| /* Copyright (c) 2015, Google Inc. |
| * |
| * Permission to use, copy, modify, and/or distribute this software for any |
| * purpose with or without fee is hereby granted, provided that the above |
| * copyright notice and this permission notice appear in all copies. |
| * |
| * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES |
| * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF |
| * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY |
| * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES |
| * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION |
| * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN |
| * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ |
| |
| /* A 64-bit implementation of the NIST P-224 elliptic curve point multiplication |
| * |
| * Inspired by Daniel J. Bernstein's public domain nistp224 implementation |
| * and Adam Langley's public domain 64-bit C implementation of curve25519. */ |
| |
| #include <openssl/base.h> |
| |
| #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \ |
| !defined(OPENSSL_SMALL) |
| |
| #include <openssl/bn.h> |
| #include <openssl/ec.h> |
| #include <openssl/err.h> |
| #include <openssl/mem.h> |
| |
| #include <string.h> |
| |
| #include "internal.h" |
| #include "../internal.h" |
| |
| |
| typedef uint8_t u8; |
| typedef uint64_t u64; |
| typedef int64_t s64; |
| |
| /* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 |
| * using 64-bit coefficients called 'limbs', and sometimes (for multiplication |
| * results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + |
| * 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-limb |
| * representation is an 'felem'; a 7-widelimb representation is a 'widefelem'. |
| * Even within felems, bits of adjacent limbs overlap, and we don't always |
| * reduce the representations: we ensure that inputs to each felem |
| * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60, and |
| * fit into a 128-bit word without overflow. The coefficients are then again |
| * partially reduced to obtain an felem satisfying a_i < 2^57. We only reduce |
| * to the unique minimal representation at the end of the computation. */ |
| |
| typedef uint64_t limb; |
| typedef uint128_t widelimb; |
| |
| typedef limb felem[4]; |
| typedef widelimb widefelem[7]; |
| |
| /* Field element represented as a byte arrary. 28*8 = 224 bits is also the |
| * group order size for the elliptic curve, and we also use this type for |
| * scalars for point multiplication. */ |
| typedef u8 felem_bytearray[28]; |
| |
| /* Precomputed multiples of the standard generator |
| * Points are given in coordinates (X, Y, Z) where Z normally is 1 |
| * (0 for the point at infinity). |
| * For each field element, slice a_0 is word 0, etc. |
| * |
| * The table has 2 * 16 elements, starting with the following: |
| * index | bits | point |
| * ------+---------+------------------------------ |
| * 0 | 0 0 0 0 | 0G |
| * 1 | 0 0 0 1 | 1G |
| * 2 | 0 0 1 0 | 2^56G |
| * 3 | 0 0 1 1 | (2^56 + 1)G |
| * 4 | 0 1 0 0 | 2^112G |
| * 5 | 0 1 0 1 | (2^112 + 1)G |
| * 6 | 0 1 1 0 | (2^112 + 2^56)G |
| * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G |
| * 8 | 1 0 0 0 | 2^168G |
| * 9 | 1 0 0 1 | (2^168 + 1)G |
| * 10 | 1 0 1 0 | (2^168 + 2^56)G |
| * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G |
| * 12 | 1 1 0 0 | (2^168 + 2^112)G |
| * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G |
| * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G |
| * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G |
| * followed by a copy of this with each element multiplied by 2^28. |
| * |
| * The reason for this is so that we can clock bits into four different |
| * locations when doing simple scalar multiplies against the base point, |
| * and then another four locations using the second 16 elements. */ |
| static const felem g_pre_comp[2][16][3] = { |
| {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, |
| {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, |
| {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, |
| {1, 0, 0, 0}}, |
| {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, |
| {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, |
| {1, 0, 0, 0}}, |
| {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, |
| {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, |
| {1, 0, 0, 0}}, |
| {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, |
| {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, |
| {1, 0, 0, 0}}, |
| {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, |
| {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, |
| {1, 0, 0, 0}}, |
| {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, |
| {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, |
| {1, 0, 0, 0}}, |
| {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, |
| {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, |
| {1, 0, 0, 0}}, |
| {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, |
| {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, |
| {1, 0, 0, 0}}, |
| {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, |
| {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, |
