2])<<0) + |
| 615 uint64(in[8])*(uint64(in2[1])<<0) |
| 616 tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) + |
| 617 uint64(in[3])*(uint64(in2[7])<<1) + |
| 618 uint64(in[4])*(uint64(in2[6])<<0) + |
| 619 uint64(in[5])*(uint64(in2[5])<<1) + |
| 620 uint64(in[6])*(uint64(in2[4])<<0) + |
| 621 uint64(in[7])*(uint64(in2[3])<<1) + |
| 622 uint64(in[8])*(uint64(in2[2])<<0) |
| 623 tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) + |
| 624 uint64(in[4])*(uint64(in2[7])<<0) + |
| 625 uint64(in[5])*(uint64(in2[6])<<0) + |
| 626 uint64(in[6])*(uint64(in2[5])<<0) + |
| 627 uint64(in[7])*(uint64(in2[4])<<0) + |
| 628 uint64(in[8])*(uint64(in2[3])<<0) |
| 629 tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) + |
| 630 uint64(in[5])*(uint64(in2[7])<<1) + |
| 631 uint64(in[6])*(uint64(in2[6])<<0) + |
| 632 uint64(in[7])*(uint64(in2[5])<<1) + |
| 633 uint64(in[8])*(uint64(in2[4])<<0) |
| 634 tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) + |
| 635 uint64(in[6])*(uint64(in2[7])<<0) + |
| 636 uint64(in[7])*(uint64(in2[6])<<0) + |
| 637 uint64(in[8])*(uint64(in2[5])<<0) |
| 638 tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) + |
| 639 uint64(in[7])*(uint64(in2[7])<<1) + |
| 640 uint64(in[8])*(uint64(in2[6])<<0) |
| 641 tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) + |
| 642 uint64(in[8])*(uint64(in2[7])<<0) |
| 643 tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0) |
| 644 |
| 645 p256ReduceDegree(out, tmp) |
| 646 } |
| 647 |
| 648 func p256Assign(out, in *[p256Limbs]uint32) { |
| 649 *out = *in |
| 650 } |
| 651 |
| 652 // p256Invert calculates |out| = |in|^{-1} |
| 653 // |
| 654 // Based on Fermat's Little Theorem: |
| 655 // a^p = a (mod p) |
| 656 // a^{p-1} = 1 (mod p) |
| 657 // a^{p-2} = a^{-1} (mod p) |
| 658 func p256Invert(out, in *[p256Limbs]uint32) { |
| 659 var ftmp, ftmp2 [p256Limbs]uint32 |
| 660 |
| 661 // each e_I will hold |in|^{2^I - 1} |
| 662 var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32 |
| 663 |
| 664 p256Square(&ftmp, in) // 2^1 |
| 665 p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0 |
| 666 p256Assign(&e2, &ftmp) |
| 667 p256Square(&ftmp, &ftmp) // 2^3 - 2^1 |
| 668 p256Square(&ftmp, &ftmp) // 2^4 - 2^2 |
| 669 p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0 |
| 670 p256Assign(&e4, &ftmp) |
| 671 p256Square(&ftmp, &ftmp) // 2^5 - 2^1 |
| 672 p256Square(&ftmp, &ftmp) // 2^6 - 2^2 |
| 673 p256Square(&ftmp, &ftmp) // 2^7 - 2^3 |
| 674 p256Square(&ftmp, &ftmp) // 2^8 - 2^4 |
| 675 p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0 |
| 676 p256Assign(&e8, &ftmp) |
| 677 for i := 0; i < 8; i++ { |
| 678 p256Square(&ftmp, &ftmp) |
| 679 } // 2^16 - 2^8 |
| 680 p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0 |
| 681 p256Assign(&e16, &ftmp) |
| 682 for i := 0; i < 16; i++ { |
| 683 p256Square(&ftmp, &ftmp) |
