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| 1 /* This Source Code Form is subject to the terms of the Mozilla Public | |
| 2 * License, v. 2.0. If a copy of the MPL was not distributed with this | |
| 3 * file, You can obtain one at http://mozilla.org/MPL/2.0/. */ | |
| 4 | |
| 5 /* A 32-bit implementation of the NIST P-256 elliptic curve. */ | |
| 6 | |
| 7 #include <string.h> | |
| 8 | |
| 9 #include "prtypes.h" | |
| 10 #include "mpi.h" | |
| 11 #include "mpi-priv.h" | |
| 12 #include "ecp.h" | |
| 13 | |
| 14 typedef PRUint8 u8; | |
| 15 typedef PRUint32 u32; | |
| 16 typedef PRUint64 u64; | |
| 17 | |
| 18 /* Our field elements are represented as nine, unsigned 32-bit words. Freebl's | |
| 19 * MPI library calls them digits, but here they are called limbs, which is | |
| 20 * GMP's terminology. | |
| 21 * | |
| 22 * The value of an felem (field element) is: | |
| 23 * x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228) | |
| 24 * | |
| 25 * That is, each limb is alternately 29 or 28-bits wide in little-endian | |
| 26 * order. | |
| 27 * | |
| 28 * This means that an felem hits 2**257, rather than 2**256 as we would like. A | |
| 29 * 28, 29, ... pattern would cause us to hit 2**256, but that causes problems | |
| 30 * when multiplying as terms end up one bit short of a limb which would require | |
| 31 * much bit-shifting to correct. | |
| 32 * | |
| 33 * Finally, the values stored in an felem are in Montgomery form. So the value | |
| 34 * |y| is stored as (y*R) mod p, where p is the P-256 prime and R is 2**257. | |
| 35 */ | |
| 36 typedef u32 limb; | |
| 37 #define NLIMBS 9 | |
| 38 typedef limb felem[NLIMBS]; | |
| 39 | |
| 40 static const limb kBottom28Bits = 0xfffffff; | |
| 41 static const limb kBottom29Bits = 0x1fffffff; | |
| 42 | |
| 43 /* kOne is the number 1 as an felem. It's 2**257 mod p split up into 29 and | |
| 44 * 28-bit words. | |
| 45 */ | |
| 46 static const felem kOne = { | |
| 47 2, 0, 0, 0xffff800, | |
| 48 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, | |
| 49 0 | |
| 50 }; | |
| 51 static const felem kZero = {0}; | |
| 52 static const felem kP = { | |
| 53 0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, | |
| 54 0, 0, 0x200000, 0xf000000, | |
| 55 0xfffffff | |
| 56 }; | |
| 57 static const felem k2P = { | |
| 58 0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, | |
| 59 0, 0, 0x400000, 0xe000000, | |
| 60 0x1fffffff | |
| 61 }; | |
| 62 | |
| 63 /* kPrecomputed contains precomputed values to aid the calculation of scalar | |
| 64 * multiples of the base point, G. It's actually two, equal length, tables | |
| 65 * concatenated. | |
| 66 * | |
| 67 * The first table contains (x,y) felem pairs for 16 multiples of the base | |
| 68 * point, G. | |
| 69 * | |
| 70 * Index | Index (binary) | Value | |
| 71 * 0 | 0000 | 0G (all zeros, omitted) | |
| 72 * 1 | 0001 | G | |
| 73 * 2 | 0010 | 2**64G | |
| 74 * 3 | 0011 | 2**64G + G | |
| 75 * 4 | 0100 | 2**128G | |
| 76 * 5 | 0101 | 2**128G + G | |
| 77 * 6 | 0110 | 2**128G + 2**64G | |
| 78 * 7 | 0111 | 2**128G + 2**64G + G | |
| 79 * 8 | 1000 | 2**192G | |
| 80 * 9 | 1001 | 2**192G + G | |
| 81 * 10 | 1010 | 2**192G + 2**64G | |
| 82 * 11 | 1011 | 2**192G + 2**64G + G | |
| 83 * 12 | 1100 | 2**192G + 2**128G | |
| 84 * 13 | 1101 | 2**192G + 2**128G + G | |
| 85 * 14 | 1110 | 2**192G + 2**128G + 2**64G | |
| 86 * 15 | 1111 | 2**192G + 2**128G + 2**64G + G | |
| 87 * | |
| 88 * The second table follows the same style, but the terms are 2**32G, | |
| 89 * 2**96G, 2**160G, 2**224G. | |
| 90 * | |
| 91 * This is ~2KB of data. | |
| 92 */ | |
| 93 static const limb kPrecomputed[NLIMBS * 2 * 15 * 2] = { | |
| 94 0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7e
dc, 0xd4a6eab, 0x3120bee, | |
| 95 0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba
21, 0x14b10bb, 0xae3fe3, | |
| 96 0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe4907
3, 0x3fa36cc, 0x5ebcd2c, | |
| 97 0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea1244
6, 0xe1ade1e, 0xec91f22, | |
| 98 0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109
, 0xa267a00, 0xb57c050, | |
| 99 0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0
x7d6dee7, 0x2976e4b, | |
| 100 0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a
5a9, 0x843a649, 0xc3ab0fa, | |
| 101 0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11
, 0x58c43df, 0xf423fc2, | |
| 102 0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db4
0f, 0x83e277d, 0xb0dd609, | |
| 103 0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5
, 0xe10c9e, 0x33ab581, | |
| 104 0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f
, 0x48764cd, 0x76dbcca, | |
| 105 0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b2
0, 0x4ba3173, 0xc168c33, | |
| 106 0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0
, 0x65dd7ff, 0x3a1e4f6, | |
| 107 0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f07
7, 0xa6add89, 0x4894acd, | |
| 108 0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a,
0x69a8556, 0x7e7c0, | |
| 109 0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c
, 0xda0cf5b, 0x812e881, | |
| 110 0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51
, 0xc22be3e, 0xe35e65a, | |
| 111 0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9
, 0x1c5a839, 0x47a1e26, | |
| 112 0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c5
02, 0x2f32042, 0xa17769b, | |
| 113 0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a
02, 0x3fc93, 0x5620023, | |
| 114 0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c
, 0x407f75c, 0xbaab133, | |
| 115 0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea
7, 0x3293ac0, 0xcdc98aa, | |
| 116 0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16,
0x2b6fcc7, 0xf5a4e29, | |
| 117 0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72
, 0x73e1c35, 0xee70fbc, | |
| 118 0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85
, 0x27de188, 0x66f70b8, | |
| 119 0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae
914, 0x2f3ec51, 0x3826b59, | |
| 120 0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x
823d9d2, 0x8213f39, | |
| 121 0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4
a, 0xf5ddc3d, 0x3786689, | |
| 122 0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a72
9, 0x4be3499, 0x52b23aa, | |
| 123 0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb0480
35, 0xe31de66, 0xc6ecaa3, | |
| 124 0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a752
9, 0xcb7beb1, 0xb2a78a1, | |
| 125 0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff6
58, 0xe3d6511, 0xc7d76f, | |
| 126 0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c,
0x50daa90, 0xb13f72, | |
| 127 0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d3241
1, 0xb04a838, 0xd760d2d, | |
| 128 0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11
e, 0x20bca9a, 0x66f496b, | |
| 129 0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d,
0xbe985f7, 0x1acbc1a, | |
| 130 0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56
ff, 0x65ef930, 0x21dc4a, | |
| 131 0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15
f, 0x624e62e, 0xa90ae2f, | |
| 132 0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522
b, 0xdc78583, 0x40eeabb, | |
| 133 0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef3
4, 0xae2a960, 0x91b8bdc, | |
| 134 0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0
x2413c8e, 0x5425bf9, | |
| 135 0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633