| {1, 0, 0, 0}}, |
| {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, |
| {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, |
| {1, 0, 0, 0}}, |
| {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, |
| {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, |
| {1, 0, 0, 0}}, |
| {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, |
| {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, |
| {1, 0, 0, 0}}, |
| {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, |
| {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, |
| {1, 0, 0, 0}}, |
| {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, |
| {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, |
| {1, 0, 0, 0}}, |
| {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, |
| {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, |
| {1, 0, 0, 0}}}, |
| {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, |
| {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31}, |
| {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d}, |
| {1, 0, 0, 0}}, |
| {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3}, |
| {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a}, |
| {1, 0, 0, 0}}, |
| {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33}, |
| {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100}, |
| {1, 0, 0, 0}}, |
| {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5}, |
| {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea}, |
| {1, 0, 0, 0}}, |
| {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be}, |
| {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51}, |
| {1, 0, 0, 0}}, |
| {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1}, |
| {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb}, |
| {1, 0, 0, 0}}, |
| {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233}, |
| {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def}, |
| {1, 0, 0, 0}}, |
| {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae}, |
| {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45}, |
| {1, 0, 0, 0}}, |
| {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e}, |
| {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb}, |
| {1, 0, 0, 0}}, |
| {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de}, |
| {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3}, |
| {1, 0, 0, 0}}, |
| {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05}, |
| {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58}, |
| {1, 0, 0, 0}}, |
| {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb}, |
| {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0}, |
| {1, 0, 0, 0}}, |
| {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9}, |
| {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea}, |
| {1, 0, 0, 0}}, |
| {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba}, |
| {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405}, |
| {1, 0, 0, 0}}, |
| {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e}, |
| {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e}, |
| {1, 0, 0, 0}}}}; |
| |
| /* Helper functions to convert field elements to/from internal representation */ |
| static void bin28_to_felem(felem out, const u8 in[28]) { |
| out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff; |
| out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff; |
| out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff; |
| out[3] = (*((const uint64_t *)(in + 20))) >> 8; |
| } |
| |
| static void felem_to_bin28(u8 out[28], const felem in) { |
| size_t i; |
| for (i = 0; i < 7; ++i) { |
| out[i] = in[0] >> (8 * i); |
| out[i + 7] = in[1] >> (8 * i); |
| out[i + 14] = in[2] >> (8 * i); |
| out[i + 21] = in[3] >> (8 * i); |
| } |
| } |
| |
| /* To preserve endianness when using BN_bn2bin and BN_bin2bn */ |
| static void flip_endian(u8 *out, const u8 *in, size_t len) { |
| size_t i; |
| for (i = 0; i < len; ++i) { |
| out[i] = in[len - 1 - i]; |
| } |
| } |
| |
| /* From OpenSSL BIGNUM to internal representation */ |
| static int BN_to_felem(felem out, const BIGNUM *bn) { |
| /* BN_bn2bin eats leading zeroes */ |
| felem_bytearray b_out; |
| memset(b_out, 0, sizeof(b_out)); |
| size_t num_bytes = BN_num_bytes(bn); |
| if (num_bytes > sizeof(b_out) || |
| BN_is_negative(bn)) { |
| OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); |
| return 0; |
| } |
| |
| felem_bytearray b_in; |
| num_bytes = BN_bn2bin(bn, b_in); |
| flip_endian(b_out, b_in, num_bytes); |
| bin28_to_felem(out, b_out); |
| return 1; |
| } |
| |
| /* From internal representation to OpenSSL BIGNUM */ |
| static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) { |
| felem_bytearray b_in, b_out; |
| felem_to_bin28(b_in, in); |
| flip_endian(b_out, b_in, sizeof(b_out)); |
| return BN_bin2bn(b_out, sizeof(b_out), out); |
| } |
| |
| /* Field operations, using the internal representation of field elements. |
| * NB! These operations are specific to our point multiplication and cannot be |
| * expected to be correct in general - e.g., multiplication with a large scalar |
| * will cause an overflow. */ |
| |
| static void felem_assign(felem out, const felem in) { |