| 684 } // 2^32 - 2^16 |
| 685 p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0 |
| 686 p256Assign(&e32, &ftmp) |
| 687 for i := 0; i < 32; i++ { |
| 688 p256Square(&ftmp, &ftmp) |
| 689 } // 2^64 - 2^32 |
| 690 p256Assign(&e64, &ftmp) |
| 691 p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0 |
| 692 for i := 0; i < 192; i++ { |
| 693 p256Square(&ftmp, &ftmp) |
| 694 } // 2^256 - 2^224 + 2^192 |
| 695 |
| 696 p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0 |
| 697 for i := 0; i < 16; i++ { |
| 698 p256Square(&ftmp2, &ftmp2) |
| 699 } // 2^80 - 2^16 |
| 700 p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0 |
| 701 for i := 0; i < 8; i++ { |
| 702 p256Square(&ftmp2, &ftmp2) |
| 703 } // 2^88 - 2^8 |
| 704 p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0 |
| 705 for i := 0; i < 4; i++ { |
| 706 p256Square(&ftmp2, &ftmp2) |
| 707 } // 2^92 - 2^4 |
| 708 p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0 |
| 709 p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1 |
| 710 p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2 |
| 711 p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0 |
| 712 p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1 |
| 713 p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2 |
| 714 p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3 |
| 715 |
| 716 p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3 |
| 717 } |
| 718 |
| 719 // p256Scalar3 sets out=3*out. |
| 720 // |
| 721 // On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 722 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 723 func p256Scalar3(out *[p256Limbs]uint32) { |
| 724 var carry uint32 |
| 725 |
| 726 for i := 0; ; i++ { |
| 727 out[i] *= 3 |
| 728 out[i] += carry |
| 729 carry = out[i] >> 29 |
| 730 out[i] &= bottom29Bits |
| 731 |
| 732 i++ |
| 733 if i == p256Limbs { |
| 734 break |
| 735 } |
| 736 |
| 737 out[i] *= 3 |
| 738 out[i] += carry |
| 739 carry = out[i] >> 28 |
| 740 out[i] &= bottom28Bits |
| 741 } |
| 742 |
| 743 p256ReduceCarry(out, carry) |
| 744 } |
| 745 |
| 746 // p256Scalar4 sets out=4*out. |
| 747 // |
| 748 // On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 749 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 750 func p256Scalar4(out *[p256Limbs]uint32) { |
| 751 var carry, nextCarry uint32 |
| 752 |
| 753 for i := 0; ; i++ { |
| 754 nextCarry = out[i] >> 27 |
| 755 out[i] <<= 2 |
| 756 out[i] &= bottom29Bits |
| 757 out[i] += carry |
| 758 carry = nextCarry + (out[i] >> 29) |
| 759 out[i] &= bottom29Bits |
| 760 |
| 761 i++ |
| 762 if i == p256Limbs { |
| 763 break |
| 764 } |
| 765 nextCarry = out[i] >> 26 |
| 766 out[i] <<= 2 |
| 767 out[i] &= bottom28Bits |
| 768 out[i] += carry |
| 769 carry = nextCarry + (out[i] >> 28) |
| 770 out[i] &= bottom28Bits |
| 771 } |
| 772 |
