diff --git a/cbits/p256/p256.h b/cbits/include32/p256/p256.h similarity index 98% rename from cbits/p256/p256.h rename to cbits/include32/p256/p256.h index 8d44e9d..61acc1c 100644 --- a/cbits/p256/p256.h +++ b/cbits/include32/p256/p256.h @@ -40,6 +40,11 @@ extern "C" { #define P256_NDIGITS 8 #define P256_NBYTES 32 +// n' such as n * n' = -1 mod (2^32) +#define P256_MONTGOMERY_FACTOR 0xEE00BC4F + +#define P256_LITERAL(lo,hi) (lo), (hi) + typedef int cryptonite_p256_err; typedef uint32_t cryptonite_p256_digit; typedef int32_t cryptonite_p256_sdigit; diff --git a/cbits/include32/p256/p256_gf.h b/cbits/include32/p256/p256_gf.h new file mode 100644 index 0000000..ca8a716 --- /dev/null +++ b/cbits/include32/p256/p256_gf.h @@ -0,0 +1,779 @@ +/* + * Copyright 2013 The Android Open Source Project + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * * Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * * Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * * Neither the name of Google Inc. nor the names of its contributors may + * be used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY Google Inc. ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF + * MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO + * EVENT SHALL Google Inc. BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, + * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, + * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; + * OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, + * WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR + * OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF + * ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +// This is an implementation of the P256 finite field. It's written to be +// portable and still constant-time. +// +// WARNING: Implementing these functions in a constant-time manner is far from +// obvious. Be careful when touching this code. +// +// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background. + +#include +#include + +#include +#include + +#include "p256/p256.h" + +typedef uint8_t u8; +typedef uint32_t u32; +typedef int32_t s32; +typedef uint64_t u64; + +/* Our field elements are represented as nine 32-bit limbs. + * + * The value of an felem (field element) is: + * x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228) + * + * That is, each limb is alternately 29 or 28-bits wide in little-endian + * order. + * + * This means that an felem hits 2**257, rather than 2**256 as we would like. A + * 28, 29, ... pattern would cause us to hit 2**256, but that causes problems + * when multiplying as terms end up one bit short of a limb which would require + * much bit-shifting to correct. + * + * Finally, the values stored in an felem are in Montgomery form. So the value + * |y| is stored as (y*R) mod p, where p is the P-256 prime and R is 2**257. + */ +typedef u32 limb; +#define NLIMBS 9 +typedef limb felem[NLIMBS]; + +static const limb kBottom28Bits = 0xfffffff; +static const limb kBottom29Bits = 0x1fffffff; + +/* kOne is the number 1 as an felem. It's 2**257 mod p split up into 29 and + * 28-bit words. */ +static const felem kOne = { + 2, 0, 0, 0xffff800, + 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, + 0 +}; +static const felem kZero = {0}; +static const felem kP = { + 0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, + 0, 0, 0x200000, 0xf000000, + 0xfffffff +}; +static const felem k2P = { + 0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, + 0, 0, 0x400000, 0xe000000, + 0x1fffffff +}; +/* kPrecomputed contains precomputed values to aid the calculation of scalar + * multiples of the base point, G. It's actually two, equal length, tables + * concatenated. + * + * The first table contains (x,y) felem pairs for 16 multiples of the base + * point, G. + * + * Index | Index (binary) | Value + * 0 | 0000 | 0G (all zeros, omitted) + * 1 | 0001 | G + * 2 | 0010 | 2**64G + * 3 | 0011 | 2**64G + G + * 4 | 0100 | 2**128G + * 5 | 0101 | 2**128G + G + * 6 | 0110 | 2**128G + 2**64G + * 7 | 0111 | 2**128G + 2**64G + G + * 8 | 1000 | 2**192G + * 9 | 1001 | 2**192G + G + * 10 | 1010 | 2**192G + 2**64G + * 11 | 1011 | 2**192G + 2**64G + G + * 12 | 1100 | 2**192G + 2**128G + * 13 | 1101 | 2**192G + 2**128G + G + * 14 | 1110 | 2**192G + 2**128G + 2**64G + * 15 | 1111 | 2**192G + 2**128G + 2**64G + G + * + * The second table follows the same style, but the terms are 2**32G, + * 2**96G, 2**160G, 2**224G. + * + * This is ~2KB of data. */ +static const limb kPrecomputed[NLIMBS * 2 * 15 * 2] = { + 0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7edc, 0xd4a6eab, 0x3120bee, + 0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba21, 0x14b10bb, 0xae3fe3, + 0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe49073, 0x3fa36cc, 0x5ebcd2c, + 0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea12446, 0xe1ade1e, 0xec91f22, + 0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109, 0xa267a00, 0xb57c050, + 0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0x7d6dee7, 0x2976e4b, + 0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a5a9, 0x843a649, 0xc3ab0fa, + 0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11, 0x58c43df, 0xf423fc2, + 0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db40f, 0x83e277d, 0xb0dd609, + 0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5, 0xe10c9e, 0x33ab581, + 0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f, 0x48764cd, 0x76dbcca, + 0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b20, 0x4ba3173, 0xc168c33, + 0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0, 0x65dd7ff, 0x3a1e4f6, + 0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f077, 0xa6add89, 0x4894acd, + 0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a, 0x69a8556, 0x7e7c0, + 0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c, 0xda0cf5b, 0x812e881, + 0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51, 0xc22be3e, 0xe35e65a, + 0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9, 0x1c5a839, 0x47a1e26, + 0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c502, 0x2f32042, 0xa17769b, + 0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a02, 0x3fc93, 0x5620023, + 0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c, 0x407f75c, 0xbaab133, + 0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea7, 0x3293ac0, 0xcdc98aa, + 0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16, 0x2b6fcc7, 0xf5a4e29, + 0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72, 0x73e1c35, 0xee70fbc, + 0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85, 0x27de188, 0x66f70b8, + 0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae914, 0x2f3ec51, 0x3826b59, + 0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x823d9d2, 0x8213f39, + 0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4a, 0xf5ddc3d, 0x3786689, + 0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a729, 0x4be3499, 0x52b23aa, + 0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb048035, 0xe31de66, 0xc6ecaa3, + 0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a7529, 0xcb7beb1, 0xb2a78a1, + 0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff658, 0xe3d6511, 0xc7d76f, + 0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c, 0x50daa90, 0xb13f72, + 0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d32411, 0xb04a838, 0xd760d2d, + 0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11e, 0x20bca9a, 0x66f496b, + 0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d, 0xbe985f7, 0x1acbc1a, + 0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56ff, 0x65ef930, 0x21dc4a, + 0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15f, 0x624e62e, 0xa90ae2f, + 0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522b, 0xdc78583, 0x40eeabb, + 0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef34, 0xae2a960, 0x91b8bdc, + 0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0x2413c8e, 0x5425bf9, + 0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633, 0x7c91952, 0xd806dce, + 0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef73, 0x8956f34, 0xe4b5cf2, + 0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7, 0x627b614, 0x7371cca, + 0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc9, 0x9c19bf2, 0x5882229, + 0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b3, 0xe85ff25, 0x408ef57, + 0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113, 0xa4a1769, 0x11fbc6c, + 0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b7, 0x4acbad9, 0x5efc5fa, + 0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc, 0x7bf0fa9, 0x957651, + 0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57, 0xf2ecaac, 0xca86dec, + 0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c12d, 0xf20bd46, 0x1951fa7, + 0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74, 0x99bb618, 0x2db944c, + 0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e74779, 0x576138, 0x9587927, + 0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782d, 0xfc72e0b, 0x701b298, + 0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5d8, 0xf858d3a, 0x942eea8, + 0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a1, 0x8395659, 0x52ed4e2, + 0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c0, 0x6bdf55a, 0x4e4457d, + 0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747b, 0x878558d, 0x7d29aa4, + 0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d7, 0xa5bef68, 0xb7b30d8, + 0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78, +}; + + +/* Field element operations: */ + +/* NON_ZERO_TO_ALL_ONES returns: + * 0xffffffff for 0 < x <= 2**31 + * 0 for x == 0 or x > 2**31. + * + * x must be a u32 or an equivalent type such as limb. */ +#define NON_ZERO_TO_ALL_ONES(x) ((((u32)(x) - 1) >> 31) - 1) + +/* felem_reduce_carry adds a multiple of p in order to cancel |carry|, + * which is a term at 2**257. + * + * On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28. + * On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29. */ +static void felem_reduce_carry(felem inout, limb carry) { + const u32 carry_mask = NON_ZERO_TO_ALL_ONES(carry); + + inout[0] += carry << 1; + inout[3] += 0x10000000 & carry_mask; + /* carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the + * previous line therefore this doesn't underflow. */ + inout[3] -= carry << 11; + inout[4] += (0x20000000 - 1) & carry_mask; + inout[5] += (0x10000000 - 1) & carry_mask; + inout[6] += (0x20000000 - 1) & carry_mask; + inout[6] -= carry << 22; + /* This may underflow if carry is non-zero but, if so, we'll fix it in the + * next line. */ + inout[7] -= 1 & carry_mask; + inout[7] += carry << 25; +} + +/* felem_sum sets out = in+in2. + * + * On entry, in[i]+in2[i] must not overflow a 32-bit word. + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 */ +static void felem_sum(felem out, const felem in, const felem in2) { + limb carry = 0; + unsigned i; + + for (i = 0;; i++) { + out[i] = in[i] + in2[i]; + out[i] += carry; + carry = out[i] >> 29; + out[i] &= kBottom29Bits; + + i++; + if (i == NLIMBS) + break; + + out[i] = in[i] + in2[i]; + out[i] += carry; + carry = out[i] >> 28; + out[i] &= kBottom28Bits; + } + + felem_reduce_carry(out, carry); +} + +#define two31m3 (((limb)1) << 31) - (((limb)1) << 3) +#define two30m2 (((limb)1) << 30) - (((limb)1) << 2) +#define two30p13m2 (((limb)1) << 30) + (((limb)1) << 13) - (((limb)1) << 2) +#define two31m2 (((limb)1) << 31) - (((limb)1) << 2) +#define two31p24m2 (((limb)1) << 31) + (((limb)1) << 24) - (((limb)1) << 2) +#define two30m27m2 (((limb)1) << 30) - (((limb)1) << 27) - (((limb)1) << 2) + +/* zero31 is 0 