, 0x7c91952, 0xd806dce, | |
| 136 0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef7
3, 0x8956f34, 0xe4b5cf2, | |
| 137 0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7
, 0x627b614, 0x7371cca, | |
| 138 0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc
9, 0x9c19bf2, 0x5882229, | |
| 139 0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b
3, 0xe85ff25, 0x408ef57, | |
| 140 0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113
, 0xa4a1769, 0x11fbc6c, | |
| 141 0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b
7, 0x4acbad9, 0x5efc5fa, | |
| 142 0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc
, 0x7bf0fa9, 0x957651, | |
| 143 0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57,
0xf2ecaac, 0xca86dec, | |
| 144 0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c1
2d, 0xf20bd46, 0x1951fa7, | |
| 145 0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74
, 0x99bb618, 0x2db944c, | |
| 146 0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e7477
9, 0x576138, 0x9587927, | |
| 147 0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782
d, 0xfc72e0b, 0x701b298, | |
| 148 0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5
d8, 0xf858d3a, 0x942eea8, | |
| 149 0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a
1, 0x8395659, 0x52ed4e2, | |
| 150 0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c
0, 0x6bdf55a, 0x4e4457d, | |
| 151 0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747
b, 0x878558d, 0x7d29aa4, | |
| 152 0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d
7, 0xa5bef68, 0xb7b30d8, | |
| 153 0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f5195
1, 0x9d0c177, 0x1c49a78, | |
| 154 }; | |
| 155 | |
| 156 /* Field element operations: | |
| 157 */ | |
| 158 | |
| 159 /* NON_ZERO_TO_ALL_ONES returns: | |
| 160 * 0xffffffff for 0 < x <= 2**31 | |
| 161 * 0 for x == 0 or x > 2**31. | |
| 162 * | |
| 163 * x must be a u32 or an equivalent type such as limb. | |
| 164 */ | |
| 165 #define NON_ZERO_TO_ALL_ONES(x) ((((u32)(x) - 1) >> 31) - 1) | |
| 166 | |
| 167 /* felem_reduce_carry adds a multiple of p in order to cancel |carry|, | |
| 168 * which is a term at 2**257. | |
| 169 * | |
| 170 * On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28. | |
| 171 * On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29. | |
| 172 */ | |
| 173 static void felem_reduce_carry(felem inout, limb carry) | |
| 174 { | |
| 175 const u32 carry_mask = NON_ZERO_TO_ALL_ONES(carry); | |
| 176 | |
| 177 inout[0] += carry << 1; | |
| 178 inout[3] += 0x10000000 & carry_mask; | |
| 179 /* carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the | |
| 180 * previous line therefore this doesn't underflow. | |
| 181 */ | |
| 182 inout[3] -= carry << 11; | |
| 183 inout[4] += (0x20000000 - 1) & carry_mask; | |
| 184 inout[5] += (0x10000000 - 1) & carry_mask; | |
| 185 inout[6] += (0x20000000 - 1) & carry_mask; | |
| 186 inout[6] -= carry << 22; | |
| 187 /* This may underflow if carry is non-zero but, if so, we'll fix it in the | |
| 188 * next line. | |
| 189 */ | |
| 190 inout[7] -= 1 & carry_mask; | |
| 191 inout[7] += carry << 25; | |
| 192 } | |
| 193 | |
| 194 /* felem_sum sets out = in+in2. | |
| 195 * | |
| 196 * On entry, in[i]+in2[i] must not overflow a 32-bit word. | |
| 197 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 | |
| 198 */ | |
| 199 static void felem_sum(felem out, const felem in, const felem in2) | |
| 200 { | |
| 201 limb carry = 0; | |
| 202 unsigned int i; | |
| 203 for (i = 0;; i++) { | |
| 204 out[i] = in[i] + in2[i]; | |
| 205 out[i] += carry; | |
| 206 carry = out[i] >> 29; | |
| 207 out[i] &= kBottom29Bits; | |
| 208 | |
| 209 i++; | |
| 210 if (i == NLIMBS) | |
| 211 break; | |
| 212 | |
| 213 out[i] = in[i] + in2[i]; | |
| 214 out[i] += carry; | |
| 215 carry = out[i] >> 28; | |
| 216 out[i] &= kBottom28Bits; | |
| 217 } | |
| 218 | |
| 219 felem_reduce_carry(out, carry); | |
| 220 } | |
| 221 | |
| 222 #define two31m3 (((limb)1) << 31) - (((limb)1) << 3) | |
| 223 #define two30m2 (((limb)1) << 30) - (((limb)1) << 2) | |
| 224 #define two30p13m2 (((limb)1) << 30) + (((limb)1) << 13) - (((limb)1) << 2) | |
| 225 #define two31m2 (((limb)1) << 31) - (((limb)1) << 2) | |
| 226 #define two31p24m2 (((limb)1) << 31) + (((limb)1) << 24) - (((limb)1) << 2) | |
| 227 #define two30m27m2 (((limb)1) << 30) - (((limb)1) << 27) - (((limb)1) << 2) | |
| 228 | |
| 229 /* zero31 is 0 mod p. | |
| 230 */ | |
| 231 static const felem zero31 = { | |
| 232 two31m3, two30m2, two31m2, two30p13m2, | |
| 233 two31m2, two30m2, two31p24m2, two30m27m2, | |
| 234 two31m2 | |
| 235 }; | |
| 236 | |
| 237 /* felem_diff sets out = in-in2. | |
| 238 * | |
| 239 * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and | |
| 240 * in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. | |
| 241 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 242 */ | |
| 243 static void felem_diff(felem out, const felem in, const felem in2) | |
| 244 { | |
| 245 limb carry = 0; | |
| 246 unsigned int i; | |
| 247 | |
| 248 for (i = 0;; i++) { | |
| 249 out[i] = in[i] - in2[i]; | |
| 250 out[i] += zero31[i]; | |
| 251 out[i] += carry; | |
| 252 carry = out[i] >> 29; | |
| 253 out[i] &= kBottom29Bits; | |
| 254 | |
| 255 i++; | |
| 256 if (i == NLIMBS) | |
| 257 break; | |
| 258 | |
| 259 out[i] = in[i] - in2[i]; | |
| 260 out[i] += zero31[i]; | |
| 261 out[i] += carry; | |
| 262 carry = out[i] >> 28; | |
| 263 out[i] &= kBottom28Bits; | |
| 264 } | |
| 265 | |
| 266 felem_reduce_carry(out, carry); | |
| 267 } | |
| 268 | |
| 269 /* felem_reduce_degree sets out = tmp/R mod p where tmp contains 64-bit words | |
| 270 * with the same 29,28,... bit positions as an felem. | |
| 271 * | |
| 272 * The values in felems are in Montgomery form: x*R mod p where R = 2**257. | |
| 273 * Since we just multiplied two Montgomery values together, the result is | |
| 274 * x*y*R*R mod p. We wish to divide by R in order for the result also to be | |
| 275 * in Montgomery form. | |
| 276 * | |
| 277 * On entry: tmp[i] < 2**64 | |
| 278 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 | |
| 279 */ | |
| 280 static void felem_reduce_degree(felem out, u64 tmp[17]) | |
| 281 { | |
| 282 /* The following table may be helpful when reading this code: | |
| 283 * | |
| 284 * Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10... | |
| 285 * Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29 | |
| 286 * Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285 | |
| 287 * (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285 | |
| 288 */ | |
| 289 limb tmp2[18], carry, x, xMask; | |
| 290 unsigned int i; | |
| 291 | |
| 292 /* tmp contains 64-bit words with the same 29,28,29-bit positions as an | |
| 293 * felem. So the top of an element of tmp might overlap with another | |
| 294 * element two positions down. The following loop eliminates this | |
| 295 * overlap. | |
| 296 */ | |
| 297 tmp2[0] = tmp[0] & kBottom29Bits; | |
| 298 | |
| 299 /* In the following we use "(limb) tmp[x]" and "(limb) (tmp[x]>>32)" to try | |
| 300 * and hint to the compiler that it can do a single-word shift by selecting | |
| 301 * the right register rather than doing a double-word shift and truncating | |
| 302 * afterwards. | |
| 303 */ | |
| 304 tmp2[1] = ((limb) tmp[0]) >> 29; | |
| 305 tmp2[1] |= (((limb) (tmp[0] >> 32)) << 3) & kBottom28Bits; | |
| 306 tmp2[1] += ((limb) tmp[1]) & kBottom28Bits; | |