| out[0] = in[0]; |
| out[1] = in[1]; |
| out[2] = in[2]; |
| out[3] = in[3]; |
| } |
| |
| /* Sum two field elements: out += in */ |
| static void felem_sum(felem out, const felem in) { |
| out[0] += in[0]; |
| out[1] += in[1]; |
| out[2] += in[2]; |
| out[3] += in[3]; |
| } |
| |
| /* Get negative value: out = -in */ |
| /* Assumes in[i] < 2^57 */ |
| static void felem_neg(felem out, const felem in) { |
| static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2); |
| static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2); |
| static const limb two58m42m2 = |
| (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2); |
| |
| /* Set to 0 mod 2^224-2^96+1 to ensure out > in */ |
| out[0] = two58p2 - in[0]; |
| out[1] = two58m42m2 - in[1]; |
| out[2] = two58m2 - in[2]; |
| out[3] = two58m2 - in[3]; |
| } |
| |
| /* Subtract field elements: out -= in */ |
| /* Assumes in[i] < 2^57 */ |
| static void felem_diff(felem out, const felem in) { |
| static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2); |
| static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2); |
| static const limb two58m42m2 = |
| (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2); |
| |
| /* Add 0 mod 2^224-2^96+1 to ensure out > in */ |
| out[0] += two58p2; |
| out[1] += two58m42m2; |
| out[2] += two58m2; |
| out[3] += two58m2; |
| |
| out[0] -= in[0]; |
| out[1] -= in[1]; |
| out[2] -= in[2]; |
| out[3] -= in[3]; |
| } |
| |
| /* Subtract in unreduced 128-bit mode: out -= in */ |
| /* Assumes in[i] < 2^119 */ |
| static void widefelem_diff(widefelem out, const widefelem in) { |
| static const widelimb two120 = ((widelimb)1) << 120; |
| static const widelimb two120m64 = |
| (((widelimb)1) << 120) - (((widelimb)1) << 64); |
| static const widelimb two120m104m64 = |
| (((widelimb)1) << 120) - (((widelimb)1) << 104) - (((widelimb)1) << 64); |
| |
| /* Add 0 mod 2^224-2^96+1 to ensure out > in */ |
| out[0] += two120; |
| out[1] += two120m64; |
| out[2] += two120m64; |
| out[3] += two120; |
| out[4] += two120m104m64; |
| out[5] += two120m64; |
| out[6] += two120m64; |
| |
| out[0] -= in[0]; |
| out[1] -= in[1]; |
| out[2] -= in[2]; |
| out[3] -= in[3]; |
| out[4] -= in[4]; |
| out[5] -= in[5]; |
| out[6] -= in[6]; |
| } |
| |
| /* Subtract in mixed mode: out128 -= in64 */ |
| /* in[i] < 2^63 */ |
| static void felem_diff_128_64(widefelem out, const felem in) { |
| static const widelimb two64p8 = (((widelimb)1) << 64) + (((widelimb)1) << 8); |
| static const widelimb two64m8 = (((widelimb)1) << 64) - (((widelimb)1) << 8); |
| static const widelimb two64m48m8 = |
| (((widelimb)1) << 64) - (((widelimb)1) << 48) - (((widelimb)1) << 8); |
| |
| /* Add 0 mod 2^224-2^96+1 to ensure out > in */ |
| out[0] += two64p8; |
| out[1] += two64m48m8; |
| out[2] += two64m8; |
| out[3] += two64m8; |
| |
| out[0] -= in[0]; |
| out[1] -= in[1]; |
| out[2] -= in[2]; |
| out[3] -= in[3]; |
| } |
| |
| /* Multiply a field element by a scalar: out = out * scalar |
| * The scalars we actually use are small, so results fit without overflow */ |
| static void felem_scalar(felem out, const limb scalar) { |
| out[0] *= scalar; |
| out[1] *= scalar; |
| out[2] *= scalar; |
| out[3] *= scalar; |
| } |
| |
| /* Multiply an unreduced field element by a scalar: out = out * scalar |
| * The scalars we actually use are small, so results fit without overflow */ |
| static void widefelem_scalar(widefelem out, const widelimb scalar) { |
| out[0] *= scalar; |
| out[1] *= scalar; |
| out[2] *= scalar; |
| out[3] *= scalar; |
| out[4] *= scalar; |
| out[5] *= scalar; |
| out[6] *= scalar; |
| } |
| |
| /* Square a field element: out = in^2 */ |
| static void felem_square(widefelem out, const felem in) { |
| limb tmp0, tmp1, tmp2; |
| tmp0 = 2 * in[0]; |
| tmp1 = 2 * in[1]; |
| tmp2 = 2 * in[2]; |
| out[0] = ((widelimb)in[0]) * in[0]; |
| out[1] = ((widelimb)in[0]) * tmp1; |
| out[2] = ((widelimb)in[0]) * tmp2 + ((widelimb)in[1]) * in[1]; |
| out[3] = ((widelimb)in[3]) * tmp0 + ((widelimb)in[1]) * tmp2; |
| out[4] = ((widelimb)in[3]) * tmp1 + ((widelimb)in[2]) * in[2]; |
| out[5] = ((widelimb)in[3]) * tmp2; |
| out[6] = ((widelimb)in[3]) * in[3]; |
| } |
| |
| /* Multiply two field elements: out = in1 * in2 */ |
| static void felem_mul(widefelem out, const felem in1, const felem in2) { |
| out[0] = ((widelimb)in1[0]) * in2[0]; |
| out[1] = ((widelimb)in1[0]) * in2[1] + ((widelimb)in1[1]) * in2[0]; |
| out[2] = ((widelimb)in1[0]) * in2[2] + ((widelimb)in1[1]) * in2[1] + |
| ((widelimb)in1[2]) * in2[0]; |
| out[3] = ((widelimb)in1[0]) * in2[3] + ((widelimb)in1[1]) * in2[2] + |
| ((widelimb)in1[2]) * in2[1] + ((widelimb)in1[3]) * in2[0]; |
| out[4] = ((widelimb)in1[1]) * in2[3] + ((widelimb)in1[2]) * in2[2] + |
| ((widelimb)in1[3]) * in2[1]; |
| out[5] = ((widelimb)in1[2]) * in2[3] + ((widelimb)in1[3]) * in2[2]; |
| out[6] = ((widelimb)in1[3]) * in2[3]; |
| } |
| |
| /* Reduce seven 128-bit coefficients to four 64-bit coefficients. |
| * Requires in[i] < 2^126, |
| * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */ |
| static void felem_reduce(felem out, const widefelem in) { |
| static const widelimb two127p15 = |