mod p. */ +static const felem zero31 = { two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2 }; + +/* felem_diff sets out = in-in2. + * + * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and + * in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ +static void felem_diff(felem out, const felem in, const felem in2) { + limb carry = 0; + unsigned i; + + for (i = 0;; i++) { + out[i] = in[i] - in2[i]; + out[i] += zero31[i]; + out[i] += carry; + carry = out[i] >> 29; + out[i] &= kBottom29Bits; + + i++; + if (i == NLIMBS) + break; + + out[i] = in[i] - in2[i]; + out[i] += zero31[i]; + out[i] += carry; + carry = out[i] >> 28; + out[i] &= kBottom28Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_reduce_degree sets out = tmp/R mod p where tmp contains 64-bit words + * with the same 29,28,... bit positions as an felem. + * + * The values in felems are in Montgomery form: x*R mod p where R = 2**257. + * Since we just multiplied two Montgomery values together, the result is + * x*y*R*R mod p. We wish to divide by R in order for the result also to be + * in Montgomery form. + * + * On entry: tmp[i] < 2**64 + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 */ +static void felem_reduce_degree(felem out, u64 tmp[17]) { + /* The following table may be helpful when reading this code: + * + * Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10... + * Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29 + * Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285 + * (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285 */ + limb tmp2[18], carry, x, xMask; + unsigned i; + + /* tmp contains 64-bit words with the same 29,28,29-bit positions as an + * felem. So the top of an element of tmp might overlap with another + * element two positions down. The following loop eliminates this + * overlap. */ + tmp2[0] = (limb)(tmp[0] & kBottom29Bits); + + /* In the following we use "(limb) tmp[x]" and "(limb) (tmp[x]>>32)" to try + * and hint to the compiler that it can do a single-word shift by selecting + * the right register rather than doing a double-word shift and truncating + * afterwards. */ + tmp2[1] = ((limb) tmp[0]) >> 29; + tmp2[1] |= (((limb)(tmp[0] >> 32)) << 3) & kBottom28Bits; + tmp2[1] += ((limb) tmp[1]) & kBottom28Bits; + carry = tmp2[1] >> 28; + tmp2[1] &= kBottom28Bits; + + for (i = 2; i < 17; i++) { + tmp2[i] = ((limb)(tmp[i - 2] >> 32)) >> 25; + tmp2[i] += ((limb)(tmp[i - 1])) >> 28; + tmp2[i] += (((limb)(tmp[i - 1] >> 32)) << 4) & kBottom29Bits; + tmp2[i] += ((limb) tmp[i]) & kBottom29Bits; + tmp2[i] += carry; + carry = tmp2[i] >> 29; + tmp2[i] &= kBottom29Bits; + + i++; + if (i == 17) + break; + tmp2[i] = ((limb)(tmp[i - 2] >> 32)) >> 25; + tmp2[i] += ((limb)(tmp[i - 1])) >> 29; + tmp2[i] += (((limb)(tmp[i - 1] >> 32)) << 3) & kBottom28Bits; + tmp2[i] += ((limb) tmp[i]) & kBottom28Bits; + tmp2[i] += carry; + carry = tmp2[i] >> 28; + tmp2[i] &= kBottom28Bits; + } + + tmp2[17] = ((limb)(tmp[15] >> 32)) >> 25; + tmp2[17] += ((limb)(tmp[16])) >> 29; + tmp2[17] += (((limb)(tmp[16] >> 32)) << 3); + tmp2[17] += carry; + + /* Montgomery elimination of terms. + * + * Since R is 2**257, we can divide by R with a bitwise shift if we can + * ensure that the right-most 257 bits are all zero. We can make that true by + * adding multiplies of p without affecting the value. + * + * So we eliminate limbs from right to left. Since the bottom 29 bits of p + * are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0. + * We can do that for 8 further limbs and then right shift to eliminate the + * extra factor of R. */ + for (i = 0;; i += 2) { + tmp2[i + 1] += tmp2[i] >> 29; + x = tmp2[i] & kBottom29Bits; + xMask = NON_ZERO_TO_ALL_ONES(x); + tmp2[i] = 0; + + /* The bounds calculations for this loop are tricky. Each iteration of + * the loop eliminates two words by adding values to words to their + * right. + * + * The following table contains the amounts added to each word (as an + * offset from the value of i at the top of the loop). The amounts are + * accounted for from the first and second half of the loop separately + * and are written as, for example, 28 to mean a value <2**28. + * + * Word: 3 4 5 6 7 8 9 10 + * Added in top half: 28 11 29 21 29 28 + * 28 29 + * 29 + * Added in bottom half: 29 10 28 21 28 28 + * 29 + * + * The value that is currently offset 7 will be offset 5 for the next + * iteration and then offset 3 for the iteration after that. Therefore + * the total value added will be the values added at 7, 5 and 3. + * + * The following table accumulates these values. The sums at the bottom + * are written as, for example, 29+28, to mean a value < 2**29+2**28. + * + * Word: 3 4 5 6 7 8 9 10 11 12 13 + * 28 11 10 29 21 29 28 28 28 28 28 + * 29 28 11 28 29 28 29 28 29 28 + * 29 28 21 21 29 21 29 21 + * 10 29 28 21 28 21 28 + * 28 29 28 29 28 29 28 + * 11 10 29 10 29 10 + * 29 28 11 28 11 + * 29 29 + * -------------------------------------------- + * 30+ 31+ 30+ 31+ 30+ + * 28+ 29+ 28+ 29+ 21+ + * 21+ 28+ 21+ 28+ 10 + * 10 21+ 10 21+ + * 11 11 + * + * So the greatest amount is added to tmp2[10] and tmp2[12]. If + * tmp2[10/12] has an initial value of <2**29, then the maximum value + * will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32, + * as required. */ + tmp2[i + 3] += (x << 10) & kBottom28Bits; + tmp2[i + 4] += (x >> 18); + + tmp2[i + 6] += (x << 21) & kBottom29Bits; + tmp2[i + 7] += x >> 8; + + /* At position 200, which is the starting bit position for word 7, we + * have a factor of 0xf000000 = 2**28 - 2**24. */ + tmp2[i + 7] += 0x10000000 & xMask; + /* Word 7 is 28 bits wide, so the 2**28 term exactly hits word 8. */ + tmp2[i + 8] += (x - 1) & xMask; + tmp2[i + 7] -= (x << 24) & kBottom28Bits; + tmp2[i + 8] -= x >> 4; + + tmp2[i + 8] += 0x20000000 & xMask; + tmp2[i + 8] -= x; + tmp2[i + 8] += (x << 28) & kBottom29Bits; + tmp2[i + 9] += ((x >> 1) - 1) & xMask; + + if (i+1 == NLIMBS) + break; + tmp2[i + 2] += tmp2[i + 1] >> 28; + x = tmp2[i + 1] & kBottom28Bits; + xMask = NON_ZERO_TO_ALL_ONES(x); + tmp2[i + 1] = 0; + + tmp2[i + 4] += (x << 11) & kBottom29Bits; + tmp2[i + 5] += (x >> 18); + + tmp2[i + 7] += (x << 21) & kBottom28Bits; + tmp2[i + 8] += x >> 7; + + /* At position 199, which is the starting bit of the 8th word when + * dealing with a context starting on an odd word, we have a factor of + * 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th + * word from i+1 is i+8. */ + tmp2[i + 8] += 0x20000000 & xMask; + tmp2[i + 9] += (x - 1) & xMask; + tmp2[i + 8] -= (x << 25) & kBottom29Bits; + tmp2[i + 9] -= x >> 4; + + tmp2[i + 9] += 0x10000000 & xMask; + tmp2[i + 9] -= x; + tmp2[i + 10] += (x - 1) & xMask; + } + + /* We merge the right shift with a carry chain. The words above 2**257 have + * widths of 28,29,... which we need to correct when copying them down. */ + carry = 0; + for (i = 0; i < 8; i++) { + /* The maximum value of tmp2[i + 9] occurs on the first iteration and + * is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is + * therefore safe. */ + out[i] = tmp2[i + 9]; + out[i] += carry; + out[i] += (tmp2[i + 10] << 28) & kBottom29Bits; + carry = out[i] >> 29; + out[i] &= kBottom29Bits; + + i++; + out[i] = tmp2[i + 9] >> 1; + out[i] += carry; + carry = out[i] >> 28; + out[i] &= kBottom28Bits; + } + + out[8] = tmp2[17]; + out[8] += carry; + carry = out[8] >> 29; + out[8] &= kBottom29Bits; + + felem_reduce_carry(out, carry); +} + +/* felem_square sets out=in*in. + * + * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29. + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ +static void felem_square(felem out, const felem in) { + u64 tmp[17]; + + tmp[0] = ((u64) in[0]) * in[0]; + tmp[1] = ((u64) in[0]) * (in[1] << 1); + tmp[2] = ((u64) in[0]) * (in[2] << 1) + + ((u64) in[1]) * (in[1] << 1); + tmp[3] = ((u64) in[0]) * (in[3] << 1) + + ((u64) in[1]) * (in[2] << 1); + tmp[4] = ((u64) in[0]) * (in[4] << 1) + + ((u64) in[1]) * (in[3] << 2) + ((u64) in[2]) * in[2]; + tmp[5] = ((u64) in[0]) * (in[5] << 1) + ((u64) in[1]) * + (in[4] << 1) + ((u64) in[2]) * (in[3] << 1); + tmp[6] = ((u64) in[0]) * (in[6] << 1) + ((u64) in[1]) * + (in[5] << 2) + ((u64) in[2]) * (in[4] << 1) + + ((u64) in[3]) * (in[3] << 1); + tmp[7] = ((u64) in[0]) * (in[7] << 1) + ((u64) in[1]) * + (in[6] << 1) + ((u64) in[2]) * (in[5] << 1) + + ((u64) in[3]) * (in[4] << 1); + /* tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60, + * which is < 2**64 as required. */ + tmp[8] = ((u64) in[0]) * (in[8] << 1) + ((u64) in[1]) * + (in[7] << 2) + ((u64) in[2]) * (in[6] << 1) + + ((u64) in[3]) * (in[5] << 2) + ((u64) in[4]) * in[4]; + tmp[9] = ((u64) in[1]) * (in[8] << 1) + ((u64) in[2]) * + (in[7] << 1) + ((u64) in[3]) * (in[6] << 1) + + ((u64) in[4]) * (in[5] << 1); + tmp[10] = ((u64) in[2]) * (in[8] << 1) + ((u64) in[3]) * + (in[7] << 2) + ((u64) in[4]) * (in[6] << 1) + + ((u64) in[5]) * (in[5] << 1); + tmp[11] = ((u64) in[3]) * (in[8] << 1) + ((u64) in[4]) * + (in[7] << 1) + ((u64) in[5]) * (in[6] << 1); + tmp[12] = ((u64) in[4]) * (in[8] << 1) + + ((u64) in[5]) * (in[7] << 2) + ((u64) in[6]) * in[6]; + tmp[13] = ((u64) in[5]) * (in[8] << 1) + + ((u64) in[6]) * (in[7] << 1); + tmp[14] = ((u64) in[6]) * (in[8] << 1) + + ((u64) in[7]) * (in[7] << 1); + tmp[15] = ((u64) in[7]) * (in[8] << 1); + tmp[16] = ((u64) in[8]) * in[8]; + + felem_reduce_degree(out, tmp); +} + +/* felem_mul sets out=in*in2. + * + * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and + * in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ +static void felem_mul(felem out, const felem in, const felem in2) { + u64 tmp[17]; + + tmp[0] = ((u64) in[0]) * in2[0]; + tmp[1] = ((u64) in[0]) * (in2[1] << 0) + + ((u64) in[1]) * (in2[0] << 0); + tmp[2] = ((u64) in[0]) * (in2[2] << 0) + ((u64) in[1]) * + (in2[1] << 1) + ((u64) in[2]) * (in2[0] << 0); + tmp[3] = ((u64) in[0]) * (in2[3] << 0) + ((u64) in[1]) * + (in2[2] << 0) + ((u64) in[2]) * (in2[1] << 0) + + ((u64) in[3]) * (in2[0] << 0); + tmp[4] = ((u64) in[0]) * (in2[4] << 0) + ((u64) in[1]) * + (in2[3] << 1) + ((u64) in[2]) * (in2[2] << 0) + + ((u64) in[3]) * (in2[1] << 1) + + ((u64) in[4]) * (in2[0] << 0); + tmp[5] = ((u64) in[0]) * (in2[5] << 0) + ((u64) in[1]) * + (in2[4] << 0) + ((u64) in[2]) * (in2[3] << 0) + + ((u64) in[3]) * (in2[2] << 0) + ((u64) in[4]) * + (in2[1] << 0) + ((u64) in[5]) * (in2[0] << 0); + tmp[6] = ((u64) in[0]) * (in2[6] << 0) + ((u64) in[1]) * + (in2[5] << 1) + ((u64) in[2]) * (in2[4] << 0) + + ((u64) in[3]) * (in2[3] << 1) + ((u64) in[4]) * + (in2[2] << 0) + ((u64) in[5]) * (in2[1] << 1) + + ((u64) in[6]) * (in2[0] << 0); + tmp[7] = ((u64) in[0]) * (in2[7] << 0) + ((u64) in[1]) * + (in2[6] << 0) + ((u64) in[2]) * (in2[5] << 0) + + ((u64) in[3]) * (in2[4] << 0) + ((u64) in[4]) * + (in2[3] << 0) + ((u64) in[5]) * (in2[2] << 0) + + ((u64) in[6]) * (in2[1] << 0) + + ((u64) in[7]) * (in2[0] << 0); + /* tmp[8] has the greatest value but doesn't overflow. See logic in + * felem_square. */ + tmp[8] = ((u64) in[0]) * (in2[8] << 0) + ((u64) in[1]) * + (in2[7] << 1) + ((u64) in[2]) * (in2[6] << 0) + + ((u64) in[3]) * (in2[5] << 1) + ((u64) in[4]) * + (in2[4] << 0) + ((u64) in[5]) * (in2[3] << 1) + + ((u64) in[6]) * (in2[2] << 0) + ((u64) in[7]) * + (in2[1] << 1) + ((u64) in[8]) * (in2[0] << 0); + tmp[9] = ((u64) in[1]) * (in2[8] << 0) + ((u64) in[2]) * + (in2[7] << 0) + ((u64) in[3]) * (in2[6] << 0) + + ((u64) in[4]) * (in2[5] << 0) + ((u64) in[5]) * + (in2[4] << 0) + ((u64) in[6]) * (in2[3] << 0) + + ((u64) in[7]) * (in2[2] << 0) + + ((u64) in[8]) * (in2[1] << 0); + tmp[10] = ((u64) in[2]) * (in2[8] << 0) + ((u64) in[3]) * + (in2[7] << 