| 307 carry = tmp2[1] >> 28; | |
| 308 tmp2[1] &= kBottom28Bits; | |
| 309 | |
| 310 for (i = 2; i < 17; i++) { | |
| 311 tmp2[i] = ((limb) (tmp[i - 2] >> 32)) >> 25; | |
| 312 tmp2[i] += ((limb) (tmp[i - 1])) >> 28; | |
| 313 tmp2[i] += (((limb) (tmp[i - 1] >> 32)) << 4) & kBottom29Bits; | |
| 314 tmp2[i] += ((limb) tmp[i]) & kBottom29Bits; | |
| 315 tmp2[i] += carry; | |
| 316 carry = tmp2[i] >> 29; | |
| 317 tmp2[i] &= kBottom29Bits; | |
| 318 | |
| 319 i++; | |
| 320 if (i == 17) | |
| 321 break; | |
| 322 tmp2[i] = ((limb) (tmp[i - 2] >> 32)) >> 25; | |
| 323 tmp2[i] += ((limb) (tmp[i - 1])) >> 29; | |
| 324 tmp2[i] += (((limb) (tmp[i - 1] >> 32)) << 3) & kBottom28Bits; | |
| 325 tmp2[i] += ((limb) tmp[i]) & kBottom28Bits; | |
| 326 tmp2[i] += carry; | |
| 327 carry = tmp2[i] >> 28; | |
| 328 tmp2[i] &= kBottom28Bits; | |
| 329 } | |
| 330 | |
| 331 tmp2[17] = ((limb) (tmp[15] >> 32)) >> 25; | |
| 332 tmp2[17] += ((limb) (tmp[16])) >> 29; | |
| 333 tmp2[17] += (((limb) (tmp[16] >> 32)) << 3); | |
| 334 tmp2[17] += carry; | |
| 335 | |
| 336 /* Montgomery elimination of terms: | |
| 337 * | |
| 338 * Since R is 2**257, we can divide by R with a bitwise shift if we can | |
| 339 * ensure that the right-most 257 bits are all zero. We can make that true | |
| 340 * by adding multiplies of p without affecting the value. | |
| 341 * | |
| 342 * So we eliminate limbs from right to left. Since the bottom 29 bits of p | |
| 343 * are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0. | |
| 344 * We can do that for 8 further limbs and then right shift to eliminate the | |
| 345 * extra factor of R. | |
| 346 */ | |
| 347 for (i = 0;; i += 2) { | |
| 348 tmp2[i + 1] += tmp2[i] >> 29; | |
| 349 x = tmp2[i] & kBottom29Bits; | |
| 350 xMask = NON_ZERO_TO_ALL_ONES(x); | |
| 351 tmp2[i] = 0; | |
| 352 | |
| 353 /* The bounds calculations for this loop are tricky. Each iteration of | |
| 354 * the loop eliminates two words by adding values to words to their | |
| 355 * right. | |
| 356 * | |
| 357 * The following table contains the amounts added to each word (as an | |
| 358 * offset from the value of i at the top of the loop). The amounts are | |
| 359 * accounted for from the first and second half of the loop separately | |
| 360 * and are written as, for example, 28 to mean a value <2**28. | |
| 361 * | |
| 362 * Word: 3 4 5 6 7 8 9 10 | |
| 363 * Added in top half: 28 11 29 21 29 28 | |
| 364 * 28 29 | |
| 365 * 29 | |
| 366 * Added in bottom half: 29 10 28 21 28 28 | |
| 367 * 29 | |
| 368 * | |
| 369 * The value that is currently offset 7 will be offset 5 for the next | |
| 370 * iteration and then offset 3 for the iteration after that. Therefore | |
| 371 * the total value added will be the values added at 7, 5 and 3. | |
| 372 * | |
| 373 * The following table accumulates these values. The sums at the bottom | |
| 374 * are written as, for example, 29+28, to mean a value < 2**29+2**28. | |
| 375 * | |
| 376 * Word: 3 4 5 6 7 8 9 10 11 12 13 | |
| 377 * 28 11 10 29 21 29 28 28 28 28 28 | |
| 378 * 29 28 11 28 29 28 29 28 29 28 | |
| 379 * 29 28 21 21 29 21 29 21 | |
| 380 * 10 29 28 21 28 21 28 | |
| 381 * 28 29 28 29 28 29 28 | |
| 382 * 11 10 29 10 29 10 | |
| 383 * 29 28 11 28 11 | |
| 384 * 29 29 | |
| 385 * -------------------------------------------- | |
| 386 * 30+ 31+ 30+ 31+ 30+ | |
| 387 * 28+ 29+ 28+ 29+ 21+ | |
| 388 * 21+ 28+ 21+ 28+ 10 | |
| 389 * 10 21+ 10 21+ | |
| 390 * 11 11 | |
| 391 * | |
| 392 * So the greatest amount is added to tmp2[10] and tmp2[12]. If | |
| 393 * tmp2[10/12] has an initial value of <2**29, then the maximum value | |
| 394 * will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32, | |
| 395 * as required. | |
| 396 */ | |
| 397 tmp2[i + 3] += (x << 10) & kBottom28Bits; | |
| 398 tmp2[i + 4] += (x >> 18); | |
| 399 | |
| 400 tmp2[i + 6] += (x << 21) & kBottom29Bits; | |
| 401 tmp2[i + 7] += x >> 8; | |
| 402 | |
| 403 /* At position 200, which is the starting bit position for word 7, we | |
| 404 * have a factor of 0xf000000 = 2**28 - 2**24. | |
| 405 */ | |
| 406 tmp2[i + 7] += 0x10000000 & xMask; | |
| 407 /* Word 7 is 28 bits wide, so the 2**28 term exactly hits word 8. */ | |
| 408 tmp2[i + 8] += (x - 1) & xMask; | |
| 409 tmp2[i + 7] -= (x << 24) & kBottom28Bits; | |
| 410 tmp2[i + 8] -= x >> 4; | |
| 411 | |
| 412 tmp2[i + 8] += 0x20000000 & xMask; | |
| 413 tmp2[i + 8] -= x; | |
| 414 tmp2[i + 8] += (x << 28) & kBottom29Bits; | |
| 415 tmp2[i + 9] += ((x >> 1) - 1) & xMask; | |
| 416 | |
| 417 if (i+1 == NLIMBS) | |
| 418 break; | |
| 419 tmp2[i + 2] += tmp2[i + 1] >> 28; | |
| 420 x = tmp2[i + 1] & kBottom28Bits; | |
| 421 xMask = NON_ZERO_TO_ALL_ONES(x); | |
| 422 tmp2[i + 1] = 0; | |
| 423 | |
| 424 tmp2[i + 4] += (x << 11) & kBottom29Bits; | |
| 425 tmp2[i + 5] += (x >> 18); | |
| 426 | |
| 427 tmp2[i + 7] += (x << 21) & kBottom28Bits; | |
| 428 tmp2[i + 8] += x >> 7; | |
| 429 | |
| 430 /* At position 199, which is the starting bit of the 8th word when | |
| 431 * dealing with a context starting on an odd word, we have a factor of | |
| 432 * 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th | |
| 433 * word from i+1 is i+8. | |
| 434 */ | |
| 435 tmp2[i + 8] += 0x20000000 & xMask; | |
| 436 tmp2[i + 9] += (x - 1) & xMask; | |
| 437 tmp2[i + 8] -= (x << 25) & kBottom29Bits; | |
| 438 tmp2[i + 9] -= x >> 4; | |
| 439 | |
| 440 tmp2[i + 9] += 0x10000000 & xMask; | |
| 441 tmp2[i + 9] -= x; | |
| 442 tmp2[i + 10] += (x - 1) & xMask; | |
| 443 } | |
| 444 | |
| 445 /* We merge the right shift with a carry chain. The words above 2**257 have | |
| 446 * widths of 28,29,... which we need to correct when copying them down. | |
| 447 */ | |
| 448 carry = 0; | |
| 449 for (i = 0; i < 8; i++) { | |
| 450 /* The maximum value of tmp2[i + 9] occurs on the first iteration and | |
| 451 * is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is | |
| 452 * therefore safe. | |
| 453 */ | |
| 454 out[i] = tmp2[i + 9]; | |
| 455 out[i] += carry; | |
| 456 out[i] += (tmp2[i + 10] << 28) & kBottom29Bits; | |
| 457 carry = out[i] >> 29; | |
| 458 out[i] &= kBottom29Bits; | |
| 459 | |
| 460 i++; | |
| 461 out[i] = tmp2[i + 9] >> 1; | |
| 462 out[i] += carry; | |
| 463 carry = out[i] >> 28; | |
| 464 out[i] &= kBottom28Bits; | |
| 465 } | |
| 466 | |
| 467 out[8] = tmp2[17]; | |
| 468 out[8] += carry; | |
| 469 carry = out[8] >> 29; | |
| 470 out[8] &= kBottom29Bits; | |
| 471 | |
| 472 felem_reduce_carry(out, carry); | |
| 473 } | |
| 474 | |
| 475 /* felem_square sets out=in*in. | |
| 476 * | |
| 477 * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29. | |
| 478 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 479 */ | |
| 480 static void felem_square(felem out, const felem in) | |
| 481 { | |
| 482 u64 tmp[17]; | |
| 483 | |
| 484 tmp[0] = ((u64) in[0]) * in[0]; | |
| 485 tmp[1] = ((u64) in[0]) * (in[1] << 1); | |
| 486 tmp[2] = ((u64) in[0]) * (in[2] << 1) + | |
| 487 ((u64) in[1]) * (in[1] << 1); | |
| 488 tmp[3] = ((u64) in[0]) * (in[3] << 1) + | |
| 489 ((u64) in[1]) * (in[2] << 1); | |
| 490 tmp[4] = ((u64) in[0]) * (in[4] << 1) + | |
| 491 ((u64) in[1]) * (in[3] << 2) + | |
| 492 ((u64) in[2]) * in[2]; | |
| 493 tmp[5] = ((u64) in[0]) * (in[5] << 1) + | |
| 494 ((u64) in[1]) * (in[4] << 1) + | |
| 495 ((u64) in[2]) * (in[3] << 1); | |
| 496 tmp[6] = ((u64) in[0]) * (in[6] << 1) + | |
| 497 ((u64) in[1]) * (in[5] << 2) + | |