| 773 p256ReduceCarry(out, carry) |
| 774 } |
| 775 |
| 776 // p256Scalar8 sets out=8*out. |
| 777 // |
| 778 // On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 779 // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. |
| 780 func p256Scalar8(out *[p256Limbs]uint32) { |
| 781 var carry, nextCarry uint32 |
| 782 |
| 783 for i := 0; ; i++ { |
| 784 nextCarry = out[i] >> 26 |
| 785 out[i] <<= 3 |
| 786 out[i] &= bottom29Bits |
| 787 out[i] += carry |
| 788 carry = nextCarry + (out[i] >> 29) |
| 789 out[i] &= bottom29Bits |
| 790 |
| 791 i++ |
| 792 if i == p256Limbs { |
| 793 break |
| 794 } |
| 795 nextCarry = out[i] >> 25 |
| 796 out[i] <<= 3 |
| 797 out[i] &= bottom28Bits |
| 798 out[i] += carry |
| 799 carry = nextCarry + (out[i] >> 28) |
| 800 out[i] &= bottom28Bits |
| 801 } |
| 802 |
| 803 p256ReduceCarry(out, carry) |
| 804 } |
| 805 |
| 806 // Group operations: |
| 807 // |
| 808 // Elements of the elliptic curve group are represented in Jacobian |
| 809 // coordinates: (x, y, z). An affine point (x', y') is x'=x/z**2, y'=y/z**3 in |
| 810 // Jacobian form. |
| 811 |
| 812 // p256PointDouble sets {xOut,yOut,zOut} = 2*{x,y,z}. |
| 813 // |
| 814 // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling
-dbl-2009-l |
| 815 func p256PointDouble(xOut, yOut, zOut, x, y, z *[p256Limbs]uint32) { |
| 816 var delta, gamma, alpha, beta, tmp, tmp2 [p256Limbs]uint32 |
| 817 |
| 818 p256Square(&delta, z) |
| 819 p256Square(&gamma, y) |
| 820 p256Mul(&beta, x, &gamma) |
| 821 |
| 822 p256Sum(&tmp, x, &delta) |
| 823 p256Diff(&tmp2, x, &delta) |
| 824 p256Mul(&alpha, &tmp, &tmp2) |
| 825 p256Scalar3(&alpha) |
| 826 |
| 827 p256Sum(&tmp, y, z) |
| 828 p256Square(&tmp, &tmp) |
| 829 p256Diff(&tmp, &tmp, &gamma) |
| 830 p256Diff(zOut, &tmp, &delta) |
| 831 |
| 832 p256Scalar4(&beta) |
| 833 p256Square(xOut, &alpha) |
| 834 p256Diff(xOut, xOut, &beta) |
| 835 p256Diff(xOut, xOut, &beta) |
| 836 |
| 837 p256Diff(&tmp, &beta, xOut) |
| 838 p256Mul(&tmp, &alpha, &tmp) |
| 839 p256Square(&tmp2, &gamma) |
| 840 p256Scalar8(&tmp2) |
| 841 p256Diff(yOut, &tmp, &tmp2) |
| 842 } |
| 843 |
| 844 // p256PointAddMixed sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,1}. |
| 845 // (i.e. the second point is affine.) |
| 846 // |
| 847 // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition
-add-2007-bl |
| 848 // |
| 849 // Note that this function does not handle P+P, infinity+P nor P+infinity |
| 850 // correctly. |
| 851 func p256PointAddMixed(xOut, yOut, zOut, x1, y1, z1, x2, y2 *[p256Limbs]uint32)
{ |
| 852 var z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32 |
| 853 |
| 854 p256Square(&z1z1, z1) |
| 855 p256Sum(&tmp, z1, z1) |
| 856 |
| 857 p256Mul(&u2, x2, &z1z1) |
| 858 p256Mul(&z1z1z1, z1, &z1z1) |
| 859 p256Mul(&s2, y2, &z1z1z1) |
| 860 p256Diff(&h, &u2, x1) |
| 861 p256Sum(&i, &h, &h) |