1) + ((u64) in[4]) * (in2[6] << 0) + + ((u64) in[5]) * (in2[5] << 1) + ((u64) in[6]) * + (in2[4] << 0) + ((u64) in[7]) * (in2[3] << 1) + + ((u64) in[8]) * (in2[2] << 0); + tmp[11] = ((u64) in[3]) * (in2[8] << 0) + ((u64) in[4]) * + (in2[7] << 0) + ((u64) in[5]) * (in2[6] << 0) + + ((u64) in[6]) * (in2[5] << 0) + ((u64) in[7]) * + (in2[4] << 0) + ((u64) in[8]) * (in2[3] << 0); + tmp[12] = ((u64) in[4]) * (in2[8] << 0) + ((u64) in[5]) * + (in2[7] << 1) + ((u64) in[6]) * (in2[6] << 0) + + ((u64) in[7]) * (in2[5] << 1) + + ((u64) in[8]) * (in2[4] << 0); + tmp[13] = ((u64) in[5]) * (in2[8] << 0) + ((u64) in[6]) * + (in2[7] << 0) + ((u64) in[7]) * (in2[6] << 0) + + ((u64) in[8]) * (in2[5] << 0); + tmp[14] = ((u64) in[6]) * (in2[8] << 0) + ((u64) in[7]) * + (in2[7] << 1) + ((u64) in[8]) * (in2[6] << 0); + tmp[15] = ((u64) in[7]) * (in2[8] << 0) + + ((u64) in[8]) * (in2[7] << 0); + tmp[16] = ((u64) in[8]) * (in2[8] << 0); + + felem_reduce_degree(out, tmp); +} + +static void felem_assign(felem out, const felem in) { + memcpy(out, in, sizeof(felem)); +} + +/* felem_scalar_3 sets out=3*out. + * + * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ +static void felem_scalar_3(felem out) { + limb carry = 0; + unsigned i; + + for (i = 0;; i++) { + out[i] *= 3; + out[i] += carry; + carry = out[i] >> 29; + out[i] &= kBottom29Bits; + + i++; + if (i == NLIMBS) + break; + + out[i] *= 3; + out[i] += carry; + carry = out[i] >> 28; + out[i] &= kBottom28Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_scalar_4 sets out=4*out. + * + * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ +static void felem_scalar_4(felem out) { + limb carry = 0, next_carry; + unsigned i; + + for (i = 0;; i++) { + next_carry = out[i] >> 27; + out[i] <<= 2; + out[i] &= kBottom29Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 29); + out[i] &= kBottom29Bits; + + i++; + if (i == NLIMBS) + break; + + next_carry = out[i] >> 26; + out[i] <<= 2; + out[i] &= kBottom28Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 28); + out[i] &= kBottom28Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_scalar_8 sets out=8*out. + * + * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. + * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ +static void felem_scalar_8(felem out) { + limb carry = 0, next_carry; + unsigned i; + + for (i = 0;; i++) { + next_carry = out[i] >> 26; + out[i] <<= 3; + out[i] &= kBottom29Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 29); + out[i] &= kBottom29Bits; + + i++; + if (i == NLIMBS) + break; + + next_carry = out[i] >> 25; + out[i] <<= 3; + out[i] &= kBottom28Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 28); + out[i] &= kBottom28Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_is_zero_vartime returns 1 iff |in| == 0. It takes a variable amount of + * time depending on the value of |in|. */ +static char felem_is_zero_vartime(const felem in) { + limb carry; + int i; + limb tmp[NLIMBS]; + + felem_assign(tmp, in); + + /* First, reduce tmp to a minimal form. */ + do { + carry = 0; + for (i = 0;; i++) { + tmp[i] += carry; + carry = tmp[i] >> 29; + tmp[i] &= kBottom29Bits; + + i++; + if (i == NLIMBS) + break; + + tmp[i] += carry; + carry = tmp[i] >> 28; + tmp[i] &= kBottom28Bits; + } + + felem_reduce_carry(tmp, carry); + } while (carry); + + /* tmp < 2**257, so the only possible zero values are 0, p and 2p. */ + return memcmp(tmp, kZero, sizeof(tmp)) == 0 || + memcmp(tmp, kP, sizeof(tmp)) == 0 || + memcmp(tmp, k2P, sizeof(tmp)) == 0; +} + + +/* Montgomery operations: */ + +#define kRDigits {2, 0, 0, 0xfffffffe, 0xffffffff, 0xffffffff, 0xfffffffd, 1} // 2^257 mod p256.p + +#define kRInvDigits {0x80000000, 1, 0xffffffff, 0, 0x80000001, 0xfffffffe, 1, 0x7fffffff} // 1 / 2^257 mod p256.p + +static const cryptonite_p256_int kR = { kRDigits }; +static const cryptonite_p256_int kRInv = { kRInvDigits }; + +/* to_montgomery sets out = R*in. */ +static void to_montgomery(felem out, const cryptonite_p256_int* in) { + cryptonite_p256_int in_shifted; + int i; + + cryptonite_p256_init(&in_shifted); + cryptonite_p256_modmul(&cryptonite_SECP256r1_p, in, 0, &kR, &in_shifted); + + for (i = 0; i < NLIMBS; i++) { + if ((i & 1) == 0) { + out[i] = P256_DIGIT(&in_shifted, 0) & kBottom29Bits; + cryptonite_p256_shr(&in_shifted, 29, &in_shifted); + } else { + out[i] = P256_DIGIT(&in_shifted, 0) & kBottom28Bits; + cryptonite_p256_shr(&in_shifted, 28, &in_shifted); + } + } + + cryptonite_p256_clear(&in_shifted); +} + +/* from_montgomery sets out=in/R. */ +static void from_montgomery(cryptonite_p256_int* out, const felem in) { + cryptonite_p256_int result, tmp; + int i, top; + + cryptonite_p256_init(&result); + cryptonite_p256_init(&tmp); + + cryptonite_p256_add_d(&tmp, in[NLIMBS - 1], &result); + for (i = NLIMBS - 2; i >= 0; i--) { + if ((i & 1) == 0) { + top = cryptonite_p256_shl(&result, 29, &tmp); + } else { + top = cryptonite_p256_shl(&result, 28, &tmp); + } + top |= cryptonite_p256_add_d(&tmp, in[i], &result); + } + + cryptonite_p256_modmul(&cryptonite_SECP256r1_p, &kRInv, top, &result, out); + + cryptonite_p256_clear(&result); + cryptonite_p256_clear(&tmp); +} diff --git a/cbits/include64/p256/p256.h b/cbits/include64/p256/p256.h new file mode 100644 index 0000000..c043957 --- /dev/null +++ b/cbits/include64/p256/p256.h @@ -0,0 +1,167 @@ +/* + * Copyright 2013 The Android Open Source Project + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * * Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * * Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * * Neither the name of Google Inc. nor the names of its contributors may + * be used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY Google Inc. ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF + * MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO + * EVENT SHALL Google Inc. BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, + * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, + * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; + * OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, + * WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR + * OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF + * ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +#ifndef SYSTEM_CORE_INCLUDE_MINCRYPT_LITE_P256_H_ +#define SYSTEM_CORE_INCLUDE_MINCRYPT_LITE_P256_H_ + +// Collection of routines manipulating 256 bit unsigned integers. +// Just enough to implement ecdsa-p256 and related algorithms. + +#include + +#ifdef __cplusplus +extern "C" { +#endif + +#define P256_BITSPERDIGIT 64 +#define P256_NDIGITS 4 +#define P256_NBYTES 32 + +// n' such as n * n' = -1 mod (2^64) +#define P256_MONTGOMERY_FACTOR 0xCCD1C8AAEE00BC4F + +#define P256_LITERAL(lo,hi) (((uint32_t) (lo)) + (((uint64_t) (hi)) << 32)) + +typedef int cryptonite_p256_err; +typedef uint64_t cryptonite_p256_digit; +typedef int64_t cryptonite_p256_sdigit; +typedef __uint128_t cryptonite_p256_ddigit; +typedef __int128_t cryptonite_p256_sddigit; + +// Defining cryptonite_p256_int as struct to leverage struct assigment. +typedef struct { + cryptonite_p256_digit a[P256_NDIGITS]; +} cryptonite_p256_int; + +extern const cryptonite_p256_int cryptonite_SECP256r1_n; // Curve order +extern const cryptonite_p256_int cryptonite_SECP256r1_p; // Curve prime +extern const cryptonite_p256_int cryptonite_SECP256r1_b; // Curve param + +// Initialize a cryptonite_p256_int to zero. +void cryptonite_p256_init(cryptonite_p256_int* a); + +// Clear a cryptonite_p256_int to zero. +void cryptonite_p256_clear(cryptonite_p256_int* a); + +// Return bit. Index 0 is least significant. +int cryptonite_p256_get_bit(const cryptonite_p256_int* a, int index); + +// b := a % MOD +void cryptonite_p256_mod( + const cryptonite_p256_int* MOD, + const cryptonite_p256_int* a, + cryptonite_p256_int* b); + +// c := a * (top_b | b) % MOD +void cryptonite_p256_modmul( + const cryptonite_p256_int* MOD, + const cryptonite_p256_int* a, + const cryptonite_p256_digit top_b, + const cryptonite_p256_int* b, + cryptonite_p256_int* c); + +// b := 1 / a % MOD +// MOD best be SECP256r1_n +void cryptonite_p256_modinv( + const cryptonite_p256_int* MOD, + const cryptonite_p256_int* a, + cryptonite_p256_int* b); + +// b := 1 / a % MOD +// MOD best be SECP256r1_n +// Faster than cryptonite_p256_modinv() +void cryptonite_p256_modinv_vartime( + const cryptonite_p256_int* MOD, + const cryptonite_p256_int* a, + cryptonite_p256_int* b); + +// b := a << (n % P256_BITSPERDIGIT) +// Returns the bits shifted out of most significant digit. +cryptonite_p256_digit cryptonite_p256_shl(const cryptonite_p256_int* a, int n, cryptonite_p256_int* b); + +// b := a >> (n % P256_BITSPERDIGIT) +void cryptonite_p256_shr(const cryptonite_p256_int* a, int n, cryptonite_p256_int* b); + +int cryptonite_p256_is_zero(const cryptonite_p256_int* a); +int cryptonite_p256_is_odd(const cryptonite_p256_int* a); +int cryptonite_p256_is_even(const cryptonite_p256_int* a); + +// Returns -1, 0 or 1. +int cryptonite_p256_cmp(const cryptonite_p256_int* a, const cryptonite_p256_int *b); + +// c: = a - b +// Returns -1 on borrow. +int cryptonite_p256_sub(const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c); + +// c := a + b +// Returns 1 on carry. +int cryptonite_p256_add(const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c); + +// c := a + (single digit)b +// Returns carry 1 on carry. +int cryptonite_p256_add_d(const cryptonite_p256_int* a, cryptonite_p256_digit b, cryptonite_p256_int* c); + +// ec routines. + +// {out_x,out_y} := nG +void cryptonite_p256_base_point_mul(const cryptonite_p256_int *n, + cryptonite_p256_int *out_x, + cryptonite_p256_int *out_y); + +// {out_x,out_y} := n{in_x,in_y} +void cryptonite_p256_point_mul(const cryptonite_p256_int *n, + const cryptonite_p256_int *in_x, + const cryptonite_p256_int *in_y, + cryptonite_p256_int *out_x, + cryptonite_p256_int *out_y); + +// {out_x,out_y} := n1G + n2{in_x,in_y} +void cryptonite_p256_points_mul_vartime( + const cryptonite_p256_int *n1, const cryptonite_p256_int *n2, + const cryptonite_p256_int *in_x, const cryptonite_p256_int *in_y, + cryptonite_p256_int *out_x, cryptonite_p256_int *out_y); + +// Return whether point {x,y} is on curve. +int cryptonite_p256_is_valid_point(const cryptonite_p256_int* x, const cryptonite_p256_int* y); + +// Outputs big-endian binary form. No leading zero skips. +void cryptonite_p256_to_bin(const cryptonite_p256_int* src, uint8_t dst[P256_NBYTES]); + +// Reads from big-endian binary form, +// thus pre-pad with leading zeros if short. +void cryptonite_p256_from_bin(const uint8_t src[P256_NBYTES], cryptonite_p256_int* dst); + +#define P256_DIGITS(x) ((x)->a) +#define P256_DIGIT(x,y) ((x)->a[y]) + +#define P256_ZERO {{0}} +#define P256_ONE {{1}} + +#ifdef __cplusplus +} +#endif + +#endif // SYSTEM_CORE_INCLUDE_MINCRYPT_LITE_P256_H_ diff --git a/cbits/include64/p256/p256_gf.h b/cbits/include64/p256/p256_gf.h new file mode 100644 index 0000000..28b869a --- /dev/null +++ b/cbits/include64/p256/p256_gf.h @@ -0,0 +1,713 @@ +/* + * Copyright 2013 The Android Open Source Project + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are met: + * * Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * * Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * * Neither the name of Google Inc. nor the names of its contributors may + * be used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY Google Inc. ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF + * MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO + * EVENT SHALL Google Inc. BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, + * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, + * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; + * OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, + * WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR + * OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF + * ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +// This is an implementation of the P256 finite field. It's written to be +// portable and still constant-time. +// +// WARNING: Implementing these functions in a constant-time manner is far from +// obvious. Be careful when touching this code. +// +// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background. + +#include +#include + +#include +#include + +#include "p256/p256.h" + +typedef uint8_t u8; +typedef uint32_t u32; +typedef uint64_t u64; +typedef int64_t s64; +typedef __uint128_t u128; + +/* Our field elements are represented as five 64-bit limbs. + * + * The value of an felem (field element) is: + * x[0] + (x[1] * 2**51) + (x[2] * 2**103) + ... + (x[4] * 2**206) + * + * That is, each limb is alternately 51 or 52-bits wide in little-endian + * order. + * + * This means that an felem hits 2**257, rather than 2**256 as we would like. + * + * Finally, the values stored in an felem are in Montgomery form. So the value + * |y| is stored as (y*R) mod p, where p is the P-256 prime and R is 2**257. + */ +typedef u64 limb; +#define NLIMBS 5 +typedef limb felem[NLIMBS]; + +static const limb kBottom51Bits = 0x7ffffffffffff; +static const limb kBottom52Bits = 0xfffffffffffff; + +/* kOne is the number 1 as an felem. It's 2**257 mod p split up into 51 and + * 52-bit words. */ +static const felem kOne = { + 2, 0xfc00000000000, 0x7ffffffffffff, 0xfff7fffffffff, 0x7ffff +}; +static const felem kZero = {0}; +static const felem kP = { + 0x7ffffffffffff, 0x1fffffffffff, 0, 0x4000000000, 0x3fffffffc0000 +}; +static const felem k2P = { + 0x7fffffffffffe, 0x3fffffffffff, 0, 0x8000000000, 0x7fffffff80000 +}; +/* kPrecomputed contains precomputed values to aid the calculation of scalar + * multiples of the base point, G. It's actually two, equal length, tables + * concatenated. + * + * The first table contains (x,y) felem pairs for 16 multiples of the base + * point, G. + * + * Index | Index (binary) | Value + * 0 | 0000 | 0G (all zeros, omitted) + * 1 | 0001 | G + * 2 | 0010 | 2**64G + * 3 | 0011 | 2**64G + G + * 4 | 0100 | 2**128G + * 5 | 0101 | 2**128G + G + * 6 | 0110 | 2**128G + 2**64G + * 7 | 0111 | 2**128G + 2**64G + G + * 8 | 1000 | 2**192G + * 9 | 1001 | 2**192G + G + * 10 | 1010 | 2**192G + 2**64G + * 11 | 1011 | 2**192G + 2**64G + G + * 12 | 1100 | 2**192G + 2**128G + * 13 | 1101 | 2**192G + 2**128G + G + * 14 | 1110 | 2**192G + 2**128G + 2**64G + * 15 | 1111 | 2**192G + 2**128G + 2**64G + G + * + * The second table follows the same style, but the terms are 2**32G, + * 2**96G, 2**160G, 2**224G. + * + * This is ~2KB of data. */ +static const limb kPrecomputed[NLIMBS * 2 * 15 * 2] = { + 0x661a831522878, 0xf17fb6d805e79, 0x5889441d6ea57, 0xae33cfdb995bb, 0xc482fbb529ba, + 0x4a6af9d2aac15, 0x90e867917377c, 0x487cc962d2ae3, 0xec2a97443446e, 0x2b8ff8c52c42, + 0x45f8a2d41a576, 0xb06988d2653e4, 0x718b22c357305, 0x33fc920e79d2b, 0x17af34b0fe8db, + 0x38e17eb402f2f, 0x3382558649705, 0x47f6d48f482d1, 0x7bd42488d9b83, 0x3b247c8b86b78, + 0x4d08fc26f7778, 0x7a29a82fb2795, 0x75cd18f90d11a, 0xad8e213b0bc, 0x2d5f0142899e8, + 0x506f98098fb57, 0x2f0c98301e4aa, 0x39b30dd5cf67d, 0x9c146498ab13c, 0xa5db92df5b7b, + 0x184897fc4124a, 0xe3f73a19d8aa, 0x4e1c18e47066b, 0x27b2d4b52eaee, 0x30eac3ea10e99, + 0x4e74546e2e7d5, 0x1f4dde2d97a1d, 0x6ead0f88e1200, 0x7dec87c220f02, 0x3d08ff096310f, + 0x23e5659633ffa, 0x6ec648f08c722, 0x3172a3806ea35, 0xf6e5b681eb3c5, 0x2c3758260f89d, + 0x38dca4fd1da12, 0xf06067b78830d, 0x3194be87a068c, 0x78893c7eb602b, 0xcead60438432, + 0x6ee69a56a67ab, 0xd886f77701895, 0x67b0a4d9cee2b, 0x3586bbf3e4d53, 0x1db6f32921d93, + 0x260756ca4b366, 0x4f40e9d2039fa, 0x4f3f09f5a82bf, 0xccde2d641e8cd, 0x305a30cd2e8c5, + 0x471c235cb5439, 0xab279cd962f5a, 0x17e1fb6e2dd94, 0xfe64589800a77, 0xe8793d99775f, + 0x48c62f4e614aa, 0xbf76ef20eb2a4, 0x669c672556c, 0x24683e0eff056, 0x12252b369ab76, + 0x821de9f162d5, 0xf911ec99a95be, 0x6721f065c906b, 0x58d452035c736, 0x1f9f01a6a15, + 0x6135009b7d8d3, 0xdaeeeb417dfc0, 0x63865fea0ee17, 0x6e0a304b939d6, 0x204ba2076833d, + 0x4ade586f35669, 0x2c1077e34611a, 0x5b1a3bea3b81a, 0xf97d018a22c8b, 0x38d7996b08af8, + 0x6ea62baeb7aa0, 0xebdcbd9ef2670, 0x35dc8fe0df3fe, 0xe458309d20c24, 0x11e87898716a0, + 0x7c44bab7cb456, 0xd64d3cf1bb64, 0x189bff1bf9e66, 0xb5218a049311, 0x285dda6cbcc81, + 0x3238dcafd8c7c, 0x607736c8de0, 0xdb83d99508b1, 0x4e1a0d404cd81, 0x1588008c00ff2, + 0x16b8b36722b27, 0x876609c3f3f1a, 0x66b72ef0e17d6, 0x705f8a279d568, 0x2eaac4cd01fdd, + 0x1171ce9705fe9, 0xffc79cd3264ee, 0x700c8ab4b80f0, 0x208d3d4f57a1, 0x337262a8ca4eb, + 0x297fd01d843fd, 0xa90956fa097f8, 0x529759fdb3845, 0x1d78c5e2d0397, 0x3d6938a4adbf3, + 0x16d5853560b66, 0xf138946b9a430, 0x2ab79f4dea6a0, 0xd42053ee43ae1, 0x3b9c3ef1cf870, + 0x598934ad81baf, 0x5f1821b1d07a7, 0x416bb3a973ff3, 0x23f07bd0a047a, 0x19bdc2e09f786, + 0x56dc9981cd51f, 0xfbace23c8cd65, 0x673bd3bf5b52e, 0x46a95d229fd61, 0xe09ad64bcfb1, + 0xe5292b91f17d, 0xfeefcd8afc287, 0x58f52b0a58711, 0x4800f20c201ef, 0x2084fce608f67, + 0x12ba0b128ae0b, 0x5977ae17030b4, 0x101126ee420f6, 0xf70823495c6bd, 0xde19a27d7770, + 0x5c6ac852260e8, 0x9d22950ac4356, 0x441cca955246c, 0x660a34e5332d9, 0x14ac8ea92f8d2, + 0x6b6d7709f307e, 0x67d7e13879db, 0x2ea8626f9fbbd, 0x99609006a4b40, 0x31bb2a8f8c779, + 0x10c04828ea335, 0xae9acdcbc080a, 0x617af2342607a, 0xc7494ea53e553, 0x2ca9e2872defa, + 0x6c399fab21f1f, 0xab139b245e758, 0x3ad933dcba589, 0x4797fecb08811, 0x31f5dbf8f594, + 0x7dc6361cc7a69, 0xc8a7953ead3f9, 0x79ed693d18015, 0x418a024999a6a, 0x2c4fdc9436aa, + 0x1eb98cb06aa75, 0x2989592796a9c, 0x11194821e425, 0xe27a648228388, 0x35d834b6c12a0, + 0x541807713b532, 0x7ae0a1008aaee, 0x7017a29bcb5e, 0x6b193c23c315c, 0x19bd25ac82f2a, + 0x6a01a43eef294, 0xddf5b5fd84f19, 0x33f5ba081c016, 0xdeb052d1bc082, 0x6b2f06afa617, + 0x7ca1eda6a939f, 0xbdeb35997b50c, 0x47f2d1bccda5, 0xc2ff4adfed667, 0x87712997be4, + 0x21fc2e2b37659, 0xf7d62cd5ed951, 0x27fa9cbdf7efa, 0xba25582bf3a6b, 0x2a42b8bd89398, + 0x6d377d07eecd2, 0x9ca1df5af387, 0x1109e3427e2ba, 0xce4aa4572a19, 0x103baaef71e16, + 0x2c3b2dfde328a, 0xbec4b4a30e1ef, 0x37d92a86204f3, 0x806cfde68eb39, 0x246e2f72b8aa5, + 0x68d3de93462a9, 0x53b8acba6bbc3, 0x2492a70fa1696, 0x38c62d5760f55, 0x15096fe4904f2, + 0x4e44e9bed3e3a, 0xb28bfd79cc9bc, 0x6a77513839320, 0x480dcec6739db, 0x3601b739f2465, + 0x43c348e2a7e1, 0xe448106327879, 0x175d9cae1b0ed, 0xd3b89dee743b8, 0x392d73ca255bc, + 0x32946db0d3a18, 0x9261b09907cc, 0x5ba517a755722, 0x51f24fdaf5184, 0x1cdc732989ed8, + 0x2f7806ba16694, 0xae0c9f029f8d0, 0xd8b45102ce1, 0xca1c7db9316d6, 0x162088a67066f, + 0x39de35b2b4162, 0xa19f550d88ae9, 0x7921b27026cde, 0x94b936b66e900, 0x1023bd5fa17fc, + 0x436837814cfa4, 0x29113492283c4, 0x66d1cdd8b51d8, 0xa540702278eb2, 0x47ef1b29285d, + 0x587b50917e50e, 0xb4cda75bab3b, 0x112520b0a9886, 0x66b9ac16fee49, 0x17bf17e92b2eb, + 0x2456a2f150ed7, 0xfa214412d0280, 0x3ca7dd947fe5b, 0xa72c28598d58a, 0x255d945efc3e, + 0x2873f04e0f215, 0x74178fd1af57b, 0x788848b5b2d6, 0xb1ffafaae0db6, 0x32a1b7b3cbb2a, + 0x4bd9935d6b2da, 0x9c08f24ad30a5, 0x4e58407a80f, 0x1b3a3825a5b17, 0x6547e9fc82f5, + 0x47484aa3656c3, 0x6ee43f341a494, 0x64a98f87adea2, 0x619b3f8e95f01, 0xb6e513266ed8, + 0x421c2a673090, 0xa1c1de32348c7, 0x55b85c3a1e8a3, 0xe05ce8ef330b4, 0x2561e49c15d84, + 0x40aa2d33130fa, 0x12b827d35866f, 0xfe4cf62c8ddb, 0x2fa0ef05bb28d, 0x1c06ca63f1cb8, + 0x32a971863863b, 0xff6fc86830da1, 0x71e7b25a14cf3, 0xea9c5ebb1373a, 0x250bbaa3e1634, + 0x5b5ffeda5b765, 0xf25d2a746331b, 0x115e3a3f43632, 0x67303af43c9d5, 0x14bb538a0e559, + 0x75623687d43b7, 0xa349674a4b38d, 0x613c61829ffc6, 0x689828d8110c7, 0x139115f5af7d5, + 0xf1d856152289, 0x45cbe967168ab, 0x51f38e1680901, 0x34808e8f652b0, 0x1f4a6a921e156, + 0x35dfaf3d8341f, 0xf53ace725cb63, 0x3d86a54eef35b, 0xa103aabaffe2c, 0x2decc36296fbd, + 0x510282be73d6f, 0xd4e6365db206a, 0x4bdc5f5bb8bf3, 0xde7ea32a3aee7, 0x71269e274305, +}; + + +/* Field element operations: */ + +/* NON_ZERO_TO_ALL_ONES returns: + * 0xffffffffffffffff for 0 < x <= 2**63 + * 0 for x == 0 or x > 2**63. + * + * x must be a u64 or an equivalent type such as limb. */ +#define NON_ZERO_TO_ALL_ONES(x) ((((u64)(x) - 1) >> 63) - 1) + +/* felem_reduce_carry adds a multiple of p in order to cancel |carry|, + * which is a term at 2**257. + * + * On entry: carry < 2**6, inout[0,2,...] < 2**51, inout[1,3,...] < 2**52. + * On exit: inout[0,2,..] < 2**52, inout[1,3,...] < 2**53. */ +static void felem_reduce_carry(felem inout, limb carry) { + const u64 carry_mask = NON_ZERO_TO_ALL_ONES(carry); + + inout[0] += carry << 1; + inout[1] += 0x10000000000000 & carry_mask; + /* carry < 2**6 thus (carry << 46) < 2**52 and we added 2**52 in the + * previous line therefore this doesn't underflow. */ + inout[1] -= carry << 46; + inout[2] += (0x8000000000000 - 1) & carry_mask; + inout[3] += (0x10000000000000 - 1) & carry_mask; + inout[3] -= carry << 39; + /* This may underflow if carry is non-zero but, if so, we'll fix it in the + * next line. */ + inout[4] -= 1 & carry_mask; + inout[4] += carry << 19; +} + +/* felem_sum sets out = in+in2. + * + * On entry, in[i]+in2[i] must not overflow a 64-bit word. + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53 */ +static void felem_sum(felem out, const felem in, const felem in2) { + limb carry = 0; + unsigned i; + + for (i = 0;; i++) { + out[i] = in[i] + in2[i]; + out[i] += carry; + carry = out[i] >> 51; + out[i] &= kBottom51Bits; + + i++; + if (i == NLIMBS) + break; + + out[i] = in[i] + in2[i]; + out[i] += carry; + carry = out[i] >> 52; + out[i] &= kBottom52Bits; + } + + felem_reduce_carry(out, carry); +} + +#define two53m3 (((limb)1) << 53) - (((limb)1) << 3) +#define two54m52p48m2 (((limb)1) << 54) - (((limb)1) << 52) + (((limb)1) << 48) - (((limb)1) << 2) +#define two53m2p0 (((limb)1) << 53) - (((limb)1) << 2) + (((limb)1) << 0) +#define two54m52p41m2 (((limb)1) << 54) - (((limb)1) << 52) + (((limb)1) << 41) - (((limb)1) << 2) +#define two53m21m2p0 (((limb)1) << 53) - (((limb)1) << 21) - (((limb)1) << 2) + (((limb)1) << 0) + +/* zero53 is 0 mod p. */ +static const felem zero53 = { two53m3, two54m52p48m2, two53m2p0, two54m52p41m2, two53m21m2p0 }; + +/* felem_diff sets out = in-in2. + * + * On entry: in[0,2,...] < 2**52, in[1,3,...] < 2**53 and + * in2[0,2,...] < 2**52, in2[1,3,...] < 2**53. + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53. */ +static void felem_diff(felem out, const felem in, const felem in2) { + limb carry = 0; + unsigned i; + + for (i = 0;; i++) { + out[i] = in[i] - in2[i]; + out[i] += zero53[i]; + out[i] += carry; + carry = out[i] >> 51; + out[i] &= kBottom51Bits; + + i++; + if (i == NLIMBS) + break; + + out[i] = in[i] - in2[i]; + out[i] += zero53[i]; + out[i] += carry; + carry = out[i] >> 52; + out[i] &= kBottom52Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_reduce_degree sets out = tmp/R mod p where tmp contains 64-bit words + * with the same 51,52,... bit positions as an felem. + * + * The values in felems are in Montgomery form: x*R mod p where R = 2**257. + * Since we just multiplied two Montgomery values together, the result is + * x*y*R*R mod p. We wish to divide by R in order for the result also to be + * in Montgomery form. + * + * On entry: tmp[i] < 2**128 + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53 */ +static void felem_reduce_degree(felem out, u128 tmp[9]) { + /* The following table may be helpful when reading this code: + * + * Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 + * Width (bits): 51| 52| 51| 52| 51| 52| 51| 52| 51| 52| 51 + * Start bit: 0 | 51|103|154|206|257|309|360|412|463|515 + * (odd phase): 0 | 52|103|155|206|258|309|361|412|464|515 */ + limb tmp2[10], carry, x, xShiftedMask; + unsigned i; + + /* tmp contains 128-bit words with the same 51,52,51-bit positions as an + * felem. So the top of an element of tmp might overlap with another + * element two positions down. The following loop eliminates this + * overlap. */ + tmp2[0] = (limb)(tmp[0] & kBottom51Bits); + + /* In the following we use "(limb) tmp[x]" and "(limb) (tmp[x]>>64)" to try + * and hint to the compiler that it can do a single-word shift by selecting + * the right register rather than doing a double-word shift and truncating + * afterwards. */ + tmp2[1] = ((limb) tmp[0]) >> 51; + tmp2[1] |= (((limb)(tmp[0] >> 64)) << 13) & kBottom52Bits; + tmp2[1] += ((limb) tmp[1]) & kBottom52Bits; + carry = tmp2[1] >> 52; + tmp2[1] &= kBottom52Bits; + + for (i = 2; i < 9; i++) { + tmp2[i] = ((limb)(tmp[i - 2] >> 64)) >> 39; + tmp2[i] += ((limb)(tmp[i - 1])) >> 52; + tmp2[i] += (((limb)(tmp[i - 1] >> 64)) << 12) & kBottom51Bits; + tmp2[i] += ((limb) tmp[i]) & kBottom51Bits; + tmp2[i] += carry; + carry = tmp2[i] >> 51; + tmp2[i] &= kBottom51Bits; + + i++; + if (i == 9) + break; + tmp2[i] = ((limb)(tmp[i - 2] >> 64)) >> 39; + tmp2[i] += ((limb)(tmp[i - 1])) >> 51; + tmp2[i] += (((limb)(tmp[i - 1] >> 64)) << 13) & kBottom52Bits; + tmp2[i] += ((limb) tmp[i]) & kBottom52Bits; + tmp2[i] += carry; + carry = tmp2[i] >> 52; + tmp2[i] &= kBottom52Bits; + } + + tmp2[9] = ((limb)(tmp[7] >> 64)) >> 39; + tmp2[9] += ((limb)(tmp[8])) >> 51; + tmp2[9] += (((limb)(tmp[8] >> 64)) << 13); + tmp2[9] += carry; + + /* Montgomery elimination of terms. + * + * Since R is 2**257, we can divide by R with a bitwise shift if we can + * ensure that the right-most 257 bits are all zero. We can make that true by + * adding multiplies of p without affecting the value. + * + * So we eliminate limbs from right to left. Since the bottom 51 bits of p + * are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0. + * We can do that for 8 further limbs and then right shift to eliminate the + * extra factor of R. */ + for (i = 0;; i += 2) { + tmp2[i + 1] += tmp2[i] >> 51; + x = tmp2[i] & kBottom51Bits; + xShiftedMask = NON_ZERO_TO_ALL_ONES(x >> 1); + tmp2[i] = 0; + + /* The bounds calculations for this loop are tricky. Each iteration of + * the loop eliminates two words by adding values to words to their + * right. + * + * The following table contains the amounts added to each word (as an + * offset from the value of i at the top of the loop). The amounts are + * accounted for from the first and second half of the loop separately + * and are written as, for example, 51 to mean a value <2**51. + * + * Word: 1 2 3 4 5 6 + * Added in top half: 52 44 52 37 50 + * 51 + * 51 + * Added in bottom half: 51 45 51 38 50 + * 52 + * 52 + * + * The value that is currently offset 5 will be offset 3 for the next + * iteration and then offset 1 for the iteration after that. Therefore + * the total value added will be the values added at 5, 3 and 1. + * + * The following table accumulates these values. The sums at the bottom + * are written as, for example, 53+45, to mean a value < 2**53+2**45. + * + * Word: 1 2 3 4 5 6 7 8 9 + * 52 44 52 37 50 50 50 50 50 + * 51 45 51 38 37 38 37 + * 52 51 52 51 52 51 + * 51 52 51 52 51 + * 44 52 51 52 + * 51 45 44 + * 52 + * ------------------------------------ + * 53+ 53+ 54+ 52+ 53+ 52+ + * 45 44+ 50+ 51+ 52+ 50+ + * 37 45+ 50+ 50+ 37 + * 38 44+ 38 + * 37 + * + * So the greatest amount is added to tmp2[5]. If tmp2[5] has an initial + * value of <2**52, then the maximum value will be < 2**54 + 2**52 + 2**50 + + * 2**45 + 2**38, which is < 2**64, as required. */ + tmp2[i + 1] += (x << 45) & kBottom52Bits; + tmp2[i + 2] += x >> 7; + + tmp2[i + 3] += (x << 38) & kBottom52Bits; + tmp2[i + 4] += x >> 14; + + /* On tmp2[i + 4], when x < 2**1, the subtraction with (x << 18) will not + * underflow because it is balanced with the (x << 50) term. On the next + * word tmp2[i + 5], terms with (x >> 1) and (x >> 33) are both zero and + * there is no underflow either. + * + * When x >= 2**1, we add 2**51 to tmp2[i + 4] to avoid an underflow. + * Removing 1 from tmp2[i + 5] is safe because (x >> 1) - (x >> 33) is + * strictly positive. + */ + tmp2[i + 4] += 0x8000000000000 & xShiftedMask; + tmp2[i + 5] -= 1 & xShiftedMask; + + tmp2[i + 4] -= (x << 18) & kBottom51Bits; + tmp2[i + 4] += (x << 50) & kBottom51Bits; + tmp2[i + 5] += (x >> 1) - (x >> 33); + + if (i+1 == NLIMBS) + break; + tmp2[i + 2] += tmp2[i + 1] >> 52; + x = tmp2[i + 1] & kBottom52Bits; + xShiftedMask = NON_ZERO_TO_ALL_ONES(x >> 2); + tmp2[i + 1] = 0; + + tmp2[i + 2] += (x << 44) & kBottom51Bits; + tmp2[i + 3] += x >> 7; + + tmp2[i + 4] += (x << 37) & kBottom51Bits; + tmp2[i + 5] += x >> 14; + + /* On tmp2[i + 5], when x < 2**2, the subtraction with (x << 18) will not + * underflow because it is balanced with the (x << 50) term. On the next + * word tmp2[i + 6], terms with (x >> 2) and (x >> 34) are both zero and + * there is no underflow either. + * + * When x >= 2**2, we add 2**52 to tmp2[i + 5] to avoid an underflow. + * Removing 1 from tmp2[i + 6] is safe because (x >> 2) - (x >> 34) is + * stricly positive. + */ + tmp2[i + 5] += 0x10000000000000 & xShiftedMask; + tmp2[i + 6] -= 1 & xShiftedMask; + + tmp2[i + 5] -= (x << 18) & kBottom52Bits; + tmp2[i + 5] += (x << 50) & kBottom52Bits; + tmp2[i + 6] += (x >> 2) - (x >> 34); + } + + /* We merge the right shift with a carry chain. The words above 2**257 have + * widths of 52,51,... which we need to correct when copying them down. */ + carry = 0; + for (i = 0; i < 4; i++) { + out[i] = tmp2[i + 5]; + out[i] += carry; + carry = out[i] >> 51; + out[i] &= kBottom51Bits; + + i++; + out[i] = tmp2[i + 5] << 1; + out[i] += carry; + carry = out[i] >> 52; + out[i] &= kBottom52Bits; + } + + out[4] = tmp2[9]; + out[4] += carry; + carry = out[4] >> 51; + out[4] &= kBottom51Bits; + + felem_reduce_carry(out, carry); +} + +/* felem_square sets out=in*in. + * + * On entry: in[0,2,...] < 2**52, in[1,3,...] < 2**53. + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53. */ +static void felem_square(felem out, const felem in) { + u128 tmp[9], x1x1, x3x3; + + x1x1 = ((u128) in[1]) * in[1]; + x3x3 = ((u128) in[3]) * in[3]; + + tmp[0] = ((u128) in[0]) * (in[0] << 0); + tmp[1] = ((u128) in[0]) * (in[1] << 1) + ((x1x1 & 1) << 51); + tmp[2] = ((u128) in[0]) * (in[2] << 1) + (x1x1 >> 1); + tmp[3] = ((u128) in[0]) * (in[3] << 1) + + ((u128) in[1]) * (in[2] << 1); + tmp[4] = ((u128) in[0]) * (in[4] << 1) + + ((u128) in[1]) * (in[3] << 0) + + ((u128) in[2]) * (in[2] << 0); + tmp[5] = ((u128) in[1]) * (in[4] << 1) + + ((u128) in[2]) * (in[3] << 1) + ((x3x3 & 1) << 51); + tmp[6] = ((u128) in[2]) * (in[4] << 1) + (x3x3 >> 1); + tmp[7] = ((u128) in[3]) * (in[4] << 1); + tmp[8] = ((u128) in[4]) * (in[4] << 0); + + felem_reduce_degree(out, tmp); +} + +/* felem_mul sets out=in*in2. + * + * On entry: in[0,2,...] < 2**52, in[1,3,...] < 2**53 and + * in2[0,2,...] < 2**52, in2[1,3,...] < 2**53. + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53. */ +static void felem_mul(felem out, const felem in, const felem in2) { + u128 tmp[9], x1y1, x1y3, x3y1, x3y3; + + x1y1 = ((u128) in[1]) * in2[1]; + x1y3 = ((u128) in[1]) * in2[3]; + x3y1 = ((u128) in[3]) * in2[1]; + x3y3 = ((u128) in[3]) * in2[3]; + + tmp[0] = ((u128) in[0]) * in2[0]; + tmp[1] = ((u128) in[0]) * in2[1] + + ((u128) in[1]) * in2[0] + ((x1y1 & 1) << 51); + tmp[2] = ((u128) in[0]) * in2[2] + (x1y1 >> 1) + + ((u128) in[2]) * in2[0]; + tmp[3] = ((u128) in[0]) * in2[3] + + ((u128) in[1]) * in2[2] + + ((u128) in[2]) * in2[1] + ((x1y3 & 1) << 51) + + ((u128) in[3]) * in2[0] + ((x3y1 & 1) << 51); + tmp[4] = ((u128) in[0]) * in2[4] + (x1y3 >> 1) + + ((u128) in[2]) * in2[2] + (x3y1 >> 1) + + ((u128) in[4]) * in2[0]; + tmp[5] = ((u128) in[1]) * in2[4] + + ((u128) in[2]) * in2[3] + + ((u128) in[3]) * in2[2] + + ((u128) in[4]) * in2[1] + ((x3y3 & 1) << 51); + tmp[6] = ((u128) in[2]) * in2[4] + (x3y3 >> 1) + + ((u128) in[4]) * in2[2]; + tmp[7] = ((u128) in[3]) * in2[4] + + ((u128) in[4]) * in2[3]; + tmp[8] = ((u128) in[4]) * in2[4]; + + felem_reduce_degree(out, tmp); +} + +static void felem_assign(felem out, const felem in) { + memcpy(out, in, sizeof(felem)); +} + +/* felem_scalar_3 sets out=3*out. + * + * On entry: out[0,2,...] < 2**52, out[1,3,...] < 2**53. + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53. */ +static void felem_scalar_3(felem out) { + limb carry = 0; + unsigned i; + + for (i = 0;; i++) { + out[i] *= 3; + out[i] += carry; + carry = out[i] >> 51; + out[i] &= kBottom51Bits; + + i++; + if (i == NLIMBS) + break; + + out[i] *= 3; + out[i] += carry; + carry = out[i] >> 52; + out[i] &= kBottom52Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_scalar_4 sets out=4*out. + * + * On entry: out[0,2,...] < 2**52, out[1,3,...] < 2**53. + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53. */ +static void felem_scalar_4(felem out) { + limb carry = 0, next_carry; + unsigned i; + + for (i = 0;; i++) { + next_carry = out[i] >> 49; + out[i] <<= 2; + out[i] &= kBottom51Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 51); + out[i] &= kBottom51Bits; + + i++; + if (i == NLIMBS) + break; + + next_carry = out[i] >> 50; + out[i] <<= 2; + out[i] &= kBottom52Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 52); + out[i] &= kBottom52Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_scalar_8 sets out=8*out. + * + * On entry: out[0,2,...] < 2**52, out[1,3,...] < 2**53. + * On exit: out[0,2,...] < 2**52, out[1,3,...] < 2**53. */ +static void felem_scalar_8(felem out) { + limb carry = 0, next_carry; + unsigned i; + + for (i = 0;; i++) { + next_carry = out[i] >> 48; + out[i] <<= 3; + out[i] &= kBottom51Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 51); + out[i] &= kBottom51Bits; + + i++; + if (i == NLIMBS) + break; + + next_carry = out[i] >> 49; + out[i] <<= 3; + out[i] &= kBottom52Bits; + out[i] += carry; + carry = next_carry + (out[i] >> 52); + out[i] &= kBottom52Bits; + } + + felem_reduce_carry(out, carry); +} + +/* felem_is_zero_vartime returns 1 iff |in| == 0. It takes a variable amount of + * time depending on the value of |in|. */ +static char felem_is_zero_vartime(const felem in) { + limb carry; + int i; + limb tmp[NLIMBS]; + + felem_assign(tmp, in); + + /* First, reduce tmp to a minimal form. */ + do { + carry = 0; + for (i = 0;; i++) { + tmp[i] += carry; + carry = tmp[i] >> 51; + tmp[i] &= kBottom51Bits; + + i++; + if (i == NLIMBS) + break; + + tmp[i] += carry; + carry = tmp[i] >> 52; + tmp[i] &= kBottom52Bits; + } + + felem_reduce_carry(tmp, carry); + } while (carry); + + /* tmp < 2**257, so the only possible zero values are 0, p and 2p. */ + return memcmp(tmp, kZero, sizeof(tmp)) == 0 || + memcmp(tmp, kP, sizeof(tmp)) == 0 || + memcmp(tmp, k2P, sizeof(tmp)) == 0; +} + + +/* Montgomery operations: */ + +#define kRDigits {2, 0xfffffffe00000000, 0xffffffffffffffff, 0x1fffffffd} // 2^257 mod p256.p + +#define kRInvDigits {0x180000000, 0xffffffff, 0xfffffffe80000001, 0x7fffffff00000001} // 1 / 2^257 mod p256.p + +static const cryptonite_p256_int kR = { kRDigits }; +static const cryptonite_p256_int kRInv = { kRInvDigits }; + +/* to_montgomery