| (((widelimb)1) << 127) + (((widelimb)1) << 15); |
| static const widelimb two127m71 = |
| (((widelimb)1) << 127) - (((widelimb)1) << 71); |
| static const widelimb two127m71m55 = |
| (((widelimb)1) << 127) - (((widelimb)1) << 71) - (((widelimb)1) << 55); |
| widelimb output[5]; |
| |
| /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */ |
| output[0] = in[0] + two127p15; |
| output[1] = in[1] + two127m71m55; |
| output[2] = in[2] + two127m71; |
| output[3] = in[3]; |
| output[4] = in[4]; |
| |
| /* Eliminate in[4], in[5], in[6] */ |
| output[4] += in[6] >> 16; |
| output[3] += (in[6] & 0xffff) << 40; |
| output[2] -= in[6]; |
| |
| output[3] += in[5] >> 16; |
| output[2] += (in[5] & 0xffff) << 40; |
| output[1] -= in[5]; |
| |
| output[2] += output[4] >> 16; |
| output[1] += (output[4] & 0xffff) << 40; |
| output[0] -= output[4]; |
| |
| /* Carry 2 -> 3 -> 4 */ |
| output[3] += output[2] >> 56; |
| output[2] &= 0x00ffffffffffffff; |
| |
| output[4] = output[3] >> 56; |
| output[3] &= 0x00ffffffffffffff; |
| |
| /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */ |
| |
| /* Eliminate output[4] */ |
| output[2] += output[4] >> 16; |
| /* output[2] < 2^56 + 2^56 = 2^57 */ |
| output[1] += (output[4] & 0xffff) << 40; |
| output[0] -= output[4]; |
| |
| /* Carry 0 -> 1 -> 2 -> 3 */ |
| output[1] += output[0] >> 56; |
| out[0] = output[0] & 0x00ffffffffffffff; |
| |
| output[2] += output[1] >> 56; |
| /* output[2] < 2^57 + 2^72 */ |
| out[1] = output[1] & 0x00ffffffffffffff; |
| output[3] += output[2] >> 56; |
| /* output[3] <= 2^56 + 2^16 */ |
| out[2] = output[2] & 0x00ffffffffffffff; |
| |
| /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, |
| * out[3] <= 2^56 + 2^16 (due to final carry), |
| * so out < 2*p */ |
| out[3] = output[3]; |
| } |
| |
| /* Reduce to unique minimal representation. |
| * Requires 0 <= in < 2*p (always call felem_reduce first) */ |
| static void felem_contract(felem out, const felem in) { |
| static const int64_t two56 = ((limb)1) << 56; |
| /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */ |
| /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */ |
| int64_t tmp[4], a; |
| tmp[0] = in[0]; |
| tmp[1] = in[1]; |
| tmp[2] = in[2]; |
| tmp[3] = in[3]; |
| /* Case 1: a = 1 iff in >= 2^224 */ |
| a = (in[3] >> 56); |
| tmp[0] -= a; |
| tmp[1] += a << 40; |
| tmp[3] &= 0x00ffffffffffffff; |
| /* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and |
| * the lower part is non-zero */ |
| a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) | |
| (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63); |
| a &= 0x00ffffffffffffff; |
| /* turn a into an all-one mask (if a = 0) or an all-zero mask */ |
| a = (a - 1) >> 63; |
| /* subtract 2^224 - 2^96 + 1 if a is all-one */ |
| tmp[3] &= a ^ 0xffffffffffffffff; |
| tmp[2] &= a ^ 0xffffffffffffffff; |
| tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; |
| tmp[0] -= 1 & a; |
| |
| /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must |
| * be non-zero, so we only need one step */ |
| a = tmp[0] >> 63; |
| tmp[0] += two56 & a; |
| tmp[1] -= 1 & a; |
| |
| /* carry 1 -> 2 -> 3 */ |
| tmp[2] += tmp[1] >> 56; |
| tmp[1] &= 0x00ffffffffffffff; |
| |
| tmp[3] += tmp[2] >> 56; |
| tmp[2] &= 0x00ffffffffffffff; |
| |
| /* Now 0 <= out < p */ |
| out[0] = tmp[0]; |
| out[1] = tmp[1]; |
| out[2] = tmp[2]; |
| out[3] = tmp[3]; |
| } |
| |
| /* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field |
| * elements are reduced to in < 2^225, so we only need to check three cases: 0, |
| * 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 */ |
| static limb felem_is_zero(const felem in) { |
| limb zero = in[0] | in[1] | in[2] | in[3]; |
| zero = (((int64_t)(zero)-1) >> 63) & 1; |
| |
| limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) | |
| (in[2] ^ 0x00ffffffffffffff) | |
| (in[3] ^ 0x00ffffffffffffff); |
| two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1; |
| limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) | |
| (in[2] ^ 0x00ffffffffffffff) | |
| (in[3] ^ 0x01ffffffffffffff); |
| two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1; |
| return (zero | two224m96p1 | two225m97p2); |
| } |
| |
| /* Invert a field element */ |
| /* Computation chain copied from djb's code */ |
| static void felem_inv(felem out, const felem in) { |
| felem ftmp, ftmp2, ftmp3, ftmp4; |
| widefelem tmp; |
| size_t i; |
| |
| felem_square(tmp, in); |
| felem_reduce(ftmp, tmp); /* 2 */ |
| felem_mul(tmp, in, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^2 - 1 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^3 - 2 */ |
| felem_mul(tmp, in, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^3 - 1 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp2, tmp); /* 2^4 - 2 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); /* 2^5 - 4 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); /* 2^6 - 8 */ |
| felem_mul(tmp, ftmp2, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^6 - 1 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp2, tmp); /* 2^7 - 2 */ |
| for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); |
| } |
| felem_mul(tmp, ftmp2, ftmp); |