| 862 p256Square(&i, &i) |
| 863 p256Mul(&j, &h, &i) |
| 864 p256Diff(&r, &s2, y1) |
| 865 p256Sum(&r, &r, &r) |
| 866 p256Mul(&v, x1, &i) |
| 867 |
| 868 p256Mul(zOut, &tmp, &h) |
| 869 p256Square(&rr, &r) |
| 870 p256Diff(xOut, &rr, &j) |
| 871 p256Diff(xOut, xOut, &v) |
| 872 p256Diff(xOut, xOut, &v) |
| 873 |
| 874 p256Diff(&tmp, &v, xOut) |
| 875 p256Mul(yOut, &tmp, &r) |
| 876 p256Mul(&tmp, y1, &j) |
| 877 p256Diff(yOut, yOut, &tmp) |
| 878 p256Diff(yOut, yOut, &tmp) |
| 879 } |
| 880 |
| 881 // p256PointAdd sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,z2}. |
| 882 // |
| 883 // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition
-add-2007-bl |
| 884 // |
| 885 // Note that this function does not handle P+P, infinity+P nor P+infinity |
| 886 // correctly. |
| 887 func p256PointAdd(xOut, yOut, zOut, x1, y1, z1, x2, y2, z2 *[p256Limbs]uint32) { |
| 888 var z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp [
p256Limbs]uint32 |
| 889 |
| 890 p256Square(&z1z1, z1) |
| 891 p256Square(&z2z2, z2) |
| 892 p256Mul(&u1, x1, &z2z2) |
| 893 |
| 894 p256Sum(&tmp, z1, z2) |
| 895 p256Square(&tmp, &tmp) |
| 896 p256Diff(&tmp, &tmp, &z1z1) |
| 897 p256Diff(&tmp, &tmp, &z2z2) |
| 898 |
| 899 p256Mul(&z2z2z2, z2, &z2z2) |
| 900 p256Mul(&s1, y1, &z2z2z2) |
| 901 |
| 902 p256Mul(&u2, x2, &z1z1) |
| 903 p256Mul(&z1z1z1, z1, &z1z1) |
| 904 p256Mul(&s2, y2, &z1z1z1) |
| 905 p256Diff(&h, &u2, &u1) |
| 906 p256Sum(&i, &h, &h) |
| 907 p256Square(&i, &i) |
| 908 p256Mul(&j, &h, &i) |
| 909 p256Diff(&r, &s2, &s1) |
| 910 p256Sum(&r, &r, &r) |
| 911 p256Mul(&v, &u1, &i) |
| 912 |
| 913 p256Mul(zOut, &tmp, &h) |
| 914 p256Square(&rr, &r) |
| 915 p256Diff(xOut, &rr, &j) |
| 916 p256Diff(xOut, xOut, &v) |
| 917 p256Diff(xOut, xOut, &v) |
| 918 |
| 919 p256Diff(&tmp, &v, xOut) |
| 920 p256Mul(yOut, &tmp, &r) |
| 921 p256Mul(&tmp, &s1, &j) |
| 922 p256Diff(yOut, yOut, &tmp) |
| 923 p256Diff(yOut, yOut, &tmp) |
| 924 } |
| 925 |
| 926 // p256CopyConditional sets out=in if mask = 0xffffffff in constant time. |
| 927 // |
| 928 // On entry: mask is either 0 or 0xffffffff. |
| 929 func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) { |
| 930 for i := 0; i < p256Limbs; i++ { |
| 931 tmp := mask & (in[i] ^ out[i]) |
| 932 out[i] ^= tmp |
| 933 } |
| 934 } |
| 935 |
| 936 // p256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table. |
| 937 // On entry: index < 16, table[0] must be zero. |
| 938 func p256SelectAffinePoint(xOut, yOut *[p256Limbs]uint32, table []uint32, index
uint32) { |
| 939 for i := range xOut { |
| 940 xOut[i] = 0 |
| 941 } |
| 942 for i := range yOut { |
| 943 yOut[i] = 0 |
| 944 } |
| 945 |
| 946 for i := uint32(1); i < 16; i++ { |
| 947 mask := i ^ index |
| 948 mask |= mask >> 2 |
| 949 mask |= mask >> 1 |
| 950 mask &= 1 |
| 951 mask-- |
| 952 for j := range xOut { |