sets out = R*in. */ +static void to_montgomery(felem out, const cryptonite_p256_int* in) { + cryptonite_p256_int in_shifted; + int i; + + cryptonite_p256_init(&in_shifted); + cryptonite_p256_modmul(&cryptonite_SECP256r1_p, in, 0, &kR, &in_shifted); + + for (i = 0; i < NLIMBS; i++) { + if ((i & 1) == 0) { + out[i] = P256_DIGIT(&in_shifted, 0) & kBottom51Bits; + cryptonite_p256_shr(&in_shifted, 51, &in_shifted); + } else { + out[i] = P256_DIGIT(&in_shifted, 0) & kBottom52Bits; + cryptonite_p256_shr(&in_shifted, 52, &in_shifted); + } + } + + cryptonite_p256_clear(&in_shifted); +} + +/* from_montgomery sets out=in/R. */ +static void from_montgomery(cryptonite_p256_int* out, const felem in) { + cryptonite_p256_int result, tmp; + int i, top; + + cryptonite_p256_init(&result); + cryptonite_p256_init(&tmp); + + cryptonite_p256_add_d(&tmp, in[NLIMBS - 1], &result); + for (i = NLIMBS - 2; i >= 0; i--) { + if ((i & 1) == 0) { + top = cryptonite_p256_shl(&result, 51, &tmp); + } else { + top = cryptonite_p256_shl(&result, 52, &tmp); + } + top += cryptonite_p256_add_d(&tmp, in[i], &result); + } + + cryptonite_p256_modmul(&cryptonite_SECP256r1_p, &kRInv, top, &result, out); + + cryptonite_p256_clear(&result); + cryptonite_p256_clear(&tmp); +} diff --git a/cbits/p256/p256.c b/cbits/p256/p256.c index 8dad6ef..4d79c0e 100644 --- a/cbits/p256/p256.c +++ b/cbits/p256/p256.c @@ -25,7 +25,7 @@ */ // This is an implementation of the P256 elliptic curve group. It's written to -// be portable 32-bit, although it's still constant-time. +// be portable and still constant-time. // // WARNING: Implementing these functions in a constant-time manner is far from // obvious. Be careful when touching this code. @@ -40,14 +40,16 @@ #include "p256/p256.h" const cryptonite_p256_int cryptonite_SECP256r1_n = // curve order - {{0xfc632551, 0xf3b9cac2, 0xa7179e84, 0xbce6faad, -1, -1, 0, -1}}; + {{P256_LITERAL(0xfc632551, 0xf3b9cac2), P256_LITERAL(0xa7179e84, 0xbce6faad), + P256_LITERAL(-1, -1), P256_LITERAL(0, -1)}}; const cryptonite_p256_int cryptonite_SECP256r1_p = // curve field size - {{-1, -1, -1, 0, 0, 0, 1, -1 }}; + {{P256_LITERAL(-1, -1), P256_LITERAL(-1, 0), + P256_LITERAL(0, 0), P256_LITERAL(1, -1) }}; const cryptonite_p256_int cryptonite_SECP256r1_b = // curve b - {{0x27d2604b, 0x3bce3c3e, 0xcc53b0f6, 0x651d06b0, - 0x769886bc, 0xb3ebbd55, 0xaa3a93e7, 0x5ac635d8}}; + {{P256_LITERAL(0x27d2604b, 0x3bce3c3e), P256_LITERAL(0xcc53b0f6, 0x651d06b0), + P256_LITERAL(0x769886bc, 0xb3ebbd55), P256_LITERAL(0xaa3a93e7, 0x5ac635d8)}}; void cryptonite_p256_init(cryptonite_p256_int* a) { memset(a, 0, sizeof(*a)); @@ -61,9 +63,10 @@ int cryptonite_p256_get_bit(const cryptonite_p256_int* scalar, int bit) { } int cryptonite_p256_is_zero(const cryptonite_p256_int* a) { - int i, result = 0; + cryptonite_p256_digit result = 0; + int i = 0; for (i = 0; i < P256_NDIGITS; ++i) result |= P256_DIGIT(a, i); - return !result; + return result == 0; } // top, c[] += a[] * b @@ -167,6 +170,10 @@ void cryptonite_p256_modmul(const cryptonite_p256_int* MOD, // top can be any value at this point. // Guestimate reducer as top * MOD, since msw of MOD is -1. top_reducer = mulAdd(MOD, top, 0, reducer); +#if P256_BITSPERDIGIT > 32 + // Correction when msw of MOD has only high 32 bits set + top_reducer += mulAdd(MOD, top >> 32, 0, reducer); +#endif // Subtract reducer from top | tmp. top = subTop(top_reducer, reducer, top, tmp + i); @@ -229,7 +236,7 @@ static void cryptonite_p256_shr1(const cryptonite_p256_int* a, int highbit, cryp P256_DIGIT(b, i) = accu; } P256_DIGIT(b, i) = (P256_DIGIT(a, i) >> 1) | - (highbit << (P256_BITSPERDIGIT - 1)); + (((cryptonite_p256_sdigit) highbit) << (P256_BITSPERDIGIT - 1)); } // Return -1, 0, 1 for a < b, a == b or a > b respectively. @@ -359,31 +366,32 @@ int cryptonite_p256_is_valid_point(const cryptonite_p256_int* x, const cryptonit } void cryptonite_p256_from_bin(const uint8_t src[P256_NBYTES], cryptonite_p256_int* dst) { - int i; + int i, n; const uint8_t* p = &src[0]; for (i = P256_NDIGITS - 1; i >= 0; --i) { - P256_DIGIT(dst, i) = - (p[0] << 24) | - (p[1] << 16) | - (p[2] << 8) | - p[3]; - p += 4; + cryptonite_p256_digit dig = 0; + n = P256_BITSPERDIGIT; + while (n > 0) { + n -= 8; + dig |= ((cryptonite_p256_digit) *(p++)) << n; + } + P256_DIGIT(dst, i) = dig; } } void cryptonite_p256_to_bin(const cryptonite_p256_int* src, uint8_t dst[P256_NBYTES]) { - int i; + int i, n; uint8_t* p = &dst[0]; for (i = P256_NDIGITS -1; i >= 0; --i) { const cryptonite_p256_digit dig = P256_DIGIT(src, i); - p[0] = dig >> 24; - p[1] = dig >> 16; - p[2] = dig >> 8; - p[3] = dig; - p += 4; + n = P256_BITSPERDIGIT; + while (n > 0) { + n -= 8; + *(p++) = dig >> n; + } } } @@ -395,6 +403,7 @@ void cryptonite_p256_to_bin(const cryptonite_p256_int* src, uint8_t dst[P256_NBY // c = a + b mod MOD void cryptonite_p256e_modadd(const cryptonite_p256_int* MOD, const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c) { + assert(c); /* avoid repeated checks inside inlined cryptonite_p256_add */ cryptonite_p256_digit top = cryptonite_p256_add(a, b, c); top = subM(MOD, top, P256_DIGITS(c), -1); top = subM(MOD, top, P256_DIGITS(c), MSB_COMPLEMENT(top)); @@ -403,15 +412,13 @@ void cryptonite_p256e_modadd(const cryptonite_p256_int* MOD, const cryptonite_p2 // c = a - b mod MOD void cryptonite_p256e_modsub(const cryptonite_p256_int* MOD, const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c) { + assert(c); /* avoid repeated checks inside inlined cryptonite_p256_sub */ cryptonite_p256_digit top = cryptonite_p256_sub(a, b, c); top = addM(MOD, top, P256_DIGITS(c), ~MSB_COMPLEMENT(top)); top = subM(MOD, top, P256_DIGITS(c), MSB_COMPLEMENT(top)); addM(MOD, 0, P256_DIGITS(c), top); } -// n' such as n * n' = -1 mod (2^32) -#define MONTGOMERY_FACTOR 0xEE00BC4F - #define NTH_DOUBLE_THEN_ADD(i, a, nth, b, out) \ cryptonite_p256e_montmul(a, a, out); \ for (i = 1; i < nth; i++) \ @@ -419,8 +426,8 @@ void cryptonite_p256e_modsub(const cryptonite_p256_int* MOD, const cryptonite_p2 cryptonite_p256e_montmul(out, b, out); const cryptonite_p256_int cryptonite_SECP256r1_r2 = // r^2 mod n - {{0xBE79EEA2, 0x83244C95, 0x49BD6FA6, 0x4699799C, - 0x2B6BEC59, 0x2845B239, 0xF3D95620, 0x66E12D94}}; + {{P256_LITERAL(0xBE79EEA2, 0x83244C95), P256_LITERAL(0x49BD6FA6, 0x4699799C), + P256_LITERAL(0x2B6BEC59, 0x2845B239), P256_LITERAL(0xF3D95620, 0x66E12D94)}}; const cryptonite_p256_int cryptonite_SECP256r1_one = {{1}}; @@ -443,7 +450,7 @@ static void cryptonite_p256e_montmul(const cryptonite_p256_int* a, const crypton } accum[j] = chain; - mand = accum[0] * MONTGOMERY_FACTOR; + mand = accum[0] * P256_MONTGOMERY_FACTOR; chain = 0; mier = P256_DIGITS(&cryptonite_SECP256r1_n); for (j=0; j -#include - -#include -#include - -#include "p256/p256.h" - -typedef uint8_t u8; -typedef uint32_t u32; -typedef int32_t s32; -typedef uint64_t u64; - -/* Our field elements are represented as nine 32-bit limbs. - * - * The value of an felem (field element) is: - * x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228) - * - * That is, each limb is alternately 29 or 28-bits wide in little-endian - * order. - * - * This means that an felem hits 2**257, rather than 2**256 as we would like. A - * 28, 29, ... pattern would cause us to hit 2**256, but that causes problems - * when multiplying as terms end up one bit short of a limb which would require - * much bit-shifting to correct. - * - * Finally, the values stored in an felem are in Montgomery form. So the value - * |y| is stored as (y*R) mod p, where p is the P-256 prime and R is 2**257. - */ -typedef u32 limb; -#define NLIMBS 9 -typedef limb felem[NLIMBS]; - -static const limb kBottom28Bits = 0xfffffff; -static const limb kBottom29Bits = 0x1fffffff; - -/* kOne is the number 1 as an felem. It's 2**257 mod p split up into 29 and - * 28-bit words. */ -static const felem kOne = { - 2, 0, 0, 0xffff800, - 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, - 0 -}; -static const felem kZero = {0}; -static const felem kP = { - 0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, - 0, 0, 0x200000, 0xf000000, - 0xfffffff -}; -static const felem k2P = { - 0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, - 0, 0, 0x400000, 0xe000000, - 0x1fffffff -}; -/* kPrecomputed contains precomputed values to aid the calculation of scalar - * multiples of the base point, G. It's actually two, equal length, tables - * concatenated. - * - * The first table contains (x,y) felem pairs for 16 multiples of the base - * point, G. - * - * Index | Index (binary) | Value - * 0 | 0000 | 0G (all zeros, omitted) - * 1 | 0001 | G - * 2 | 0010 | 2**64G - * 3 | 0011 | 2**64G + G - * 4 | 0100 | 2**128G - * 5 | 0101 | 2**128G + G - * 6 | 0110 | 2**128G + 2**64G - * 7 | 0111 | 2**128G + 2**64G + G - * 8 | 1000 | 2**192G - * 9 | 1001 | 2**192G + G - * 10 | 1010 | 2**192G + 2**64G - * 11 | 1011 | 2**192G + 2**64G + G - * 12 | 1100 | 2**192G + 2**128G - * 13 | 1101 | 2**192G + 2**128G + G - * 14 | 1110 | 2**192G + 2**128G + 2**64G - * 15 | 1111 | 2**192G + 2**128G + 2**64G + G - * - * The second table follows the same style, but the terms are 2**32G, - * 2**96G, 2**160G, 2**224G. - * - * This is ~2KB of data. */ -static const limb kPrecomputed[NLIMBS * 2 * 15 * 2] = { - 0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7edc, 0xd4a6eab, 0x3120bee, - 0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba21, 0x14b10bb, 0xae3fe3, - 0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe49073, 0x3fa36cc, 0x5ebcd2c, - 0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea12446, 0xe1ade1e, 0xec91f22, - 0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109, 0xa267a00, 0xb57c050, - 0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0x7d6dee7, 0x2976e4b, - 0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a5a9, 0x843a649, 0xc3ab0fa, - 0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11, 0x58c43df, 0xf423fc2, - 0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db40f, 0x83e277d, 0xb0dd609, - 0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5, 0xe10c9e, 0x33ab581, - 0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f, 0x48764cd, 0x76dbcca, - 0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b20, 0x4ba3173, 0xc168c33, - 0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0, 0x65dd7ff, 0x3a1e4f6, - 0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f077, 0xa6add89, 0x4894acd, - 0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a, 0x69a8556, 0x7e7c0, - 0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c, 0xda0cf5b, 0x812e881, - 0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51, 0xc22be3e, 0xe35e65a, - 0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9, 0x1c5a839, 0x47a1e26, - 0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c502, 0x2f32042, 0xa17769b, - 0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a02, 0x3fc93, 0x5620023, - 0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c, 0x407f75c, 0xbaab133, - 0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea7, 0x3293ac0, 0xcdc98aa, - 0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16, 0x2b6fcc7, 0xf5a4e29, - 0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72, 0x73e1c35, 0xee70fbc, - 0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85, 0x27de188, 0x66f70b8, - 0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae914, 0x2f3ec51, 0x3826b59, - 0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x823d9d2, 0x8213f39, - 0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4a, 0xf5ddc3d, 0x3786689, - 