| felem_reduce(ftmp2, tmp); /* 2^12 - 1 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp3, tmp); /* 2^13 - 2 */ |
| for (i = 0; i < 11; ++i) {/* 2^24 - 2^12 */ |
| felem_square(tmp, ftmp3); |
| felem_reduce(ftmp3, tmp); |
| } |
| felem_mul(tmp, ftmp3, ftmp2); |
| felem_reduce(ftmp2, tmp); /* 2^24 - 1 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp3, tmp); /* 2^25 - 2 */ |
| for (i = 0; i < 23; ++i) {/* 2^48 - 2^24 */ |
| felem_square(tmp, ftmp3); |
| felem_reduce(ftmp3, tmp); |
| } |
| felem_mul(tmp, ftmp3, ftmp2); |
| felem_reduce(ftmp3, tmp); /* 2^48 - 1 */ |
| felem_square(tmp, ftmp3); |
| felem_reduce(ftmp4, tmp); /* 2^49 - 2 */ |
| for (i = 0; i < 47; ++i) {/* 2^96 - 2^48 */ |
| felem_square(tmp, ftmp4); |
| felem_reduce(ftmp4, tmp); |
| } |
| felem_mul(tmp, ftmp3, ftmp4); |
| felem_reduce(ftmp3, tmp); /* 2^96 - 1 */ |
| felem_square(tmp, ftmp3); |
| felem_reduce(ftmp4, tmp); /* 2^97 - 2 */ |
| for (i = 0; i < 23; ++i) {/* 2^120 - 2^24 */ |
| felem_square(tmp, ftmp4); |
| felem_reduce(ftmp4, tmp); |
| } |
| felem_mul(tmp, ftmp2, ftmp4); |
| felem_reduce(ftmp2, tmp); /* 2^120 - 1 */ |
| for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); |
| } |
| felem_mul(tmp, ftmp2, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^126 - 1 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^127 - 2 */ |
| felem_mul(tmp, ftmp, in); |
| felem_reduce(ftmp, tmp); /* 2^127 - 1 */ |
| for (i = 0; i < 97; ++i) {/* 2^224 - 2^97 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); |
| } |
| felem_mul(tmp, ftmp, ftmp3); |
| felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */ |
| } |
| |
| /* Copy in constant time: |
| * if icopy == 1, copy in to out, |
| * if icopy == 0, copy out to itself. */ |
| static void copy_conditional(felem out, const felem in, limb icopy) { |
| size_t i; |
| /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */ |
| const limb copy = -icopy; |
| for (i = 0; i < 4; ++i) { |
| const limb tmp = copy & (in[i] ^ out[i]); |
| out[i] ^= tmp; |
| } |
| } |
| |
| /* ELLIPTIC CURVE POINT OPERATIONS |
| * |
| * Points are represented in Jacobian projective coordinates: |
| * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3), |
| * or to the point at infinity if Z == 0. */ |
| |
| /* Double an elliptic curve point: |
| * (X', Y', Z') = 2 * (X, Y, Z), where |
| * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2 |
| * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2 |
| * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z |
| * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed, |
| * while x_out == y_in is not (maybe this works, but it's not tested). */ |
| static void point_double(felem x_out, felem y_out, felem z_out, |
| const felem x_in, const felem y_in, const felem z_in) { |
| widefelem tmp, tmp2; |
| felem delta, gamma, beta, alpha, ftmp, ftmp2; |
| |
| felem_assign(ftmp, x_in); |
| felem_assign(ftmp2, x_in); |
| |
| /* delta = z^2 */ |
| felem_square(tmp, z_in); |
| felem_reduce(delta, tmp); |
| |
| /* gamma = y^2 */ |
| felem_square(tmp, y_in); |
| felem_reduce(gamma, tmp); |
| |
| /* beta = x*gamma */ |
| felem_mul(tmp, x_in, gamma); |
| felem_reduce(beta, tmp); |
| |
| /* alpha = 3*(x-delta)*(x+delta) */ |
| felem_diff(ftmp, delta); |
| /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */ |
| felem_sum(ftmp2, delta); |
| /* ftmp2[i] < 2^57 + 2^57 = 2^58 */ |
| felem_scalar(ftmp2, 3); |
| /* ftmp2[i] < 3 * 2^58 < 2^60 */ |
| felem_mul(tmp, ftmp, ftmp2); |
| /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */ |
| felem_reduce(alpha, tmp); |
| |
| /* x' = alpha^2 - 8*beta */ |
| felem_square(tmp, alpha); |
| /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ |
| felem_assign(ftmp, beta); |
| felem_scalar(ftmp, 8); |
| /* ftmp[i] < 8 * 2^57 = 2^60 */ |
| felem_diff_128_64(tmp, ftmp); |
| /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ |
| felem_reduce(x_out, tmp); |
| |
| /* z' = (y + z)^2 - gamma - delta */ |
| felem_sum(delta, gamma); |
| /* delta[i] < 2^57 + 2^57 = 2^58 */ |
| felem_assign(ftmp, y_in); |
| felem_sum(ftmp, z_in); |
| /* ftmp[i] < 2^57 + 2^57 = 2^58 */ |
| felem_square(tmp, ftmp); |
| /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */ |
| felem_diff_128_64(tmp, delta); |
| /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */ |
| felem_reduce(z_out, tmp); |
| |
| /* y' = alpha*(4*beta - x') - 8*gamma^2 */ |
| felem_scalar(beta, 4); |
| /* beta[i] < 4 * 2^57 = 2^59 */ |
| felem_diff(beta, x_out); |
| /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */ |
| felem_mul(tmp, alpha, beta); |
| /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */ |
| felem_square(tmp2, gamma); |
| /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */ |
| widefelem_scalar(tmp2, 8); |
| /* tmp2[i] < 8 * 2^116 = 2^119 */ |
| widefelem_diff(tmp, tmp2); |
| /* tmp[i] < 2^119 + 2^120 < 2^121 */ |
| felem_reduce(y_out, tmp); |
| } |
| |
| /* Add two elliptic curve points: |
| * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where |
| * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 - |
| * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 |
| * Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * |
| * X_1)^2 - X_3) - |
| * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3 |
| * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) |
| * |
| * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. */ |
| |
| /* This function is not entirely constant-time: it includes a branch for |
| * checking whether the two input points are equal, (while not equal to the |
| * point at infinity). This case never happens during single point |
| * multiplication, so there is no timing leak for ECDH or ECDSA signing. */ |
| static void point_add(felem x3, felem y3, felem z3, const felem x1, |
| const felem y1, const felem z1, const int mixed, |
| const felem x2, const felem y2, const felem z2) { |
| felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out; |
| widefelem tmp, tmp2; |
| limb z1_is_zero, z2_is_zero, x_equal, y_equal; |
| |
| if (!mixed) { |
| /* ftmp2 = z2^2 */ |
| felem_square(tmp, z2); |
| felem_reduce(ftmp2, tmp); |
| |
| /* ftmp4 = z2^3 */ |
| felem_mul(tmp, ftmp2, z2); |
| felem_reduce(ftmp4, tmp); |
| |
| /* ftmp4 = z2^3*y1 */ |
| felem_mul(tmp2, ftmp4, y1); |
| felem_reduce(ftmp4, tmp2); |
| |
| /* ftmp2 = z2^2*x1 */ |
| felem_mul(tmp2, ftmp2, x1); |
| felem_reduce(ftmp2, tmp2); |
| } else { |
| /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */ |
| |
| /* ftmp4 = z2^3*y1 */ |
| felem_assign(ftmp4, y1); |
| |
| /* ftmp2 = z2^2*x1 */ |
| felem_assign(ftmp2, x1); |
| } |
| |
| /* ftmp = z1^2 */ |
| felem_square(tmp, z1); |
| felem_reduce(ftmp, tmp); |
| |
| /* ftmp3 = z1^3 */ |
| felem_mul(tmp, ftmp, z1); |
| felem_reduce(ftmp3, tmp); |
| |
| /* tmp = z1^3*y2 */ |
| felem_mul(tmp, ftmp3, y2); |
| /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ |
| |
| /* ftmp3 = z1^3*y2 - z2^3*y1 */ |
| felem_diff_128_64(tmp, ftmp4); |
| /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ |
| felem_reduce(ftmp3, tmp); |
| |
| /* tmp = z1^2*x2 */ |
| felem_mul(tmp, ftmp, x2); |
| /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ |
| |
| /* ftmp = z1^2*x2 - z2^2*x1 */ |
| felem_diff_128_64(tmp, ftmp2); |
| /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ |
| felem_reduce(ftmp, tmp); |
| |
| /* the formulae are incorrect if the points are equal |
| * so we check for this and do doubling if this happens */ |
| x_equal = felem_is_zero(ftmp); |
| y_equal = felem_is_zero(ftmp3); |
| z1_is_zero = felem_is_zero(z1); |
| z2_is_zero = felem_is_zero(z2); |
| /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */ |
| if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { |
| point_double(x3, y3, z3, x1, y1, z1); |
| return; |
| } |
| |
| /* ftmp5 = z1*z2 */ |
| if (!mixed) { |
| felem_mul(tmp, z1, z2); |
| felem_reduce(ftmp5, tmp); |
| } else { |
| /* special case z2 = 0 is handled later */ |
| felem_assign(ftmp5, z1); |
| } |
| |
| /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */ |
| felem_mul(tmp, ftmp, ftmp5); |
| felem_reduce(z_out, tmp); |
| |
| /* ftmp = (z1^2*x2 - z2^2*x1)^2 */ |
| felem_assign(ftmp5, ftmp); |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); |
| |
| /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */ |
| felem_mul(tmp, ftmp, ftmp5); |
| felem_reduce(ftmp5, tmp); |
| |
| /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ |
| felem_mul(tmp, ftmp2, ftmp); |
| felem_reduce(ftmp2, tmp); |
| |
| /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ |
| felem_mul(tmp, ftmp4, ftmp5); |
| /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ |
| |
| /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */ |
| felem_square(tmp2, ftmp3); |
| /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */ |
| |
| /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */ |
| felem_diff_128_64(tmp2, ftmp5); |
| /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */ |
| |
| /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ |
| felem_assign(ftmp5, ftmp2); |
| felem_scalar(ftmp5, 2); |
| /* ftmp5[i] < 2 * 2^57 = 2^58 */ |
| |
| /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - |
| 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ |
| felem_diff_128_64(tmp2, ftmp5); |
| /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */ |
| felem_reduce(x_out, tmp2); |
| |
| /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */ |
| felem_diff(ftmp2, x_out); |
| /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */ |
| |
| /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */ |
| felem_mul(tmp2, ftmp3, ftmp2); |
| /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */ |
| |
| /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) - |
| z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ |
| widefelem_diff(tmp2, tmp); |
| /* tmp2[i] < 2^118 + 2^120 < 2^121 */ |
| felem_reduce(y_out, tmp2); |
| |
| /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is |
| * the point at infinity, so we need to check for this separately */ |
| |
| /* if point 1 is at infinity, copy point 2 to output, and vice versa */ |
| copy_conditional(x_out, x2, z1_is_zero); |
| copy_conditional(x_out, x1, z2_is_zero); |
| copy_conditional(y_out, y2, z1_is_zero); |
| copy_conditional(y_out, y1, z2_is_zero); |
| copy_conditional(z_out, z2, z1_is_zero); |