| 498 ((u64) in[2]) * (in[4] << 1) + | |
| 499 ((u64) in[3]) * (in[3] << 1); | |
| 500 tmp[7] = ((u64) in[0]) * (in[7] << 1) + | |
| 501 ((u64) in[1]) * (in[6] << 1) + | |
| 502 ((u64) in[2]) * (in[5] << 1) + | |
| 503 ((u64) in[3]) * (in[4] << 1); | |
| 504 /* tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60, | |
| 505 * which is < 2**64 as required. | |
| 506 */ | |
| 507 tmp[8] = ((u64) in[0]) * (in[8] << 1) + | |
| 508 ((u64) in[1]) * (in[7] << 2) + | |
| 509 ((u64) in[2]) * (in[6] << 1) + | |
| 510 ((u64) in[3]) * (in[5] << 2) + | |
| 511 ((u64) in[4]) * in[4]; | |
| 512 tmp[9] = ((u64) in[1]) * (in[8] << 1) + | |
| 513 ((u64) in[2]) * (in[7] << 1) + | |
| 514 ((u64) in[3]) * (in[6] << 1) + | |
| 515 ((u64) in[4]) * (in[5] << 1); | |
| 516 tmp[10] = ((u64) in[2]) * (in[8] << 1) + | |
| 517 ((u64) in[3]) * (in[7] << 2) + | |
| 518 ((u64) in[4]) * (in[6] << 1) + | |
| 519 ((u64) in[5]) * (in[5] << 1); | |
| 520 tmp[11] = ((u64) in[3]) * (in[8] << 1) + | |
| 521 ((u64) in[4]) * (in[7] << 1) + | |
| 522 ((u64) in[5]) * (in[6] << 1); | |
| 523 tmp[12] = ((u64) in[4]) * (in[8] << 1) + | |
| 524 ((u64) in[5]) * (in[7] << 2) + | |
| 525 ((u64) in[6]) * in[6]; | |
| 526 tmp[13] = ((u64) in[5]) * (in[8] << 1) + | |
| 527 ((u64) in[6]) * (in[7] << 1); | |
| 528 tmp[14] = ((u64) in[6]) * (in[8] << 1) + | |
| 529 ((u64) in[7]) * (in[7] << 1); | |
| 530 tmp[15] = ((u64) in[7]) * (in[8] << 1); | |
| 531 tmp[16] = ((u64) in[8]) * in[8]; | |
| 532 | |
| 533 felem_reduce_degree(out, tmp); | |
| 534 } | |
| 535 | |
| 536 /* felem_mul sets out=in*in2. | |
| 537 * | |
| 538 * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and | |
| 539 * in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. | |
| 540 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 541 */ | |
| 542 static void felem_mul(felem out, const felem in, const felem in2) | |
| 543 { | |
| 544 u64 tmp[17]; | |
| 545 | |
| 546 tmp[0] = ((u64) in[0]) * in2[0]; | |
| 547 tmp[1] = ((u64) in[0]) * (in2[1] << 0) + | |
| 548 ((u64) in[1]) * (in2[0] << 0); | |
| 549 tmp[2] = ((u64) in[0]) * (in2[2] << 0) + | |
| 550 ((u64) in[1]) * (in2[1] << 1) + | |
| 551 ((u64) in[2]) * (in2[0] << 0); | |
| 552 tmp[3] = ((u64) in[0]) * (in2[3] << 0) + | |
| 553 ((u64) in[1]) * (in2[2] << 0) + | |
| 554 ((u64) in[2]) * (in2[1] << 0) + | |
| 555 ((u64) in[3]) * (in2[0] << 0); | |
| 556 tmp[4] = ((u64) in[0]) * (in2[4] << 0) + | |
| 557 ((u64) in[1]) * (in2[3] << 1) + | |
| 558 ((u64) in[2]) * (in2[2] << 0) + | |
| 559 ((u64) in[3]) * (in2[1] << 1) + | |
| 560 ((u64) in[4]) * (in2[0] << 0); | |
| 561 tmp[5] = ((u64) in[0]) * (in2[5] << 0) + | |
| 562 ((u64) in[1]) * (in2[4] << 0) + | |
| 563 ((u64) in[2]) * (in2[3] << 0) + | |
| 564 ((u64) in[3]) * (in2[2] << 0) + | |
| 565 ((u64) in[4]) * (in2[1] << 0) + | |
| 566 ((u64) in[5]) * (in2[0] << 0); | |
| 567 tmp[6] = ((u64) in[0]) * (in2[6] << 0) + | |
| 568 ((u64) in[1]) * (in2[5] << 1) + | |
| 569 ((u64) in[2]) * (in2[4] << 0) + | |
| 570 ((u64) in[3]) * (in2[3] << 1) + | |
| 571 ((u64) in[4]) * (in2[2] << 0) + | |
| 572 ((u64) in[5]) * (in2[1] << 1) + | |
| 573 ((u64) in[6]) * (in2[0] << 0); | |
| 574 tmp[7] = ((u64) in[0]) * (in2[7] << 0) + | |
| 575 ((u64) in[1]) * (in2[6] << 0) + | |
| 576 ((u64) in[2]) * (in2[5] << 0) + | |
| 577 ((u64) in[3]) * (in2[4] << 0) + | |
| 578 ((u64) in[4]) * (in2[3] << 0) + | |
| 579 ((u64) in[5]) * (in2[2] << 0) + | |
| 580 ((u64) in[6]) * (in2[1] << 0) + | |
| 581 ((u64) in[7]) * (in2[0] << 0); | |
| 582 /* tmp[8] has the greatest value but doesn't overflow. See logic in | |
| 583 * felem_square. | |
| 584 */ | |
| 585 tmp[8] = ((u64) in[0]) * (in2[8] << 0) + | |
| 586 ((u64) in[1]) * (in2[7] << 1) + | |
| 587 ((u64) in[2]) * (in2[6] << 0) + | |
| 588 ((u64) in[3]) * (in2[5] << 1) + | |
| 589 ((u64) in[4]) * (in2[4] << 0) + | |
| 590 ((u64) in[5]) * (in2[3] << 1) + | |
| 591 ((u64) in[6]) * (in2[2] << 0) + | |
| 592 ((u64) in[7]) * (in2[1] << 1) + | |
| 593 ((u64) in[8]) * (in2[0] << 0); | |
| 594 tmp[9] = ((u64) in[1]) * (in2[8] << 0) + | |
| 595 ((u64) in[2]) * (in2[7] << 0) + | |
| 596 ((u64) in[3]) * (in2[6] << 0) + | |
| 597 ((u64) in[4]) * (in2[5] << 0) + | |
| 598 ((u64) in[5]) * (in2[4] << 0) + | |
| 599 ((u64) in[6]) * (in2[3] << 0) + | |
| 600 ((u64) in[7]) * (in2[2] << 0) + | |
| 601 ((u64) in[8]) * (in2[1] << 0); | |
| 602 tmp[10] = ((u64) in[2]) * (in2[8] << 0) + | |
| 603 ((u64) in[3]) * (in2[7] << 1) + | |
| 604 ((u64) in[4]) * (in2[6] << 0) + | |
| 605 ((u64) in[5]) * (in2[5] << 1) + | |
| 606 ((u64) in[6]) * (in2[4] << 0) + | |
| 607 ((u64) in[7]) * (in2[3] << 1) + | |
| 608 ((u64) in[8]) * (in2[2] << 0); | |
| 609 tmp[11] = ((u64) in[3]) * (in2[8] << 0) + | |
| 610 ((u64) in[4]) * (in2[7] << 0) + | |
| 611 ((u64) in[5]) * (in2[6] << 0) + | |
| 612 ((u64) in[6]) * (in2[5] << 0) + | |
| 613 ((u64) in[7]) * (in2[4] << 0) + | |
| 614 ((u64) in[8]) * (in2[3] << 0); | |
| 615 tmp[12] = ((u64) in[4]) * (in2[8] << 0) + | |
| 616 ((u64) in[5]) * (in2[7] << 1) + | |
| 617 ((u64) in[6]) * (in2[6] << 0) + | |
| 618 ((u64) in[7]) * (in2[5] << 1) + | |
| 619 ((u64) in[8]) * (in2[4] << 0); | |
| 620 tmp[13] = ((u64) in[5]) * (in2[8] << 0) + | |
| 621 ((u64) in[6]) * (in2[7] << 0) + | |
| 622 ((u64) in[7]) * (in2[6] << 0) + | |
| 623 ((u64) in[8]) * (in2[5] << 0); | |
| 624 tmp[14] = ((u64) in[6]) * (in2[8] << 0) + | |
| 625 ((u64) in[7]) * (in2[7] << 1) + | |
| 626 ((u64) in[8]) * (in2[6] << 0); | |
| 627 tmp[15] = ((u64) in[7]) * (in2[8] << 0) + | |
| 628 ((u64) in[8]) * (in2[7] << 0); | |
| 629 tmp[16] = ((u64) in[8]) * (in2[8] << 0); | |
| 630 | |
| 631 felem_reduce_degree(out, tmp); | |
| 632 } | |
| 633 | |
| 634 static void felem_assign(felem out, const felem in) | |
| 635 { | |
| 636 memcpy(out, in, sizeof(felem)); | |
| 637 } | |
| 638 | |
| 639 /* felem_inv calculates |out| = |in|^{-1} | |
| 640 * | |
| 641 * Based on Fermat's Little Theorem: | |
| 642 * a^p = a (mod p) | |
| 643 * a^{p-1} = 1 (mod p) | |
| 644 * a^{p-2} = a^{-1} (mod p) | |
| 645 */ | |
| 646 static void felem_inv(felem out, const felem in) | |
| 647 { | |
| 648 felem ftmp, ftmp2; | |
| 649 /* each e_I will hold |in|^{2^I - 1} */ | |
| 650 felem e2, e4, e8, e16, e32, e64; | |
| 651 unsigned int i; | |
| 652 | |
| 653 felem_square(ftmp, in); /* 2^1 */ | |
| 654 felem_mul(ftmp, in, ftmp); /* 2^2 - 2^0 */ | |
| 655 felem_assign(e2, ftmp); | |
| 656 felem_square(ftmp, ftmp); /* 2^3 - 2^1 */ | |
| 657 felem_square(ftmp, ftmp); /* 2^4 - 2^2 */ | |
| 658 felem_mul(ftmp, ftmp, e2); /* 2^4 - 2^0 */ | |
| 659 felem_assign(e4, ftmp); | |
| 660 felem_square(ftmp, ftmp); /* 2^5 - 2^1 */ | |
| 661 felem_square(ftmp, ftmp); /* 2^6 - 2^2 */ | |
| 662 felem_square(ftmp, ftmp); /* 2^7 - 2^3 */ | |
| 663 felem_square(ftmp, ftmp); /* 2^8 - 2^4 */ | |
| 664 felem_mul(ftmp, ftmp, e4); /* 2^8 - 2^0 */ | |
| 665 felem_assign(e8, ftmp); | |
| 666 for (i = 0; i < 8; i++) { | |
| 667 felem_square(ftmp, ftmp); | |
| 668 } /* 2^16 - 2^8 */ | |
| 669 felem_mul(ftmp, ftmp, e8); /* 2^16 - 2^0 */ | |
| 670 felem_assign(e16, ftmp); | |
| 671 for (i = 0; i < 16; i++) { | |
| 672 felem_square(ftmp, ftmp); | |
| 673 } /* 2^32 - 2^16 */ | |
| 674 felem_mul(ftmp, ftmp, e16); /* 2^32 - 2^0 */ | |
| 675 felem_assign(e32, ftmp); | |
| 676 for (i = 0; i < 32; i++) { | |
| 677 felem_square(ftmp, ftmp); | |
| 678 } /* 2^64 - 2^32 */ | |
| 679 felem_assign(e64, ftmp); | |
| 680 felem_mul(ftmp, ftmp, in); /* 2^64 - 2^32 + 2^0 */ | |
| 681 for (i = 0; i < 192; i++) { | |
