diff --git a/cbits/p256/p256_ec.c b/cbits/p256/p256_ec.c index 41c183a..c2b9c74 100644 --- a/cbits/p256/p256_ec.c +++ b/cbits/p256/p256_ec.c @@ -32,573 +32,11 @@ // // 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, -}; +#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: * @@ -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/cbits/p256/p256_gf.h b/cbits/p256/p256_gf.h new file mode 100644 index 0000000..7144c9d --- /dev/null +++ b/cbits/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 +// be portable 32-bit, although it's 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); +}