| 953 xOut[j] |= table[0] & mask |
| 954 table = table[1:] |
| 955 } |
| 956 for j := range yOut { |
| 957 yOut[j] |= table[0] & mask |
| 958 table = table[1:] |
| 959 } |
| 960 } |
| 961 } |
| 962 |
| 963 // p256SelectJacobianPoint sets {out_x,out_y,out_z} to the index'th entry of |
| 964 // table. |
| 965 // On entry: index < 16, table[0] must be zero. |
| 966 func p256SelectJacobianPoint(xOut, yOut, zOut *[p256Limbs]uint32, table *[16][3]
[p256Limbs]uint32, index uint32) { |
| 967 for i := range xOut { |
| 968 xOut[i] = 0 |
| 969 } |
| 970 for i := range yOut { |
| 971 yOut[i] = 0 |
| 972 } |
| 973 for i := range zOut { |
| 974 zOut[i] = 0 |
| 975 } |
| 976 |
| 977 // The implicit value at index 0 is all zero. We don't need to perform t
hat |
| 978 // iteration of the loop because we already set out_* to zero. |
| 979 for i := uint32(1); i < 16; i++ { |
| 980 mask := i ^ index |
| 981 mask |= mask >> 2 |
| 982 mask |= mask >> 1 |
| 983 mask &= 1 |
| 984 mask-- |
| 985 for j := range xOut { |
| 986 xOut[j] |= table[i][0][j] & mask |
| 987 } |
| 988 for j := range yOut { |
| 989 yOut[j] |= table[i][1][j] & mask |
| 990 } |
| 991 for j := range zOut { |
| 992 zOut[j] |= table[i][2][j] & mask |
| 993 } |
| 994 } |
| 995 } |
| 996 |
| 997 // p256GetBit returns the bit'th bit of scalar. |
| 998 func p256GetBit(scalar *[32]uint8, bit uint) uint32 { |
| 999 return uint32(((scalar[bit>>3]) >> (bit & 7)) & 1) |
| 1000 } |
| 1001 |
| 1002 // p256ScalarBaseMult sets {xOut,yOut,zOut} = scalar*G where scalar is a |
| 1003 // little-endian number. Note that the value of scalar must be less than the |
| 1004 // order of the group. |
| 1005 func p256ScalarBaseMult(xOut, yOut, zOut *[p256Limbs]uint32, scalar *[32]uint8)
{ |
| 1006 nIsInfinityMask := ^uint32(0) |
| 1007 var pIsNoninfiniteMask, mask, tableOffset uint32 |
| 1008 var px, py, tx, ty, tz [p256Limbs]uint32 |
| 1009 |
| 1010 for i := range xOut { |
| 1011 xOut[i] = 0 |
| 1012 } |
| 1013 for i := range yOut { |
| 1014 yOut[i] = 0 |
| 1015 } |
| 1016 for i := range zOut { |
| 1017 zOut[i] = 0 |
| 1018 } |
| 1019 |
| 1020 // The loop adds bits at positions 0, 64, 128 and 192, followed by |
| 1021 // positions 32,96,160 and 224 and does this 32 times. |
| 1022 for i := uint(0); i < 32; i++ { |
| 1023 if i != 0 { |
| 1024 p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) |
| 1025 } |
| 1026 tableOffset = 0 |
| 1027 for j := uint(0); j <= 32; j += 32 { |
| 1028 bit0 := p256GetBit(scalar, 31-i+j) |
| 1029 bit1 := p256GetBit(scalar, 95-i+j) |
| 1030 bit2 := p256GetBit(scalar, 159-i+j) |
| 1031 bit3 := p256GetBit(scalar, 223-i+j) |
| 1032 index := bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3) |
| 1033 |
| 1034 p256SelectAffinePoint(&px, &py, p256Precomputed[tableOff
set:], index) |
| 1035 tableOffset += 30 * p256Limbs |
| 1036 |