| 682 felem_square(ftmp, ftmp); | |
| 683 } /* 2^256 - 2^224 + 2^192 */ | |
| 684 | |
| 685 felem_mul(ftmp2, e64, e32); /* 2^64 - 2^0 */ | |
| 686 for (i = 0; i < 16; i++) { | |
| 687 felem_square(ftmp2, ftmp2); | |
| 688 } /* 2^80 - 2^16 */ | |
| 689 felem_mul(ftmp2, ftmp2, e16); /* 2^80 - 2^0 */ | |
| 690 for (i = 0; i < 8; i++) { | |
| 691 felem_square(ftmp2, ftmp2); | |
| 692 } /* 2^88 - 2^8 */ | |
| 693 felem_mul(ftmp2, ftmp2, e8); /* 2^88 - 2^0 */ | |
| 694 for (i = 0; i < 4; i++) { | |
| 695 felem_square(ftmp2, ftmp2); | |
| 696 } /* 2^92 - 2^4 */ | |
| 697 felem_mul(ftmp2, ftmp2, e4); /* 2^92 - 2^0 */ | |
| 698 felem_square(ftmp2, ftmp2); /* 2^93 - 2^1 */ | |
| 699 felem_square(ftmp2, ftmp2); /* 2^94 - 2^2 */ | |
| 700 felem_mul(ftmp2, ftmp2, e2); /* 2^94 - 2^0 */ | |
| 701 felem_square(ftmp2, ftmp2); /* 2^95 - 2^1 */ | |
| 702 felem_square(ftmp2, ftmp2); /* 2^96 - 2^2 */ | |
| 703 felem_mul(ftmp2, ftmp2, in); /* 2^96 - 3 */ | |
| 704 | |
| 705 felem_mul(out, ftmp2, ftmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */ | |
| 706 } | |
| 707 | |
| 708 /* felem_scalar_3 sets out=3*out. | |
| 709 * | |
| 710 * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 711 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 712 */ | |
| 713 static void felem_scalar_3(felem out) | |
| 714 { | |
| 715 limb carry = 0; | |
| 716 unsigned int i; | |
| 717 | |
| 718 for (i = 0;; i++) { | |
| 719 out[i] *= 3; | |
| 720 out[i] += carry; | |
| 721 carry = out[i] >> 29; | |
| 722 out[i] &= kBottom29Bits; | |
| 723 | |
| 724 i++; | |
| 725 if (i == NLIMBS) | |
| 726 break; | |
| 727 | |
| 728 out[i] *= 3; | |
| 729 out[i] += carry; | |
| 730 carry = out[i] >> 28; | |
| 731 out[i] &= kBottom28Bits; | |
| 732 } | |
| 733 | |
| 734 felem_reduce_carry(out, carry); | |
| 735 } | |
| 736 | |
| 737 /* felem_scalar_4 sets out=4*out. | |
| 738 * | |
| 739 * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 740 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 741 */ | |
| 742 static void felem_scalar_4(felem out) | |
| 743 { | |
| 744 limb carry = 0, next_carry; | |
| 745 unsigned int i; | |
| 746 | |
| 747 for (i = 0;; i++) { | |
| 748 next_carry = out[i] >> 27; | |
| 749 out[i] <<= 2; | |
| 750 out[i] &= kBottom29Bits; | |
| 751 out[i] += carry; | |
| 752 carry = next_carry + (out[i] >> 29); | |
| 753 out[i] &= kBottom29Bits; | |
| 754 | |
| 755 i++; | |
| 756 if (i == NLIMBS) | |
| 757 break; | |
| 758 next_carry = out[i] >> 26; | |
| 759 out[i] <<= 2; | |
| 760 out[i] &= kBottom28Bits; | |
| 761 out[i] += carry; | |
| 762 carry = next_carry + (out[i] >> 28); | |
| 763 out[i] &= kBottom28Bits; | |
| 764 } | |
| 765 | |
| 766 felem_reduce_carry(out, carry); | |
| 767 } | |
| 768 | |
| 769 /* felem_scalar_8 sets out=8*out. | |
| 770 * | |
| 771 * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 772 * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. | |
| 773 */ | |
| 774 static void felem_scalar_8(felem out) | |
| 775 { | |
| 776 limb carry = 0, next_carry; | |
| 777 unsigned int i; | |
| 778 | |
| 779 for (i = 0;; i++) { | |
| 780 next_carry = out[i] >> 26; | |
| 781 out[i] <<= 3; | |
| 782 out[i] &= kBottom29Bits; | |
| 783 out[i] += carry; | |
| 784 carry = next_carry + (out[i] >> 29); | |
| 785 out[i] &= kBottom29Bits; | |
| 786 | |
| 787 i++; | |
| 788 if (i == NLIMBS) | |
| 789 break; | |
| 790 next_carry = out[i] >> 25; | |
| 791 out[i] <<= 3; | |
| 792 out[i] &= kBottom28Bits; | |
| 793 out[i] += carry; | |
| 794 carry = next_carry + (out[i] >> 28); | |
| 795 out[i] &= kBottom28Bits; | |
| 796 } | |
| 797 | |
| 798 felem_reduce_carry(out, carry); | |
| 799 } | |
| 800 | |
| 801 /* felem_is_zero_vartime returns 1 iff |in| == 0. It takes a variable amount of | |
| 802 * time depending on the value of |in|. | |
| 803 */ | |
| 804 static char felem_is_zero_vartime(const felem in) | |
| 805 { | |
| 806 limb carry; | |
| 807 int i; | |
| 808 limb tmp[NLIMBS]; | |
| 809 felem_assign(tmp, in); | |
| 810 | |
| 811 /* First, reduce tmp to a minimal form. | |
| 812 */ | |
| 813 do { | |
| 814 carry = 0; | |
| 815 for (i = 0;; i++) { | |
| 816 tmp[i] += carry; | |
| 817 carry = tmp[i] >> 29; | |
| 818 tmp[i] &= kBottom29Bits; | |
| 819 | |
| 820 i++; | |
| 821 if (i == NLIMBS) | |
| 822 break; | |
| 823 | |
| 824 tmp[i] += carry; | |
| 825 carry = tmp[i] >> 28; | |
| 826 tmp[i] &= kBottom28Bits; | |
| 827 } | |
| 828 | |
| 829 felem_reduce_carry(tmp, carry); | |
| 830 } while (carry); | |
| 831 | |
| 832 /* tmp < 2**257, so the only possible zero values are 0, p and 2p. | |
| 833 */ | |
| 834 return memcmp(tmp, kZero, sizeof(tmp)) == 0 || | |
| 835 memcmp(tmp, kP, sizeof(tmp)) == 0 || | |
| 836 memcmp(tmp, k2P, sizeof(tmp)) == 0; | |
| 837 } | |
| 838 | |
| 839 /* Group operations: | |
| 840 * | |
| 841 * Elements of the elliptic curve group are represented in Jacobian | |
| 842 * coordinates: (x, y, z). An affine point (x', y') is x'=x/z**2, y'=y/z**3 in | |
| 843 * Jacobian form. | |
| 844 */ | |
| 845 | |
| 846 /* point_double sets {x_out,y_out,z_out} = 2*{x,y,z}. | |
| 847 * | |
| 848 * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling
-dbl-2009-l | |
| 849 */ | |
| 850 static void point_double(felem x_out, felem y_out, felem z_out, | |
| 851 const felem x, const felem y, const felem z) | |
| 852 { | |
| 853 felem delta, gamma, alpha, beta, tmp, tmp2; | |
| 854 | |
| 855 felem_square(delta, z); | |
| 856 felem_square(gamma, y); | |
| 857 felem_mul(beta, x, gamma); | |
| 858 | |
| 859 felem_sum(tmp, x, delta); | |
| 860 felem_diff(tmp2, x, delta); | |
| 861 felem_mul(alpha, tmp, tmp2); | |
| 862 felem_scalar_3(alpha); | |
| 863 | |
| 864 felem_sum(tmp, y, z); | |
| 865 felem_square(tmp, tmp); | |
| 866 felem_diff(tmp, tmp, gamma); | |
| 867 felem_diff(z_out, tmp, delta); | |
| 868 | |
| 869 felem_scalar_4(beta); | |
| 870 felem_square(x_out, alpha); | |
| 871 felem_diff(x_out, x_out, beta); | |
| 872 felem_diff(x_out, x_out, beta); | |
| 873 | |
| 874 felem_diff(tmp, beta, x_out); | |
| 875 felem_mul(tmp, alpha, tmp); | |
| 876 felem_square(tmp2, gamma); | |
| 877 felem_scalar_8(tmp2); | |
| 878 felem_diff(y_out, tmp, tmp2); | |
| 879 } | |
| 880 | |
| 881 /* point_add_mixed sets {x_out,y_out,z_out} = {x1,y1,z1} + {x2,y2,1}. | |
| 882 * (i.e. the second point is affine.) | |
| 883 * | |
| 884 * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition
-add-2007-bl | |
| 885 * | |
| 886 * Note that this function does not handle P+P, infinity+P nor P+infinity | |
| 887 * correctly. | |
| 888 */ | |
| 889 static void point_add_mixed(felem x_out, felem y_out, felem z_out, | |
| 890 const felem x1, const felem y1, const felem z1, | |
| 891 const felem x2, const felem y2) | |
| 892 { | |
| 893 felem z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp; | |
| 894 | |
| 895 felem_square(z1z1, z1); | |
| 896 felem_sum(tmp, z1, z1); | |
| 897 | |
| 898 felem_mul(u2, x2, z1z1); | |
| 899 felem_mul(z1z1z1, z1, z1z1); | |
| 900 felem_mul(s2, y2, z1z1z1); | |
| 901 felem_diff(h, u2, x1); | |
| 902 felem_sum(i, h, h); | |
| 903 felem_square(i, i); | |
| 904 felem_mul(j, h, i); | |
| 905 felem_diff(r, s2, y1); | |
| 906 felem_sum(r, r, r); | |
| 907 felem_mul(v, x1, i); | |
| 908 | |
| 909 felem_mul(z_out, tmp, h); | |
| 910 felem_square(rr, r); | |
| 911 felem_diff(x_out, rr, j); | |
| 912 felem_diff(x_out, x_out, v); | |
| 913 felem_diff(x_out, x_out, v); | |
| 914 | |
| 915 felem_diff(tmp, v, x_out); | |
| 916 felem_mul(y_out, tmp, r); | |