| copy_conditional(z_out, z1, z2_is_zero); |
| felem_assign(x3, x_out); |
| felem_assign(y3, y_out); |
| felem_assign(z3, z_out); |
| } |
| |
| /* select_point selects the |idx|th point from a precomputation table and |
| * copies it to out. */ |
| static void select_point(const u64 idx, size_t size, |
| const felem pre_comp[/*size*/][3], felem out[3]) { |
| limb *outlimbs = &out[0][0]; |
| memset(outlimbs, 0, 3 * sizeof(felem)); |
| |
| size_t i; |
| for (i = 0; i < size; i++) { |
| const limb *inlimbs = &pre_comp[i][0][0]; |
| u64 mask = i ^ idx; |
| mask |= mask >> 4; |
| mask |= mask >> 2; |
| mask |= mask >> 1; |
| mask &= 1; |
| mask--; |
| size_t j; |
| for (j = 0; j < 4 * 3; j++) { |
| outlimbs[j] |= inlimbs[j] & mask; |
| } |
| } |
| } |
| |
| /* get_bit returns the |i|th bit in |in| */ |
| static char get_bit(const felem_bytearray in, size_t i) { |
| if (i >= 224) { |
| return 0; |
| } |
| return (in[i >> 3] >> (i & 7)) & 1; |
| } |
| |
| /* Interleaved point multiplication using precomputed point multiples: |
| * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], |
| * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple |
| * of the generator, using certain (large) precomputed multiples in g_pre_comp. |
| * Output point (X, Y, Z) is stored in x_out, y_out, z_out */ |
| static void batch_mul(felem x_out, felem y_out, felem z_out, |
| const felem_bytearray scalars[], |
| const size_t num_points, const u8 *g_scalar, |
| const felem pre_comp[][17][3]) { |
| felem nq[3], tmp[4]; |
| u64 bits; |
| u8 sign, digit; |
| |
| /* set nq to the point at infinity */ |
| memset(nq, 0, 3 * sizeof(felem)); |
| |
| /* Loop over all scalars msb-to-lsb, interleaving additions |
| * of multiples of the generator (two in each of the last 28 rounds) |
| * and additions of other points multiples (every 5th round). */ |
| int skip = 1; /* save two point operations in the first round */ |
| size_t i = num_points != 0 ? 220 : 27; |
| for (;;) { |
| /* double */ |
| if (!skip) { |
| point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); |
| } |
| |
| /* add multiples of the generator */ |
| if (g_scalar != NULL && i <= 27) { |
| /* first, look 28 bits upwards */ |
| bits = get_bit(g_scalar, i + 196) << 3; |
| bits |= get_bit(g_scalar, i + 140) << 2; |
| bits |= get_bit(g_scalar, i + 84) << 1; |
| bits |= get_bit(g_scalar, i + 28); |
| /* select the point to add, in constant time */ |
| select_point(bits, 16, g_pre_comp[1], tmp); |
| |
| if (!skip) { |
| point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, |
| tmp[0], tmp[1], tmp[2]); |
| } else { |
| memcpy(nq, tmp, 3 * sizeof(felem)); |
| skip = 0; |
| } |
| |
| /* second, look at the current position */ |
| bits = get_bit(g_scalar, i + 168) << 3; |
| bits |= get_bit(g_scalar, i + 112) << 2; |
| bits |= get_bit(g_scalar, i + 56) << 1; |
| bits |= get_bit(g_scalar, i); |
| /* select the point to add, in constant time */ |
| select_point(bits, 16, g_pre_comp[0], tmp); |
| point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0], |
| tmp[1], tmp[2]); |
| } |
| |
| /* do other additions every 5 doublings */ |
| if (num_points != 0 && i % 5 == 0) { |
| /* loop over all scalars */ |
| size_t num; |
| for (num = 0; num < num_points; ++num) { |
| bits = get_bit(scalars[num], i + 4) << 5; |
| bits |= get_bit(scalars[num], i + 3) << 4; |
| bits |= get_bit(scalars[num], i + 2) << 3; |
| bits |= get_bit(scalars[num], i + 1) << 2; |
| bits |= get_bit(scalars[num], i) << 1; |
| bits |= get_bit(scalars[num], i - 1); |
| ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); |
| |
| /* select the point to add or subtract */ |
| select_point(digit, 17, pre_comp[num], tmp); |
| felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */ |
| copy_conditional(tmp[1], tmp[3], sign); |
| |
| if (!skip) { |
| point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, |
| tmp[0], tmp[1], tmp[2]); |
| } else { |
| memcpy(nq, tmp, 3 * sizeof(felem)); |
| skip = 0; |
| } |
| } |
| } |
| |
| if (i == 0) { |
| break; |
| } |
| --i; |
| } |
| felem_assign(x_out, nq[0]); |
| felem_assign(y_out, nq[1]); |
| felem_assign(z_out, nq[2]); |
| } |
| |
| /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns |
| * (X', Y') = (X/Z^2, Y/Z^3) */ |
| static int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group, |
| const EC_POINT *point, |
| BIGNUM *x, BIGNUM *y, |
| BN_CTX *ctx) { |
| felem z1, z2, x_in, y_in, x_out, y_out; |
| widefelem tmp; |
| |
| if (EC_POINT_is_at_infinity(group, point)) { |
| OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); |
| return 0; |
| } |
| |
| if (!BN_to_felem(x_in, &point->X) || |
| !BN_to_felem(y_in, &point->Y) || |
| !BN_to_felem(z1, &point->Z)) { |
| return 0; |
| } |
| |
| felem_inv(z2, z1); |
| felem_square(tmp, z2); |
| felem_reduce(z1, tmp); |
| felem_mul(tmp, x_in, z1); |
| felem_reduce(x_in, tmp); |
| felem_contract(x_out, x_in); |
| if (x != NULL && !felem_to_BN(x, x_out)) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); |
| return 0; |
| } |
| |
| felem_mul(tmp, z1, z2); |
| felem_reduce(z1, tmp); |
| felem_mul(tmp, y_in, z1); |
| felem_reduce(y_in, tmp); |
| felem_contract(y_out, y_in); |