| 1037 // Since scalar is less than the order of the group, we
know that |
| 1038 // {xOut,yOut,zOut} != {px,py,1}, unless both are zero,
which we handle |
| 1039 // below. |
| 1040 p256PointAddMixed(&tx, &ty, &tz, xOut, yOut, zOut, &px,
&py) |
| 1041 // The result of pointAddMixed is incorrect if {xOut,yOu
t,zOut} is zero |
| 1042 // (a.k.a. the point at infinity). We handle that situa
tion by |
| 1043 // copying the point from the table. |
| 1044 p256CopyConditional(xOut, &px, nIsInfinityMask) |
| 1045 p256CopyConditional(yOut, &py, nIsInfinityMask) |
| 1046 p256CopyConditional(zOut, &p256One, nIsInfinityMask) |
| 1047 |
| 1048 // Equally, the result is also wrong if the point from t
he table is |
| 1049 // zero, which happens when the index is zero. We handle
that by |
| 1050 // only copying from {tx,ty,tz} to {xOut,yOut,zOut} if i
ndex != 0. |
| 1051 pIsNoninfiniteMask = nonZeroToAllOnes(index) |
| 1052 mask = pIsNoninfiniteMask & ^nIsInfinityMask |
| 1053 p256CopyConditional(xOut, &tx, mask) |
| 1054 p256CopyConditional(yOut, &ty, mask) |
| 1055 p256CopyConditional(zOut, &tz, mask) |
| 1056 // If p was not zero, then n is now non-zero. |
| 1057 nIsInfinityMask &= ^pIsNoninfiniteMask |
| 1058 } |
| 1059 } |
| 1060 } |
| 1061 |
| 1062 // p256PointToAffine converts a Jacobian point to an affine point. If the input |
| 1063 // is the point at infinity then it returns (0, 0) in constant time. |
| 1064 func p256PointToAffine(xOut, yOut, x, y, z *[p256Limbs]uint32) { |
| 1065 var zInv, zInvSq [p256Limbs]uint32 |
| 1066 |
| 1067 p256Invert(&zInv, z) |
| 1068 p256Square(&zInvSq, &zInv) |
| 1069 p256Mul(xOut, x, &zInvSq) |
| 1070 p256Mul(&zInv, &zInv, &zInvSq) |
| 1071 p256Mul(yOut, y, &zInv) |
| 1072 } |
| 1073 |
| 1074 // p256ToAffine returns a pair of *big.Int containing the affine representation |
| 1075 // of {x,y,z}. |
| 1076 func p256ToAffine(x, y, z *[p256Limbs]uint32) (xOut, yOut *big.Int) { |
| 1077 var xx, yy [p256Limbs]uint32 |
| 1078 p256PointToAffine(&xx, &yy, x, y, z) |
| 1079 return p256ToBig(&xx), p256ToBig(&yy) |
| 1080 } |
| 1081 |
| 1082 // p256ScalarMult sets {xOut,yOut,zOut} = scalar*{x,y}. |
| 1083 func p256ScalarMult(xOut, yOut, zOut, x, y *[p256Limbs]uint32, scalar *[32]uint8
) { |
| 1084 var px, py, pz, tx, ty, tz [p256Limbs]uint32 |
| 1085 var precomp [16][3][p256Limbs]uint32 |
| 1086 var nIsInfinityMask, index, pIsNoninfiniteMask, mask uint32 |
| 1087 |
| 1088 // We precompute 0,1,2,... times {x,y}. |
| 1089 precomp[1][0] = *x |
| 1090 precomp[1][1] = *y |
| 1091 precomp[1][2] = p256One |
| 1092 |
| 1093 for i := 2; i < 16; i += 2 { |
| 1094 p256PointDouble(&precomp[i][0], &precomp[i][1], &precomp[i][2],
&precomp[i/2][0], &precomp[i/2][1], &precomp[i/2][2]) |
| 1095 p256PointAddMixed(&precomp[i+1][0], &precomp[i+1][1], &precomp[i
+1][2], &precomp[i][0], &precomp[i][1], &precomp[i][2], x, y) |
| 1096 } |