| 917 felem_mul(tmp, y1, j); | |
| 918 felem_diff(y_out, y_out, tmp); | |
| 919 felem_diff(y_out, y_out, tmp); | |
| 920 } | |
| 921 | |
| 922 /* point_add sets {x_out,y_out,z_out} = {x1,y1,z1} + {x2,y2,z2}. | |
| 923 * | |
| 924 * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition
-add-2007-bl | |
| 925 * | |
| 926 * Note that this function does not handle P+P, infinity+P nor P+infinity | |
| 927 * correctly. | |
| 928 */ | |
| 929 static void point_add(felem x_out, felem y_out, felem z_out, | |
| 930 const felem x1, const felem y1, const felem z1, | |
| 931 const felem x2, const felem y2, const felem z2) | |
| 932 { | |
| 933 felem z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp; | |
| 934 | |
| 935 felem_square(z1z1, z1); | |
| 936 felem_square(z2z2, z2); | |
| 937 felem_mul(u1, x1, z2z2); | |
| 938 | |
| 939 felem_sum(tmp, z1, z2); | |
| 940 felem_square(tmp, tmp); | |
| 941 felem_diff(tmp, tmp, z1z1); | |
| 942 felem_diff(tmp, tmp, z2z2); | |
| 943 | |
| 944 felem_mul(z2z2z2, z2, z2z2); | |
| 945 felem_mul(s1, y1, z2z2z2); | |
| 946 | |
| 947 felem_mul(u2, x2, z1z1); | |
| 948 felem_mul(z1z1z1, z1, z1z1); | |
| 949 felem_mul(s2, y2, z1z1z1); | |
| 950 felem_diff(h, u2, u1); | |
| 951 felem_sum(i, h, h); | |
| 952 felem_square(i, i); | |
| 953 felem_mul(j, h, i); | |
| 954 felem_diff(r, s2, s1); | |
| 955 felem_sum(r, r, r); | |
| 956 felem_mul(v, u1, i); | |
| 957 | |
| 958 felem_mul(z_out, tmp, h); | |
| 959 felem_square(rr, r); | |
| 960 felem_diff(x_out, rr, j); | |
| 961 felem_diff(x_out, x_out, v); | |
| 962 felem_diff(x_out, x_out, v); | |
| 963 | |
| 964 felem_diff(tmp, v, x_out); | |
| 965 felem_mul(y_out, tmp, r); | |
| 966 felem_mul(tmp, s1, j); | |
| 967 felem_diff(y_out, y_out, tmp); | |
| 968 felem_diff(y_out, y_out, tmp); | |
| 969 } | |
| 970 | |
| 971 /* point_add_or_double_vartime sets {x_out,y_out,z_out} = {x1,y1,z1} + | |
| 972 * {x2,y2,z2}. | |
| 973 * | |
| 974 * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition
-add-2007-bl | |
| 975 * | |
| 976 * This function handles the case where {x1,y1,z1}={x2,y2,z2}. | |
| 977 */ | |
| 978 static void point_add_or_double_vartime( | |
| 979 felem x_out, felem y_out, felem z_out, | |
| 980 const felem x1, const felem y1, const felem z1, | |
| 981 const felem x2, const felem y2, const felem z2) | |
| 982 { | |
| 983 felem z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp; | |
| 984 char x_equal, y_equal; | |
| 985 | |
| 986 felem_square(z1z1, z1); | |
| 987 felem_square(z2z2, z2); | |
| 988 felem_mul(u1, x1, z2z2); | |
| 989 | |
| 990 felem_sum(tmp, z1, z2); | |
| 991 felem_square(tmp, tmp); | |
| 992 felem_diff(tmp, tmp, z1z1); | |
| 993 felem_diff(tmp, tmp, z2z2); | |
| 994 | |
| 995 felem_mul(z2z2z2, z2, z2z2); | |
| 996 felem_mul(s1, y1, z2z2z2); | |
| 997 | |
| 998 felem_mul(u2, x2, z1z1); | |
| 999 felem_mul(z1z1z1, z1, z1z1); | |
| 1000 felem_mul(s2, y2, z1z1z1); | |
| 1001 felem_diff(h, u2, u1); | |
| 1002 x_equal = felem_is_zero_vartime(h); | |
| 1003 felem_sum(i, h, h); | |
| 1004 felem_square(i, i); | |
| 1005 felem_mul(j, h, i); | |
| 1006 felem_diff(r, s2, s1); | |
| 1007 y_equal = felem_is_zero_vartime(r); | |
| 1008 if (x_equal && y_equal) { | |
| 1009 point_double(x_out, y_out, z_out, x1, y1, z1); | |
| 1010 return; | |
| 1011 } | |
| 1012 felem_sum(r, r, r); | |
| 1013 felem_mul(v, u1, i); | |
| 1014 | |
| 1015 felem_mul(z_out, tmp, h); | |
| 1016 felem_square(rr, r); | |
| 1017 felem_diff(x_out, rr, j); | |
| 1018 felem_diff(x_out, x_out, v); | |
| 1019 felem_diff(x_out, x_out, v); | |
| 1020 | |
| 1021 felem_diff(tmp, v, x_out); | |
| 1022 felem_mul(y_out, tmp, r); | |
| 1023 felem_mul(tmp, s1, j); | |
| 1024 felem_diff(y_out, y_out, tmp); | |
| 1025 felem_diff(y_out, y_out, tmp); | |
| 1026 } | |
| 1027 | |
| 1028 /* copy_conditional sets out=in if mask = 0xffffffff in constant time. | |
| 1029 * | |
| 1030 * On entry: mask is either 0 or 0xffffffff. | |
| 1031 */ | |
| 1032 static void copy_conditional(felem out, const felem in, limb mask) | |
| 1033 { | |
| 1034 int i; | |
| 1035 | |
| 1036 for (i = 0; i < NLIMBS; i++) { | |
| 1037 const limb tmp = mask & (in[i] ^ out[i]); | |
| 1038 out[i] ^= tmp; | |
| 1039 } | |
| 1040 } | |
| 1041 | |
| 1042 /* select_affine_point sets {out_x,out_y} to the index'th entry of table. | |
| 1043 * On entry: index < 16, table[0] must be zero. | |
| 1044 */ | |
| 1045 static void select_affine_point(felem out_x, felem out_y, | |
| 1046 const limb *table, limb index) | |
| 1047 { | |
| 1048 limb i, j; | |
| 1049 | |
| 1050 memset(out_x, 0, sizeof(felem)); | |
| 1051 memset(out_y, 0, sizeof(felem)); | |
| 1052 | |
| 1053 for (i = 1; i < 16; i++) { | |
| 1054 limb mask = i ^ index; | |
| 1055 mask |= mask >> 2; | |
| 1056 mask |= mask >> 1; | |
| 1057 mask &= 1; | |
| 1058 mask--; | |
| 1059 for (j = 0; j < NLIMBS; j++, table++) { | |
| 1060 out_x[j] |= *table & mask; | |
| 1061 } | |
| 1062 for (j = 0; j < NLIMBS; j++, table++) { | |
| 1063 out_y[j] |= *table & mask; | |
| 1064 } | |
| 1065 } | |
| 1066 } | |
| 1067 | |
| 1068 /* select_jacobian_point sets {out_x,out_y,out_z} to the index'th entry of | |
| 1069 * table. On entry: index < 16, table[0] must be zero. | |
| 1070 */ | |
| 1071 static void select_jacobian_point(felem out_x, felem out_y, felem out_z, | |
| 1072 const limb *table, limb index) | |
| 1073 { | |
| 1074 limb i, j; | |
| 1075 | |
| 1076 memset(out_x, 0, sizeof(felem)); | |
| 1077 memset(out_y, 0, sizeof(felem)); | |
| 1078 memset(out_z, 0, sizeof(felem)); | |
| 1079 | |
| 1080 /* The implicit value at index 0 is all zero. We don't need to perform that | |
| 1081 * iteration of the loop because we already set out_* to zero. | |
| 1082 */ | |
| 1083 table += 3*NLIMBS; | |
| 1084 | |
| 1085 for (i = 1; i < 16; i++) { | |
| 1086 limb mask = i ^ index; | |
| 1087 mask |= mask >> 2; | |
| 1088 mask |= mask >> 1; | |
| 1089 mask &= 1; | |
| 1090 mask--; | |
| 1091 for (j = 0; j < NLIMBS; j++, table++) { | |
| 1092 out_x[j] |= *table & mask; | |
| 1093 } | |
| 1094 for (j = 0; j < NLIMBS; j++, table++) { | |
| 1095 out_y[j] |= *table & mask; | |
| 1096 } | |
| 1097 for (j = 0; j < NLIMBS; j++, table++) { | |
| 1098 out_z[j] |= *table & mask; | |
| 1099 } | |
| 1100 } | |
| 1101 } | |
| 1102 | |
| 1103 /* get_bit returns the bit'th bit of scalar. */ | |
| 1104 static char get_bit(const u8 scalar[32], int bit) | |
| 1105 { | |
| 1106 return ((scalar[bit >> 3]) >> (bit & 7)) & 1; | |
| 1107 } | |
| 1108 | |
| 1109 /* scalar_base_mult sets {nx,ny,nz} = scalar*G where scalar is a little-endian | |
| 1110 * number. Note that the value of scalar must be less than the order of the | |
| 1111 * group. | |
| 1112 */ | |
| 1113 static void scalar_base_mult(felem nx, felem ny, felem nz, const u8 scalar[32]) | |
| 1114 { | |
| 1115 int i, j; | |
| 1116 limb n_is_infinity_mask = -1, p_is_noninfinite_mask, mask; | |
| 1117 u32 table_offset; | |
| 1118 | |
| 1119 felem px, py; | |
| 1120 felem tx, ty, tz; | |
| 1121 | |
| 1122 memset(nx, 0, sizeof(felem)); | |
| 1123 memset(ny, 0, sizeof(felem)); | |
| 1124 memset(nz, 0, sizeof(felem)); | |
| 1125 | |
| 1126 /* The loop adds bits at positions 0, 64, 128 and 192, followed by | |
| 1127 * positions 32,96,160 and 224 and does this 32 times. | |
| 1128 */ | |
| 1129 for (i = 0; i < 32; i++) { | |
| 1130 if (i) { | |
| 1131 point_double(nx, ny, nz, nx, ny, nz); | |
| 1132 } | |
| 1133 table_offset = 0; | |
| 1134 for (j = 0; j <= 32; j += 32) { | |