| if (y != NULL && !felem_to_BN(y, y_out)) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); |
| return 0; |
| } |
| |
| return 1; |
| } |
| |
| static int ec_GFp_nistp224_points_mul(const EC_GROUP *group, |
| EC_POINT *r, |
| const BIGNUM *g_scalar, |
| const EC_POINT *p_, |
| const BIGNUM *p_scalar_, |
| BN_CTX *ctx) { |
| /* TODO: This function used to take |points| and |scalars| as arrays of |
| * |num| elements. The code below should be simplified to work in terms of |
| * |p_| and |p_scalar_|. */ |
| size_t num = p_ != NULL ? 1 : 0; |
| const EC_POINT **points = p_ != NULL ? &p_ : NULL; |
| BIGNUM const *const *scalars = p_ != NULL ? &p_scalar_ : NULL; |
| |
| int ret = 0; |
| BN_CTX *new_ctx = NULL; |
| BIGNUM *x, *y, *z, *tmp_scalar; |
| felem_bytearray g_secret; |
| felem_bytearray *secrets = NULL; |
| felem(*pre_comp)[17][3] = NULL; |
| felem_bytearray tmp; |
| size_t num_points = num; |
| felem x_in, y_in, z_in, x_out, y_out, z_out; |
| const EC_POINT *p = NULL; |
| const BIGNUM *p_scalar = NULL; |
| |
| if (ctx == NULL) { |
| ctx = BN_CTX_new(); |
| new_ctx = ctx; |
| if (ctx == NULL) { |
| return 0; |
| } |
| } |
| |
| BN_CTX_start(ctx); |
| if ((x = BN_CTX_get(ctx)) == NULL || |
| (y = BN_CTX_get(ctx)) == NULL || |
| (z = BN_CTX_get(ctx)) == NULL || |
| (tmp_scalar = BN_CTX_get(ctx)) == NULL) { |
| goto err; |
| } |
| |
| if (num_points > 0) { |
| secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray)); |
| pre_comp = OPENSSL_malloc(num_points * sizeof(felem[17][3])); |
| if (secrets == NULL || |
| pre_comp == NULL) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); |
| goto err; |
| } |
| |
| /* we treat NULL scalars as 0, and NULL points as points at infinity, |
| * i.e., they contribute nothing to the linear combination */ |
| memset(secrets, 0, num_points * sizeof(felem_bytearray)); |
| memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem)); |
| size_t i; |
| for (i = 0; i < num_points; ++i) { |
| if (i == num) { |
| /* the generator */ |
| p = EC_GROUP_get0_generator(group); |
| p_scalar = g_scalar; |
| } else { |
| /* the i^th point */ |
| p = points[i]; |
| p_scalar = scalars[i]; |
| } |
| |
| if (p_scalar != NULL && p != NULL) { |
| size_t num_bytes; |
| /* reduce g_scalar to 0 <= g_scalar < 2^224 */ |
| if (BN_num_bits(p_scalar) > 224 || BN_is_negative(p_scalar)) { |
| /* this is an unusual input, and we don't guarantee |
| * constant-timeness */ |
| if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); |
| goto err; |
| } |
| num_bytes = BN_bn2bin(tmp_scalar, tmp); |
| } else { |
| num_bytes = BN_bn2bin(p_scalar, tmp); |
| } |
| |
| flip_endian(secrets[i], tmp, num_bytes); |
| /* precompute multiples */ |
| if (!BN_to_felem(x_out, &p->X) || |
| !BN_to_felem(y_out, &p->Y) || |
| !BN_to_felem(z_out, &p->Z)) { |
| goto err; |
| } |
| |
| felem_assign(pre_comp[i][1][0], x_out); |
| felem_assign(pre_comp[i][1][1], y_out); |
| felem_assign(pre_comp[i][1][2], z_out); |
| |
| size_t j; |
| for (j = 2; j <= 16; ++j) { |
| if (j & 1) { |
| point_add(pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2], |
| pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2], |
| 0, pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], |
| pre_comp[i][j - 1][2]); |
| } else { |
| point_double(pre_comp[i][j][0], pre_comp[i][j][1], |
| pre_comp[i][j][2], pre_comp[i][j / 2][0], |
| pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]); |
| } |
| } |
| } |
| } |
| } |
| |
| if (g_scalar != NULL) { |
| memset(g_secret, 0, sizeof(g_secret)); |
| size_t num_bytes; |
| /* reduce g_scalar to 0 <= g_scalar < 2^224 */ |
| if (BN_num_bits(g_scalar) > 224 || BN_is_negative(g_scalar)) { |
| /* this is an unusual input, and we don't guarantee constant-timeness */ |
| if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); |
| goto err; |
| } |
| num_bytes = BN_bn2bin(tmp_scalar, tmp); |
| } else { |
| num_bytes = BN_bn2bin(g_scalar, tmp); |
| } |
| |
| flip_endian(g_secret, tmp, num_bytes); |
| } |
| batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, |
| num_points, g_scalar != NULL ? g_secret : NULL, |
| (const felem(*)[17][3])pre_comp); |
| |
| /* reduce the output to its unique minimal representation */ |
| felem_contract(x_in, x_out); |
| felem_contract(y_in, y_out); |
| felem_contract(z_in, z_out); |
| if (!felem_to_BN(x, x_in) || |
| !felem_to_BN(y, y_in) || |
| !felem_to_BN(z, z_in)) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); |
| goto err; |
| } |
| ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); |
| |
| err: |
| BN_CTX_end(ctx); |
| BN_CTX_free(new_ctx); |
| OPENSSL_free(secrets); |
| OPENSSL_free(pre_comp); |
| return ret; |
| } |
| |
| const EC_METHOD *EC_GFp_nistp224_method(void) { |
| static const EC_METHOD ret = {ec_GFp_simple_group_init, |
| ec_GFp_simple_group_finish, |
| ec_GFp_simple_group_copy, |
| ec_GFp_simple_group_set_curve, |
| ec_GFp_nistp224_point_get_affine_coordinates, |
| ec_GFp_nistp224_points_mul, |
| 0 /* check_pub_key_order */, |
| ec_GFp_simple_field_mul, |
| ec_GFp_simple_field_sqr, |
| 0 /* field_encode */, |
| 0 /* field_decode */}; |
| |
| return &ret; |
| } |
| |
| #endif /* 64_BIT && !WINDOWS && !SMALL */ |