| 1097 |
| 1098 for i := range xOut { |
| 1099 xOut[i] = 0 |
| 1100 } |
| 1101 for i := range yOut { |
| 1102 yOut[i] = 0 |
| 1103 } |
| 1104 for i := range zOut { |
| 1105 zOut[i] = 0 |
| 1106 } |
| 1107 nIsInfinityMask = ^uint32(0) |
| 1108 |
| 1109 // We add in a window of four bits each iteration and do this 64 times. |
| 1110 for i := 0; i < 64; i++ { |
| 1111 if i != 0 { |
| 1112 p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) |
| 1113 p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) |
| 1114 p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) |
| 1115 p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) |
| 1116 } |
| 1117 |
| 1118 index = uint32(scalar[31-i/2]) |
| 1119 if (i & 1) == 1 { |
| 1120 index &= 15 |
| 1121 } else { |
| 1122 index >>= 4 |
| 1123 } |
| 1124 |
| 1125 // See the comments in scalarBaseMult about handling infinities. |
| 1126 p256SelectJacobianPoint(&px, &py, &pz, &precomp, index) |
| 1127 p256PointAdd(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py, &pz) |
| 1128 p256CopyConditional(xOut, &px, nIsInfinityMask) |
| 1129 p256CopyConditional(yOut, &py, nIsInfinityMask) |
| 1130 p256CopyConditional(zOut, &pz, nIsInfinityMask) |
| 1131 |
| 1132 pIsNoninfiniteMask = nonZeroToAllOnes(index) |
| 1133 mask = pIsNoninfiniteMask & ^nIsInfinityMask |
| 1134 p256CopyConditional(xOut, &tx, mask) |
| 1135 p256CopyConditional(yOut, &ty, mask) |
| 1136 p256CopyConditional(zOut, &tz, mask) |
| 1137 nIsInfinityMask &= ^pIsNoninfiniteMask |
| 1138 } |
| 1139 } |
| 1140 |
| 1141 // p256FromBig sets out = R*in. |
| 1142 func p256FromBig(out *[p256Limbs]uint32, in *big.Int) { |
| 1143 tmp := new(big.Int).Lsh(in, 257) |
| 1144 tmp.Mod(tmp, p256.P) |
| 1145 |
| 1146 for i := 0; i < p256Limbs; i++ { |
| 1147 if bits := tmp.Bits(); len(bits) > 0 { |
| 1148 out[i] = uint32(bits[0]) & bottom29Bits |
| 1149 } else { |
| 1150 out[i] = 0 |
| 1151 } |
| 1152 tmp.Rsh(tmp, 29) |
| 1153 |
| 1154 i++ |
| 1155 if i == p256Limbs { |
| 1156 break |
| 1157 } |
| 1158 |
| 1159 if bits := tmp.Bits(); len(bits) > 0 { |
| 1160 out[i] = uint32(bits[0]) & bottom28Bits |
| 1161 } else { |
| 1162 out[i] = 0 |
| 1163 } |
| 1164 tmp.Rsh(tmp, 28) |
| 1165 } |
| 1166 } |
| 1167 |
| 1168 // p256ToBig returns a *big.Int containing the value of in. |
| 1169 func p256ToBig(in *[p256Limbs]uint32) *big.Int { |
| 1170 result, tmp := new(big.Int), new(big.Int) |
| 1171 |
| 1172 result.SetInt64(int64(in[p256Limbs-1])) |
| 1173 for i := p256Limbs - 2; i >= 0; i-- { |
| 1174 if (i & 1) == 0 { |
| 1175 result.Lsh(result, 29) |
| 1176 } else { |
| 1177 result.Lsh(result, 28) |
| 1178 } |
| 1179 tmp.SetInt64(int64(in[i])) |
| 1180 result.Add(result, tmp) |
| 1181 } |
| 1182 |
| 1183 result.Mul(result, p256RInverse) |
| 1184 result.Mod(result, p256.P) |
| 1185 return result |
| 1186 } |
LEFT | RIGHT |