| 1135 char bit0 = get_bit(scalar, 31 - i + j); | |
| 1136 char bit1 = get_bit(scalar, 95 - i + j); | |
| 1137 char bit2 = get_bit(scalar, 159 - i + j); | |
| 1138 char bit3 = get_bit(scalar, 223 - i + j); | |
| 1139 limb index = bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3); | |
| 1140 | |
| 1141 select_affine_point(px, py, kPrecomputed + table_offset, index); | |
| 1142 table_offset += 30 * NLIMBS; | |
| 1143 | |
| 1144 /* Since scalar is less than the order of the group, we know that | |
| 1145 * {nx,ny,nz} != {px,py,1}, unless both are zero, which we handle | |
| 1146 * below. | |
| 1147 */ | |
| 1148 point_add_mixed(tx, ty, tz, nx, ny, nz, px, py); | |
| 1149 /* The result of point_add_mixed is incorrect if {nx,ny,nz} is zero | |
| 1150 * (a.k.a. the point at infinity). We handle that situation by | |
| 1151 * copying the point from the table. | |
| 1152 */ | |
| 1153 copy_conditional(nx, px, n_is_infinity_mask); | |
| 1154 copy_conditional(ny, py, n_is_infinity_mask); | |
| 1155 copy_conditional(nz, kOne, n_is_infinity_mask); | |
| 1156 | |
| 1157 /* Equally, the result is also wrong if the point from the table is | |
| 1158 * zero, which happens when the index is zero. We handle that by | |
| 1159 * only copying from {tx,ty,tz} to {nx,ny,nz} if index != 0. | |
| 1160 */ | |
| 1161 p_is_noninfinite_mask = NON_ZERO_TO_ALL_ONES(index); | |
| 1162 mask = p_is_noninfinite_mask & ~n_is_infinity_mask; | |
| 1163 copy_conditional(nx, tx, mask); | |
| 1164 copy_conditional(ny, ty, mask); | |
| 1165 copy_conditional(nz, tz, mask); | |
| 1166 /* If p was not zero, then n is now non-zero. */ | |
| 1167 n_is_infinity_mask &= ~p_is_noninfinite_mask; | |
| 1168 } | |
| 1169 } | |
| 1170 } | |
| 1171 | |
| 1172 /* point_to_affine converts a Jacobian point to an affine point. If the input | |
| 1173 * is the point at infinity then it returns (0, 0) in constant time. | |
| 1174 */ | |
| 1175 static void point_to_affine(felem x_out, felem y_out, | |
| 1176 const felem nx, const felem ny, const felem nz) { | |
| 1177 felem z_inv, z_inv_sq; | |
| 1178 felem_inv(z_inv, nz); | |
| 1179 felem_square(z_inv_sq, z_inv); | |
| 1180 felem_mul(x_out, nx, z_inv_sq); | |
| 1181 felem_mul(z_inv, z_inv, z_inv_sq); | |
| 1182 felem_mul(y_out, ny, z_inv); | |
| 1183 } | |
| 1184 | |
| 1185 /* scalar_mult sets {nx,ny,nz} = scalar*{x,y}. */ | |
| 1186 static void scalar_mult(felem nx, felem ny, felem nz, | |
| 1187 const felem x, const felem y, const u8 scalar[32]) | |
| 1188 { | |
| 1189 int i; | |
| 1190 felem px, py, pz, tx, ty, tz; | |
| 1191 felem precomp[16][3]; | |
| 1192 limb n_is_infinity_mask, index, p_is_noninfinite_mask, mask; | |
| 1193 | |
| 1194 /* We precompute 0,1,2,... times {x,y}. */ | |
| 1195 memset(precomp, 0, sizeof(felem) * 3); | |
| 1196 memcpy(&precomp[1][0], x, sizeof(felem)); | |
| 1197 memcpy(&precomp[1][1], y, sizeof(felem)); | |
| 1198 memcpy(&precomp[1][2], kOne, sizeof(felem)); | |
| 1199 | |
| 1200 for (i = 2; i < 16; i += 2) { | |
| 1201 point_double(precomp[i][0], precomp[i][1], precomp[i][2], | |
| 1202 precomp[i / 2][0], precomp[i / 2][1], precomp[i / 2][2]); | |
| 1203 | |
| 1204 point_add_mixed(precomp[i + 1][0], precomp[i + 1][1], precomp[i + 1][2], | |
| 1205 precomp[i][0], precomp[i][1], precomp[i][2], x, y); | |
| 1206 } | |
| 1207 | |
| 1208 memset(nx, 0, sizeof(felem)); | |
| 1209 memset(ny, 0, sizeof(felem)); | |
| 1210 memset(nz, 0, sizeof(felem)); | |
| 1211 n_is_infinity_mask = -1; | |
| 1212 | |
| 1213 /* We add in a window of four bits each iteration and do this 64 times. */ | |
| 1214 for (i = 0; i < 64; i++) { | |
| 1215 if (i) { | |
| 1216 point_double(nx, ny, nz, nx, ny, nz); | |
| 1217 point_double(nx, ny, nz, nx, ny, nz); | |
| 1218 point_double(nx, ny, nz, nx, ny, nz); | |
| 1219 point_double(nx, ny, nz, nx, ny, nz); | |
| 1220 } | |
| 1221 | |
| 1222 index = scalar[31 - i / 2]; | |
| 1223 if ((i & 1) == 1) { | |
| 1224 index &= 15; | |
| 1225 } else { | |
| 1226 index >>= 4; | |
| 1227 } | |
| 1228 | |
| 1229 /* See the comments in scalar_base_mult about handling infinities. */ | |
| 1230 select_jacobian_point(px, py, pz, precomp[0][0], index); | |
| 1231 point_add(tx, ty, tz, nx, ny, nz, px, py, pz); | |
| 1232 copy_conditional(nx, px, n_is_infinity_mask); | |
| 1233 copy_conditional(ny, py, n_is_infinity_mask); | |
| 1234 copy_conditional(nz, pz, n_is_infinity_mask); | |
| 1235 | |
| 1236 p_is_noninfinite_mask = NON_ZERO_TO_ALL_ONES(index); | |
| 1237 mask = p_is_noninfinite_mask & ~n_is_infinity_mask; | |
| 1238 copy_conditional(nx, tx, mask); | |
| 1239 copy_conditional(ny, ty, mask); | |
| 1240 copy_conditional(nz, tz, mask); | |
| 1241 n_is_infinity_mask &= ~p_is_noninfinite_mask; | |
| 1242 } | |
| 1243 } | |
| 1244 | |
| 1245 /* Interface with Freebl: */ | |
| 1246 | |
| 1247 /* BYTESWAP_MP_DIGIT_TO_LE swaps the bytes of a mp_digit to | |
| 1248 * little-endian order. | |
| 1249 */ | |
| 1250 #ifdef IS_BIG_ENDIAN | |
| 1251 #ifdef __APPLE__ | |
| 1252 #include <libkern/OSByteOrder.h> | |
| 1253 #define BYTESWAP32(x) OSSwapInt32(x) | |
| 1254 #define BYTESWAP64(x) OSSwapInt64(x) | |
| 1255 #else | |
| 1256 #define BYTESWAP32(x) \ | |
| 1257 (((x) >> 24) | (((x) >> 8) & 0xff00) | (((x) & 0xff00) << 8) | ((x) << 24)) | |
| 1258 #define BYTESWAP64(x) \ | |
| 1259 (((x) >> 56) | (((x) >> 40) & 0xff00) | \ | |
| 1260 (((x) >> 24) & 0xff0000) | (((x) >> 8) & 0xff000000) | \ | |
| 1261 (((x) & 0xff000000) << 8) | (((x) & 0xff0000) << 24) | \ | |
| 1262 (((x) & 0xff00) << 40) | ((x) << 56)) | |
| 1263 #endif | |
| 1264 | |
| 1265 #ifdef MP_USE_UINT_DIGIT | |
| 1266 #define BYTESWAP_MP_DIGIT_TO_LE(x) BYTESWAP32(x) | |
| 1267 #else | |
| 1268 #define BYTESWAP_MP_DIGIT_TO_LE(x) BYTESWAP64(x) | |
| 1269 #endif | |
| 1270 #endif /* IS_BIG_ENDIAN */ | |
| 1271 | |
| 1272 #ifdef MP_USE_UINT_DIGIT | |
| 1273 static const mp_digit kRInvDigits[8] = { | |
| 1274 0x80000000, 1, 0xffffffff, 0, | |
| 1275 0x80000001, 0xfffffffe, 1, 0x7fffffff | |
| 1276 }; | |
| 1277 #else | |
| 1278 static const mp_digit kRInvDigits[4] = { | |
| 1279 PR_UINT64(0x180000000), 0xffffffff, | |
| 1280 PR_UINT64(0xfffffffe80000001), PR_UINT64(0x7fffffff00000001) | |
| 1281 }; | |
| 1282 #endif | |
| 1283 #define MP_DIGITS_IN_256_BITS (32/sizeof(mp_digit)) | |
| 1284 static const mp_int kRInv = { | |
| 1285 MP_ZPOS, | |
| 1286 MP_DIGITS_IN_256_BITS, | |
| 1287 MP_DIGITS_IN_256_BITS, | |
| 1288 (mp_digit*) kRInvDigits | |
| 1289 }; | |
| 1290 | |
| 1291 static const limb kTwo28 = 0x10000000; | |
| 1292 static const limb kTwo29 = 0x20000000; | |
| 1293 | |
| 1294 /* to_montgomery sets out = R*in. */ | |
| 1295 static mp_err to_montgomery(felem out, const mp_int *in, const ECGroup *group) | |
| 1296 { | |
| 1297 /* There are no MPI functions for bitshift operations and we wish to shift | |
| 1298 * in 257 bits left so we move the digits 256-bits left and then multiply | |
| 1299 * by two. | |
| 1300 */ | |
| 1301 mp_int in_shifted; | |
| 1302 int i; | |
| 1303 mp_err res; | |
| 1304 | |
| 1305 MP_CHECKOK(mp_init(&in_shifted)); | |
| 1306 MP_CHECKOK(s_mp_pad(&in_shifted, MP_USED(in) + MP_DIGITS_IN_256_BITS)); | |
| 1307 memcpy(&MP_DIGIT(&in_shifted, MP_DIGITS_IN_256_BITS), | |
| 1308 MP_DIGITS(in), | |
| 1309 MP_USED(in)*sizeof(mp_digit)); | |
| 1310 MP_CHECKOK(mp_mul_2(&in_shifted, &in_shifted)); | |
| 1311 MP_CHECKOK(group->meth->field_mod(&in_shifted, &in_shifted, group->meth)); | |
| 1312 | |