0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a729, 0x4be3499, 0x52b23aa, - 0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb048035, 0xe31de66, 0xc6ecaa3, - 0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a7529, 0xcb7beb1, 0xb2a78a1, - 0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff658, 0xe3d6511, 0xc7d76f, - 0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c, 0x50daa90, 0xb13f72, - 0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d32411, 0xb04a838, 0xd760d2d, - 0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11e, 0x20bca9a, 0x66f496b, - 0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d, 0xbe985f7, 0x1acbc1a, - 0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56ff, 0x65ef930, 0x21dc4a, - 0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15f, 0x624e62e, 0xa90ae2f, - 0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522b, 0xdc78583, 0x40eeabb, - 0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef34, 0xae2a960, 0x91b8bdc, - 0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0x2413c8e, 0x5425bf9, - 0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633, 0x7c91952, 0xd806dce, - 0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef73, 0x8956f34, 0xe4b5cf2, - 0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7, 0x627b614, 0x7371cca, - 0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc9, 0x9c19bf2, 0x5882229, - 0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b3, 0xe85ff25, 0x408ef57, - 0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113, 0xa4a1769, 0x11fbc6c, - 0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b7, 0x4acbad9, 0x5efc5fa, - 0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc, 0x7bf0fa9, 0x957651, - 0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57, 0xf2ecaac, 0xca86dec, - 0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c12d, 0xf20bd46, 0x1951fa7, - 0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74, 0x99bb618, 0x2db944c, - 0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e74779, 0x576138, 0x9587927, - 0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782d, 0xfc72e0b, 0x701b298, - 0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5d8, 0xf858d3a, 0x942eea8, - 0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a1, 0x8395659, 0x52ed4e2, - 0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c0, 0x6bdf55a, 0x4e4457d, - 0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747b, 0x878558d, 0x7d29aa4, - 0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d7, 0xa5bef68, 0xb7b30d8, - 0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78, -}; +#include "p256/p256_gf.h" /* Field element operations: */ -/* NON_ZERO_TO_ALL_ONES returns: - * 0xffffffff for 0 < x <= 2**31 - * 0 for x == 0 or x > 2**31. - * - * x must be a u32 or an equivalent type such as limb. */ -#define NON_ZERO_TO_ALL_ONES(x) ((((u32)(x) - 1) >> 31) - 1) - -/* felem_reduce_carry adds a multiple of p in order to cancel |carry|, - * which is a term at 2**257. - * - * On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28. - * On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29. */ -static void felem_reduce_carry(felem inout, limb carry) { - const u32 carry_mask = NON_ZERO_TO_ALL_ONES(carry); - - inout[0] += carry << 1; - inout[3] += 0x10000000 & carry_mask; - /* carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the - * previous line therefore this doesn't underflow. */ - inout[3] -= carry << 11; - inout[4] += (0x20000000 - 1) & carry_mask; - inout[5] += (0x10000000 - 1) & carry_mask; - inout[6] += (0x20000000 - 1) & carry_mask; - inout[6] -= carry << 22; - /* This may underflow if carry is non-zero but, if so, we'll fix it in the - * next line. */ - inout[7] -= 1 & carry_mask; - inout[7] += carry << 25; -} - -/* felem_sum sets out = in+in2. - * - * On entry, in[i]+in2[i] must not overflow a 32-bit word. - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 */ -static void felem_sum(felem out, const felem in, const felem in2) { - limb carry = 0; - unsigned i; - - for (i = 0;; i++) { - out[i] = in[i] + in2[i]; - out[i] += carry; - carry = out[i] >> 29; - out[i] &= kBottom29Bits; - - i++; - if (i == NLIMBS) - break; - - out[i] = in[i] + in2[i]; - out[i] += carry; - carry = out[i] >> 28; - out[i] &= kBottom28Bits; - } - - felem_reduce_carry(out, carry); -} - -#define two31m3 (((limb)1) << 31) - (((limb)1) << 3) -#define two30m2 (((limb)1) << 30) - (((limb)1) << 2) -#define two30p13m2 (((limb)1) << 30) + (((limb)1) << 13) - (((limb)1) << 2) -#define two31m2 (((limb)1) << 31) - (((limb)1) << 2) -#define two31p24m2 (((limb)1) << 31) + (((limb)1) << 24) - (((limb)1) << 2) -#define two30m27m2 (((limb)1) << 30) - (((limb)1) << 27) - (((limb)1) << 2) - -/* zero31 is 0 mod p. */ -static const felem zero31 = { two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2 }; - -/* felem_diff sets out = in-in2. - * - * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and - * in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ -static void felem_diff(felem out, const felem in, const felem in2) { - limb carry = 0; - unsigned i; - - for (i = 0;; i++) { - out[i] = in[i] - in2[i]; - out[i] += zero31[i]; - out[i] += carry; - carry = out[i] >> 29; - out[i] &= kBottom29Bits; - - i++; - if (i == NLIMBS) - break; - - out[i] = in[i] - in2[i]; - out[i] += zero31[i]; - out[i] += carry; - carry = out[i] >> 28; - out[i] &= kBottom28Bits; - } - - felem_reduce_carry(out, carry); -} - -/* felem_reduce_degree sets out = tmp/R mod p where tmp contains 64-bit words - * with the same 29,28,... bit positions as an felem. - * - * The values in felems are in Montgomery form: x*R mod p where R = 2**257. - * Since we just multiplied two Montgomery values together, the result is - * x*y*R*R mod p. We wish to divide by R in order for the result also to be - * in Montgomery form. - * - * On entry: tmp[i] < 2**64 - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 */ -static void felem_reduce_degree(felem out, u64 tmp[17]) { - /* The following table may be helpful when reading this code: - * - * Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10... - * Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29 - * Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285 - * (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285 */ - limb tmp2[18], carry, x, xMask; - unsigned i; - - /* tmp contains 64-bit words with the same 29,28,29-bit positions as an - * felem. So the top of an element of tmp might overlap with another - * element two positions down. The following loop eliminates this - * overlap. */ - tmp2[0] = (limb)(tmp[0] & kBottom29Bits); - - /* In the following we use "(limb) tmp[x]" and "(limb) (tmp[x]>>32)" to try - * and hint to the compiler that it can do a single-word shift by selecting - * the right register rather than doing a double-word shift and truncating - * afterwards. */ - tmp2[1] = ((limb) tmp[0]) >> 29; - tmp2[1] |= (((limb)(tmp[0] >> 32)) << 3) & kBottom28Bits; - tmp2[1] += ((limb) tmp[1]) & kBottom28Bits; - carry = tmp2[1] >> 28; - tmp2[1] &= kBottom28Bits; - - for (i = 2; i < 17; i++) { - tmp2[i] = ((limb)(tmp[i - 2] >> 32)) >> 25; - tmp2[i] += ((limb)(tmp[i - 1])) >> 28; - tmp2[i] += (((limb)(tmp[i - 1] >> 32)) << 4) & kBottom29Bits; - tmp2[i] += ((limb) tmp[i]) & kBottom29Bits; - tmp2[i] += carry; - carry = tmp2[i] >> 29; - tmp2[i] &= kBottom29Bits; - - i++; - if (i == 17) - break; - tmp2[i] = ((limb)(tmp[i - 2] >> 32)) >> 25; - tmp2[i] += ((limb)(tmp[i - 1])) >> 29; - tmp2[i] += (((limb)(tmp[i - 1] >> 32)) << 3) & kBottom28Bits; - tmp2[i] += ((limb) tmp[i]) & kBottom28Bits; - tmp2[i] += carry; - carry = tmp2[i] >> 28; - tmp2[i] &= kBottom28Bits; - } - - tmp2[17] = ((limb)(tmp[15] >> 32)) >> 25; - tmp2[17] += ((limb)(tmp[16])) >> 29; - tmp2[17] += (((limb)(tmp[16] >> 32)) << 3); - tmp2[17] += carry; - - /* Montgomery elimination of terms. - * - * Since R is 2**257, we can divide by R with a bitwise shift if we can - * ensure that the right-most 257 bits are all zero. We can make that true by - * adding multiplies of p without affecting the value. - * - * So we eliminate limbs from right to left. Since the bottom 29 bits of p - * are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0. - * We can do that for 8 further limbs and then right shift to eliminate the - * extra factor of R. */ - for (i = 0;; i += 2) { - tmp2[i + 1] += tmp2[i] >> 29; - x = tmp2[i] & kBottom29Bits; - xMask = NON_ZERO_TO_ALL_ONES(x); - tmp2[i] = 0; - - /* The bounds calculations for this loop are tricky. Each iteration of - * the loop eliminates two words by adding values to words to their - * right. - * - * The following table contains the amounts added to each word (as an - * offset from the value of i at the top of the loop). The amounts are - * accounted for from the first and second half of the loop separately - * and are written as, for example, 28 to mean a value <2**28. - * - * Word: 3 4 5 6 7 8 9 10 - * Added in top half: 28 11 29 21 29 28 - * 28 29 - * 29 - * Added in bottom half: 29 10 28 21 28 28 - * 29 - * - * The value that is currently offset 7 will be offset 5 for the next - * iteration and then offset 3 for the iteration after that. Therefore - * the total value added will be the values added at 7, 5 and 3. - * - * The following table accumulates these values. The sums at the bottom - * are written as, for example, 29+28, to mean a value < 2**29+2**28. - * - * Word: 3 4 5 6 7 8 9 10 11 12 13 - * 28 11 10 29 21 29 28 28 28 28 28 - * 29 28 11 28 29 28 29 28 29 28 - * 29 28 21 21 29 21 29 21 - * 10 29 28 21 28 21 28 - * 28 29 28 29 28 29 28 - * 11 10 29 10 29 10 - * 29 28 11 28 11 - * 29 29 - * -------------------------------------------- - * 30+ 31+ 30+ 31+ 30+ - * 28+ 29+ 28+ 29+ 21+ - * 21+ 28+ 21+ 28+ 10 - * 10 21+ 10 21+ - * 11 11 - * - * So the greatest amount is added to tmp2[10] and tmp2[12]. If - * tmp2[10/12] has an initial value of <2**29, then the maximum value - * will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32, - * as required. */ - tmp2[i + 3] += (x << 10) & kBottom28Bits; - tmp2[i + 4] += (x >> 18); - - tmp2[i + 6] += (x << 21) & kBottom29Bits; - tmp2[i + 7] += x >> 8; - - /* At position 200, which is the starting bit position for word 7, we - * have a factor of 0xf000000 = 2**28 - 2**24. */ - tmp2[i + 7] += 0x10000000 & xMask; - /* Word 7 is 28 bits wide, so the 2**28 term exactly hits word 8. */ - tmp2[i + 8] += (x - 1) & xMask; - tmp2[i + 7] -= (x << 24) & kBottom28Bits; - tmp2[i + 8] -= x >> 4; - - tmp2[i + 8] += 0x20000000 & xMask; - tmp2[i + 8] -= x; - tmp2[i + 8] += (x << 28) & kBottom29Bits; - tmp2[i + 9] += ((x >> 1) - 1) & xMask; - - if (i+1 == NLIMBS) - break; - tmp2[i + 2] += tmp2[i + 1] >> 28; - x = tmp2[i + 1] & kBottom28Bits; - xMask = NON_ZERO_TO_ALL_ONES(x); - tmp2[i + 1] = 0; - - tmp2[i + 4] += (x << 11) & kBottom29Bits; - tmp2[i + 5] += (x >> 18); - - tmp2[i + 7] += (x << 21) & kBottom28Bits; - tmp2[i + 8] += x >> 7; - - /* At position 199, which is the starting bit of the 8th word when - * dealing with a context starting on an odd word, we have a factor of - * 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th - * word from i+1 is i+8. */ - tmp2[i + 8] += 0x20000000 & xMask; - tmp2[i + 9] += (x - 1) & xMask; - tmp2[i + 8] -= (x << 25) & kBottom29Bits; - tmp2[i + 9] -= x >> 4; - - tmp2[i + 9] += 0x10000000 & xMask; - tmp2[i + 9] -= x; - tmp2[i + 10] += (x - 1) & xMask; - } - - /* We merge the right shift with a carry chain. The words above 2**257 have - * widths of 28,29,... which we need to correct when copying them down. */ - carry = 0; - for (i = 0; i < 8; i++) { - /* The maximum value of tmp2[i + 9] occurs on the first iteration and - * is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is - * therefore safe. */ - out[i] = tmp2[i + 9]; - out[i] += carry; - out[i] += (tmp2[i + 10] << 28) & kBottom29Bits; - carry = out[i] >> 29; - out[i] &= kBottom29Bits; - - i++; - out[i] = tmp2[i + 9] >> 1; - out[i] += carry; - carry = out[i] >> 28; - out[i] &= kBottom28Bits; - } - - out[8] = tmp2[17]; - out[8] += carry; - carry = out[8] >> 29; - out[8] &= kBottom29Bits; - - felem_reduce_carry(out, carry); -} - -/* felem_square sets out=in*in. - * - * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29. - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ -static void felem_square(felem out, const felem in) { - u64 tmp[17]; - - tmp[0] = ((u64) in[0]) * in[0]; - tmp[1] = ((u64) in[0]) * (in[1] << 1); - tmp[2] = ((u64) in[0]) * (in[2] << 1) + - ((u64) in[1]) * (in[1] << 1); - tmp[3] = ((u64) in[0]) * (in[3] << 1) + - ((u64) in[1]) * (in[2] << 1); - tmp[4] = ((u64) in[0]) * (in[4] << 1) + - ((u64) in[1]) * (in[3] << 2) + ((u64) in[2]) * in[2]; - tmp[5] = ((u64) in[0]) * (in[5] << 1) + ((u64) in[1]) * - (in[4] << 1) + ((u64) in[2]) * (in[3] << 1); - tmp[6] = ((u64) in[0]) * (in[6] << 1) + ((u64) in[1]) * - (in[5] << 2) + ((u64) in[2]) * (in[4] << 1) + - ((u64) in[3]) * (in[3] << 1); - tmp[7] = ((u64) in[0]) * (in[7] << 1) + ((u64) in[1]) * - (in[6] << 1) + ((u64) in[2]) * (in[5] << 1) + - ((u64) in[3]) * (in[4] << 1); - /* tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60, - * which is < 2**64 as required. */ - tmp[8] = ((u64) in[0]) * (in[8] << 1) + ((u64) in[1]) * - (in[7] << 2) + ((u64) in[2]) * (in[6] << 1) + - ((u64) in[3]) * (in[5] << 2) + ((u64) in[4]) * in[4]; - tmp[9] = ((u64) in[1]) * (in[8] << 1) + ((u64) in[2]) * - (in[7] << 1) + ((u64) in[3]) * (in[6] << 1) + - ((u64) in[4]) * (in[5] << 1); - tmp[10] = ((u64) in[2]) * (in[8] << 1) + ((u64) in[3]) * - (in[7] << 2) + ((u64) in[4]) * (in[6] << 1) + - ((u64) in[5]) * (in[5] << 1); - tmp[11] = ((u64) in[3]) * (in[8] << 1) + ((u64) in[4]) * - (in[7] << 1) + ((u64) in[5]) * (in[6] << 1); - tmp[12] = ((u64) in[4]) * (in[8] << 1) + - ((u64) in[5]) * (in[7] << 2) + ((u64) in[6]) * in[6]; - tmp[13] = ((u64) in[5]) * (in[8] << 1) + - ((u64) in[6]) * (in[7] << 1); - tmp[14] = ((u64) in[6]) * (in[8] << 1) + - ((u64) in[7]) * (in[7] << 1); - tmp[15] = ((u64) in[7]) * (in[8] << 1); - tmp[16] = ((u64) in[8]) * in[8]; - - felem_reduce_degree(out, tmp); -} - -/* felem_mul sets out=in*in2. - * - * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and - * in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ -static void felem_mul(felem out, const felem in, const felem in2) { - u64 tmp[17]; - - tmp[0] = ((u64) in[0]) * in2[0]; - tmp[1] = ((u64) in[0]) * (in2[1] << 0) + - ((u64) in[1]) * (in2[0] << 0); - tmp[2] = ((u64) in[0]) * (in2[2] << 0) + ((u64) in[1]) * - (in2[1] << 1) + ((u64) in[2]) * (in2[0] << 0); - tmp[3] = ((u64) in[0]) * (in2[3] << 0) + ((u64) in[1]) * - (in2[2] << 0) + ((u64) in[2]) * (in2[1] << 0) + - ((u64) in[3]) * (in2[0] << 0); - tmp[4] = ((u64) in[0]) * (in2[4] << 0) + ((u64) in[1]) * - (in2[3] << 1) + ((u64) in[2]) * (in2[2] << 0) + - ((u64) in[3]) * (in2[1] << 1) + - ((u64) in[4]) * (in2[0] << 0); - tmp[5] = ((u64) in[0]) * (in2[5] << 0) + ((u64) in[1]) * - (in2[4] << 0) + ((u64) in[2]) * (in2[3] << 0) + - ((u64) in[3]) * (in2[2] << 0) + ((u64) in[4]) * - (in2[1] << 0) + ((u64) in[5]) * (in2[0] << 0); - tmp[6] = ((u64) in[0]) * (in2[6] << 0) + ((u64) in[1]) * - (in2[5] << 1) + ((u64) in[2]) * (in2[4] << 0) + - ((u64) in[3]) * (in2[3] << 1) + ((u64) in[4]) * - (in2[2] << 0) + ((u64) in[5]) * (in2[1] << 1) + - ((u64) in[6]) * (in2[0] << 0); - tmp[7] = ((u64) in[0]) * (in2[7] << 0) + ((u64) in[1]) * - (in2[6] << 0) + ((u64) in[2]) * (in2[5] << 0) + - ((u64) in[3]) * (in2[4] << 0) + ((u64) in[4]) * - (in2[3] << 0) + ((u64) in[5]) * (in2[2] << 0) + - ((u64) in[6]) * (in2[1] << 0) + - ((u64) in[7]) * (in2[0] << 0); - /* tmp[8] has the greatest value but doesn't overflow. See logic in - * felem_square. */ - tmp[8] = ((u64) in[0]) * (in2[8] << 0) + ((u64) in[1]) * - (in2[7] << 1) + ((u64) in[2]) * (in2[6] << 0) + - ((u64) in[3]) * (in2[5] << 1) + ((u64) in[4]) * - (in2[4] << 0) + ((u64) in[5]) * (in2[3] << 1) + - ((u64) in[6]) * (in2[2] << 0) + ((u64) in[7]) * - (in2[1] << 1) + ((u64) in[8]) * (in2[0] << 0); - tmp[9] = ((u64) in[1]) * (in2[8] << 0) + ((u64) in[2]) * - (in2[7] << 0) + ((u64) in[3]) * (in2[6] << 0) + - ((u64) in[4]) * (in2[5] << 0) + ((u64) in[5]) * - (in2[4] << 0) + ((u64) in[6]) * (in2[3] << 0) + - ((u64) in[7]) * (in2[2] << 0) + - ((u64) in[8]) * (in2[1] << 0); - tmp[10] = ((u64) in[2]) * (in2[8] << 0) + ((u64) in[3]) * - (in2[7] << 1) + ((u64) in[4]) * (in2[6] << 0) + - ((u64) in[5]) * (in2[5] << 1) + ((u64) in[6]) * - (in2[4] << 0) + ((u64) in[7]) * (in2[3] << 1) + - ((u64) in[8]) * (in2[2] << 0); - tmp[11] = ((u64) in[3]) * (in2[8] << 0) + ((u64) in[4]) * - (in2[7] << 0) + ((u64) in[5]) * (in2[6] << 0) + - ((u64) in[6]) * (in2[5] << 0) + ((u64) in[7]) * - (in2[4] << 0) + ((u64) in[8]) * (in2[3] << 0); - tmp[12] = ((u64) in[4]) * (in2[8] << 0) + ((u64) in[5]) * - (in2[7] << 1) + ((u64) in[6]) * (in2[6] << 0) + - ((u64) in[7]) * (in2[5] << 1) + - ((u64) in[8]) * (in2[4] << 0); - tmp[13] = ((u64) in[5]) * (in2[8] << 0) + ((u64) in[6]) * - (in2[7] << 0) + ((u64) in[7]) * (in2[6] << 0) + - ((u64) in[8]) * (in2[5] << 0); - tmp[14] = ((u64) in[6]) * (in2[8] << 0) + ((u64) in[7]) * - (in2[7] << 1) + ((u64) in[8]) * (in2[6] << 0); - tmp[15] = ((u64) in[7]) * (in2[8] << 0) + - ((u64) in[8]) * (in2[7] << 0); - tmp[16] = ((u64) in[8]) * (in2[8] << 0); - - felem_reduce_degree(out, tmp); -} - -static void felem_assign(felem out, const felem in) { - memcpy(out, in, sizeof(felem)); -} - /* felem_inv calculates |out| = |in|^{-1} * * Based on Fermat's Little Theorem: @@ -667,130 +105,6 @@ static void felem_inv(felem out, const felem in) { felem_mul(out, ftmp2, ftmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */ } -/* felem_scalar_3 sets out=3*out. - * - * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ -static void felem_scalar_3(felem out) { - limb carry = 0; - unsigned i; - - for (i = 0;; i++) { - out[i] *= 3; - out[i] += carry; - carry = out[i] >> 29; - out[i] &= kBottom29Bits; - - i++; - if (i == NLIMBS) - break; - - out[i] *= 3; - out[i] += carry; - carry = out[i] >> 28; - out[i] &= kBottom28Bits; - } - - felem_reduce_carry(out, carry); -} - -/* felem_scalar_4 sets out=4*out. - * - * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ -static void felem_scalar_4(felem out) { - limb carry = 0, next_carry; - unsigned i; - - for (i = 0;; i++) { - next_carry = out[i] >> 27; - out[i] <<= 2; - out[i] &= kBottom29Bits; - out[i] += carry; - carry = next_carry + (out[i] >> 29); - out[i] &= kBottom29Bits; - - i++; - if (i == NLIMBS) - break; - - next_carry = out[i] >> 26; - out[i] <<= 2; - out[i] &= kBottom28Bits; - out[i] += carry; - carry = next_carry + (out[i] >> 28); - out[i] &= kBottom28Bits; - } - - felem_reduce_carry(out, carry); -} - -/* felem_scalar_8 sets out=8*out. - * - * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. - * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. */ -static void felem_scalar_8(felem out) { - limb carry = 0, next_carry; - unsigned i; - - for (i = 0;; i++) { - next_carry = out[i] >> 26; - out[i] <<= 3; - out[i] &= kBottom29Bits; - out[i] += carry; - carry = next_carry + (out[i] >> 29); - out[i] &= kBottom29Bits; - - i++; - if (i == NLIMBS) - break; - - next_carry = out[i] >> 25; - out[i] <<= 3; - out[i] &= kBottom28Bits; - out[i] += carry; - carry = next_carry + (out[i] >> 28); - out[i] &= kBottom28Bits; - } - - felem_reduce_carry(out, carry); -} - -/* felem_is_zero_vartime returns 1 iff |in| == 0. It takes a variable amount of - * time depending on the value of |in|. */ -static char felem_is_zero_vartime(const felem in) { - limb carry; - int i; - limb tmp[NLIMBS]; - - felem_assign(tmp, in); - - /* First, reduce tmp to a minimal form. */ - do { - carry = 0; - for (i = 0;; i++) { - tmp[i] += carry; - carry = tmp[i] >> 29; - tmp[i] &= kBottom29Bits; - - i++; - if (i == NLIMBS) - break; - - tmp[i] += carry; - carry = tmp[i] >> 28; - tmp[i] &= kBottom28Bits; - } - - felem_reduce_carry(tmp, carry); - } while (carry); - - /* tmp < 2**257, so the only possible zero values are 0, p and 2p. */ - return memcmp(tmp, kZero, sizeof(tmp)) == 0 || - memcmp(tmp, kP, sizeof(tmp)) == 0 || - memcmp(tmp, k2P, sizeof(tmp)) == 0; -} - /* Group operations: * @@ -971,9 +285,9 @@ static void point_add_or_double_vartime( felem_diff(y_out, y_out, tmp); } -/* copy_conditional sets out=in if mask = 0xffffffff in constant time. +/* copy_conditional sets out=in if mask = -1 in constant time. * - * On entry: mask is either 0 or 0xffffffff. */ + * On entry: mask is either 0 or -1. */ static void copy_conditional(felem out, const felem in, limb mask) { int i; @@ -1168,58 +482,6 @@ static void scalar_mult(felem nx, felem ny, felem nz, const felem x, } } -#define kRDigits {2, 0, 0, 0xfffffffe, 0xffffffff, 0xffffffff, 0xfffffffd, 1} // 2^257 mod p256.p - -#define kRInvDigits {0x80000000, 1, 0xffffffff, 0, 0x80000001, 0xfffffffe, 1, 0x7fffffff} // 1 / 2^257 mod p256.p - -static const cryptonite_p256_int kR = { kRDigits }; -static const cryptonite_p256_int kRInv = { kRInvDigits }; - -/* to_montgomery sets out = R*in. */ -static void to_montgomery(felem out, const cryptonite_p256_int* in) { - cryptonite_p256_int in_shifted; - int i; - - cryptonite_p256_init(&in_shifted); - cryptonite_p256_modmul(&cryptonite_SECP256r1_p, in, 0, &kR, &in_shifted); - - for (i = 0; i < NLIMBS; i++) { - if ((i & 1) == 0) { - out[i] = P256_DIGIT(&in_shifted, 0) & kBottom29Bits; - cryptonite_p256_shr(&in_shifted, 29, &in_shifted); - } else { - out[i] = P256_DIGIT(&in_shifted, 0) & kBottom28Bits; - cryptonite_p256_shr(&in_shifted, 28, &in_shifted); - } - } - - cryptonite_p256_clear(&in_shifted); -} - -/* from_montgomery sets out=in/R. */ -static void from_montgomery(cryptonite_p256_int* out, const felem in) { - cryptonite_p256_int result, tmp; - int i, top; - - cryptonite_p256_init(&result); - cryptonite_p256_init(&tmp); - - cryptonite_p256_add_d(&tmp, in[NLIMBS - 1], &result); - for (i = NLIMBS - 2; i >= 0; i--) { - if ((i & 1) == 0) { - top = cryptonite_p256_shl(&result, 29, &tmp); - } else { - top = cryptonite_p256_shl(&result, 28, &tmp); - } - top |= cryptonite_p256_add_d(&tmp, in[i], &result); - } - - cryptonite_p256_modmul(&cryptonite_SECP256r1_p, &kRInv, top, &result, out); - - cryptonite_p256_clear(&result); - cryptonite_p256_clear(&tmp); -} - /* cryptonite_p256_base_point_mul sets {out_x,out_y} = nG, where n is < the * order of the group. */ void cryptonite_p256_base_point_mul(const cryptonite_p256_int* n, cryptonite_p256_int* out_x, cryptonite_p256_int* out_y) { diff --git a/cryptonite.cabal b/cryptonite.cabal index 637521a..c7e5742 100644 --- a/cryptonite.cabal +++ b/cryptonite.cabal @@ -50,7 +50,8 @@ extra-source-files: cbits/*.h cbits/decaf/p448/*.h cbits/decaf/ed448goldilocks/decaf_tables.c cbits/decaf/ed448goldilocks/decaf.c - cbits/p256/*.h + cbits/include32/p256/*.h + cbits/include64/p256/*.h cbits/blake2/ref/*.h cbits/blake2/sse/*.h cbits/argon2/*.h @@ -284,6 +285,11 @@ Library , cbits/decaf/include , cbits/decaf/p448 + if arch(x86_64) || arch(aarch64) + include-dirs: cbits/include64 + else + include-dirs: cbits/include32 + if arch(x86_64) || arch(aarch64) C-sources: cbits/decaf/p448/arch_ref64/f_impl.c , cbits/decaf/p448/f_generic.c