| 1313 for (i = 0;; i++) { | |
| 1314 out[i] = MP_DIGIT(&in_shifted, 0) & kBottom29Bits; | |
| 1315 MP_CHECKOK(mp_div_d(&in_shifted, kTwo29, &in_shifted, NULL)); | |
| 1316 | |
| 1317 i++; | |
| 1318 if (i == NLIMBS) | |
| 1319 break; | |
| 1320 out[i] = MP_DIGIT(&in_shifted, 0) & kBottom28Bits; | |
| 1321 MP_CHECKOK(mp_div_d(&in_shifted, kTwo28, &in_shifted, NULL)); | |
| 1322 } | |
| 1323 | |
| 1324 CLEANUP: | |
| 1325 mp_clear(&in_shifted); | |
| 1326 return res; | |
| 1327 } | |
| 1328 | |
| 1329 /* from_montgomery sets out=in/R. */ | |
| 1330 static mp_err from_montgomery(mp_int *out, const felem in, | |
| 1331 const ECGroup *group) | |
| 1332 { | |
| 1333 mp_int result, tmp; | |
| 1334 mp_err res; | |
| 1335 int i; | |
| 1336 | |
| 1337 MP_CHECKOK(mp_init(&result)); | |
| 1338 MP_CHECKOK(mp_init(&tmp)); | |
| 1339 | |
| 1340 MP_CHECKOK(mp_add_d(&tmp, in[NLIMBS-1], &result)); | |
| 1341 for (i = NLIMBS-2; i >= 0; i--) { | |
| 1342 if ((i & 1) == 0) { | |
| 1343 MP_CHECKOK(mp_mul_d(&result, kTwo29, &tmp)); | |
| 1344 } else { | |
| 1345 MP_CHECKOK(mp_mul_d(&result, kTwo28, &tmp)); | |
| 1346 } | |
| 1347 MP_CHECKOK(mp_add_d(&tmp, in[i], &result)); | |
| 1348 } | |
| 1349 | |
| 1350 MP_CHECKOK(mp_mul(&result, &kRInv, out)); | |
| 1351 MP_CHECKOK(group->meth->field_mod(out, out, group->meth)); | |
| 1352 | |
| 1353 CLEANUP: | |
| 1354 mp_clear(&result); | |
| 1355 mp_clear(&tmp); | |
| 1356 return res; | |
| 1357 } | |
| 1358 | |
| 1359 /* scalar_from_mp_int sets out_scalar=n, where n < the group order. */ | |
| 1360 static void scalar_from_mp_int(u8 out_scalar[32], const mp_int *n) | |
| 1361 { | |
| 1362 /* We require that |n| is less than the order of the group and therefore it | |
| 1363 * will fit into |out_scalar|. However, these is a timing side-channel here | |
| 1364 * that we cannot avoid: if |n| is sufficiently small it may be one or more | |
| 1365 * words too short and we'll copy less data. | |
| 1366 */ | |
| 1367 memset(out_scalar, 0, 32); | |
| 1368 #ifdef IS_LITTLE_ENDIAN | |
| 1369 memcpy(out_scalar, MP_DIGITS(n), MP_USED(n) * sizeof(mp_digit)); | |
| 1370 #else | |
| 1371 { | |
| 1372 mp_size i; | |
| 1373 mp_digit swapped[MP_DIGITS_IN_256_BITS]; | |
| 1374 for (i = 0; i < MP_USED(n); i++) { | |
| 1375 swapped[i] = BYTESWAP_MP_DIGIT_TO_LE(MP_DIGIT(n, i)); | |
| 1376 } | |
| 1377 memcpy(out_scalar, swapped, MP_USED(n) * sizeof(mp_digit)); | |
| 1378 } | |
| 1379 #endif | |
| 1380 } | |
| 1381 | |
| 1382 /* ec_GFp_nistp256_base_point_mul sets {out_x,out_y} = nG, where n is < the | |
| 1383 * order of the group. | |
| 1384 */ | |
| 1385 static mp_err ec_GFp_nistp256_base_point_mul(const mp_int *n, | |
| 1386 mp_int *out_x, mp_int *out_y, | |
| 1387 const ECGroup *group) | |
| 1388 { | |
| 1389 u8 scalar[32]; | |
| 1390 felem x, y, z, x_affine, y_affine; | |
| 1391 mp_err res; | |
| 1392 | |
| 1393 /* FIXME(agl): test that n < order. */ | |
| 1394 | |
| 1395 scalar_from_mp_int(scalar, n); | |
| 1396 scalar_base_mult(x, y, z, scalar); | |
| 1397 point_to_affine(x_affine, y_affine, x, y, z); | |
| 1398 MP_CHECKOK(from_montgomery(out_x, x_affine, group)); | |
| 1399 MP_CHECKOK(from_montgomery(out_y, y_affine, group)); | |
| 1400 | |
| 1401 CLEANUP: | |
| 1402 return res; | |
| 1403 } | |
| 1404 | |
| 1405 /* ec_GFp_nistp256_point_mul sets {out_x,out_y} = n*{in_x,in_y}, where n is < | |
| 1406 * the order of the group. | |
| 1407 */ | |
| 1408 static mp_err ec_GFp_nistp256_point_mul(const mp_int *n, | |
| 1409 const mp_int *in_x, const mp_int *in_y, | |
| 1410 mp_int *out_x, mp_int *out_y, | |
| 1411 const ECGroup *group) | |
| 1412 { | |
| 1413 u8 scalar[32]; | |
| 1414 felem x, y, z, x_affine, y_affine, px, py; | |
| 1415 mp_err res; | |
| 1416 | |
| 1417 scalar_from_mp_int(scalar, n); | |
| 1418 | |
| 1419 MP_CHECKOK(to_montgomery(px, in_x, group)); | |
| 1420 MP_CHECKOK(to_montgomery(py, in_y, group)); | |
| 1421 | |
| 1422 scalar_mult(x, y, z, px, py, scalar); | |
| 1423 point_to_affine(x_affine, y_affine, x, y, z); | |
| 1424 MP_CHECKOK(from_montgomery(out_x, x_affine, group)); | |
| 1425 MP_CHECKOK(from_montgomery(out_y, y_affine, group)); | |
| 1426 | |
| 1427 CLEANUP: | |
| 1428 return res; | |
| 1429 } | |
| 1430 | |
| 1431 /* ec_GFp_nistp256_point_mul_vartime sets {out_x,out_y} = n1*G + | |
| 1432 * n2*{in_x,in_y}, where n1 and n2 are < the order of the group. | |
| 1433 * | |
| 1434 * As indicated by the name, this function operates in variable time. This | |
| 1435 * is safe because it's used for signature validation which doesn't deal | |
| 1436 * with secrets. | |
| 1437 */ | |
| 1438 static mp_err ec_GFp_nistp256_points_mul_vartime( | |
| 1439 const mp_int *n1, const mp_int *n2, | |
| 1440 const mp_int *in_x, const mp_int *in_y, | |
| 1441 mp_int *out_x, mp_int *out_y, | |
| 1442 const ECGroup *group) | |
| 1443 { | |
| 1444 u8 scalar1[32], scalar2[32]; | |
| 1445 felem x1, y1, z1, x2, y2, z2, x_affine, y_affine, px, py; | |
| 1446 mp_err res = MP_OKAY; | |
| 1447 | |
| 1448 /* If n2 == NULL, this is just a base-point multiplication. */ | |
| 1449 if (n2 == NULL) { | |
| 1450 return ec_GFp_nistp256_base_point_mul(n1, out_x, out_y, group); | |
| 1451 } | |
| 1452 | |
| 1453 /* If n1 == nULL, this is just an arbitary-point multiplication. */ | |
| 1454 if (n1 == NULL) { | |
| 1455 return ec_GFp_nistp256_point_mul(n2, in_x, in_y, out_x, out_y, group); | |
| 1456 } | |
| 1457 | |
| 1458 /* If both scalars are zero, then the result is the point at infinity. */ | |
| 1459 if (mp_cmp_z(n1) == 0 && mp_cmp_z(n2) == 0) { | |
| 1460 mp_zero(out_x); | |
| 1461 mp_zero(out_y); | |
| 1462 return res; | |
| 1463 } | |
| 1464 | |
| 1465 scalar_from_mp_int(scalar1, n1); | |
| 1466 scalar_from_mp_int(scalar2, n2); | |
| 1467 | |
| 1468 MP_CHECKOK(to_montgomery(px, in_x, group)); | |
| 1469 MP_CHECKOK(to_montgomery(py, in_y, group)); | |
| 1470 scalar_base_mult(x1, y1, z1, scalar1); | |
| 1471 scalar_mult(x2, y2, z2, px, py, scalar2); | |
| 1472 | |
| 1473 if (mp_cmp_z(n2) == 0) { | |
| 1474 /* If n2 == 0, then {x2,y2,z2} is zero and the result is just | |
| 1475 * {x1,y1,z1}. */ | |
| 1476 } else if (mp_cmp_z(n1) == 0) { | |
| 1477 /* If n1 == 0, then {x1,y1,z1} is zero and the result is just | |
| 1478 * {x2,y2,z2}. */ | |
| 1479 memcpy(x1, x2, sizeof(x2)); | |
| 1480 memcpy(y1, y2, sizeof(y2)); | |
| 1481 memcpy(z1, z2, sizeof(z2)); | |
| 1482 } else { | |
| 1483 /* This function handles the case where {x1,y1,z1} == {x2,y2,z2}. */ | |
| 1484 point_add_or_double_vartime(x1, y1, z1, x1, y1, z1, x2, y2, z2); | |
| 1485 } | |
| 1486 | |
| 1487 point_to_affine(x_affine, y_affine, x1, y1, z1); | |
| 1488 MP_CHECKOK(from_montgomery(out_x, x_affine, group)); | |
| 1489 MP_CHECKOK(from_montgomery(out_y, y_affine, group)); | |
| 1490 | |
| 1491 CLEANUP: | |
| 1492 return res; | |
| 1493 } | |
| 1494 | |
| 1495 /* Wire in fast point multiplication for named curves. */ | |
| 1496 mp_err ec_group_set_gfp256_32(ECGroup *group, ECCurveName name) | |
| 1497 { | |
| 1498 if (name == ECCurve_NIST_P256) { | |
| 1499 group->base_point_mul = &ec_GFp_nistp256_base_point_mul; | |
| 1500 group->point_mul = &ec_GFp_nistp256_point_mul; | |
| 1501 group->points_mul = &ec_GFp_nistp256_points_mul_vartime; | |
| 1502 } | |
| 1503 return MP_OKAY; | |
| 1504 } | |
| OLD | NEW |
|---|
Chromium Code Reviews
Issue 2078763002:
Delete bundled copy of NSS and replace with README. (Closed)
Base URL: https://chromium.googlesource.com/chromium/deps/nss@master