/* crypto/ec/ecp_nistp224.c */ /* * Written by Emilia Kasper (Google) for the OpenSSL project. */ /* Copyright 2011 Google Inc. * * Licensed under the Apache License, Version 2.0 (the "License"); * * you may not use this file except in compliance with the License. * You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ /* * 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 #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128 #ifndef OPENSSL_SYS_VMS #include #else #include #endif #include #include #include "ec_lcl.h" #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1)) /* even with gcc, the typedef won't work for 32-bit platforms */ typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */ #else #error "Need GCC 3.1 or later to define type uint128_t" #endif typedef uint8_t u8; typedef uint64_t u64; typedef int64_t s64; /******************************************************************************/ /* INTERNAL REPRESENTATION OF FIELD ELEMENTS * * 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]; static const felem_bytearray nistp224_curve_params[5] = { {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, /* p */ 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0x00,0x00,0x00,0x00, 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x01}, {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, /* a */ 0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFF,0xFF, 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE}, {0xB4,0x05,0x0A,0x85,0x0C,0x04,0xB3,0xAB,0xF5,0x41, /* b */ 0x32,0x56,0x50,0x44,0xB0,0xB7,0xD7,0xBF,0xD8,0xBA, 0x27,0x0B,0x39,0x43,0x23,0x55,0xFF,0xB4}, {0xB7,0x0E,0x0C,0xBD,0x6B,0xB4,0xBF,0x7F,0x32,0x13, /* x */ 0x90,0xB9,0x4A,0x03,0xC1,0xD3,0x56,0xC2,0x11,0x22, 0x34,0x32,0x80,0xD6,0x11,0x5C,0x1D,0x21}, {0xbd,0x37,0x63,0x88,0xb5,0xf7,0x23,0xfb,0x4c,0x22, /* y */ 0xdf,0xe6,0xcd,0x43,0x75,0xa0,0x5a,0x07,0x47,0x64, 0x44,0xd5,0x81,0x99,0x85,0x00,0x7e,0x34} }; /* 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 gmul[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}}}}; /* Precomputation for the group generator. */ typedef struct { felem g_pre_comp[2][16][3]; int references; } NISTP224_PRE_COMP; const EC_METHOD *EC_GFp_nistp224_method(void) { static const EC_METHOD ret = { EC_FLAGS_DEFAULT_OCT, NID_X9_62_prime_field, ec_GFp_nistp224_group_init, ec_GFp_simple_group_finish, ec_GFp_simple_group_clear_finish, ec_GFp_nist_group_copy, ec_GFp_nistp224_group_set_curve, ec_GFp_simple_group_get_curve, ec_GFp_simple_group_get_degree, ec_GFp_simple_group_check_discriminant, ec_GFp_simple_point_init, ec_GFp_simple_point_finish, ec_GFp_simple_point_clear_finish, ec_GFp_simple_point_copy, ec_GFp_simple_point_set_to_infinity, ec_GFp_simple_set_Jprojective_coordinates_GFp, ec_GFp_simple_get_Jprojective_coordinates_GFp, ec_GFp_simple_point_set_affine_coordinates, ec_GFp_nistp224_point_get_affine_coordinates, 0 /* point_set_compressed_coordinates */, 0 /* point2oct */, 0 /* oct2point */, ec_GFp_simple_add, ec_GFp_simple_dbl, ec_GFp_simple_invert, ec_GFp_simple_is_at_infinity, ec_GFp_simple_is_on_curve, ec_GFp_simple_cmp, ec_GFp_simple_make_affine, ec_GFp_simple_points_make_affine, ec_GFp_nistp224_points_mul, ec_GFp_nistp224_precompute_mult, ec_GFp_nistp224_have_precompute_mult, ec_GFp_nist_field_mul, ec_GFp_nist_field_sqr, 0 /* field_div */, 0 /* field_encode */, 0 /* field_decode */, 0 /* field_set_to_one */ }; return &ret; } /* 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+21))) & 0x00ffffffffffffff; } static void felem_to_bin28(u8 out[28], const felem in) { unsigned 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, unsigned len) { unsigned 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) { felem_bytearray b_in; felem_bytearray b_out; unsigned num_bytes; /* BN_bn2bin eats leading zeroes */ memset(b_out, 0, sizeof b_out); num_bytes = BN_num_bytes(bn); if (num_bytes > sizeof b_out) { ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); return 0; } if (BN_is_negative(bn)) { ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); return 0; } 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 * * 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_one(felem out) { out[0] = 1; out[1] = 0; out[2] = 0; out[3] = 0; } 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]; } static void felem_square_reduce(felem out, const felem in) { widefelem tmp; felem_square(tmp, in); felem_reduce(out, tmp); } static void felem_mul_reduce(felem out, const felem in1, const felem in2) { widefelem tmp; felem_mul(tmp, in1, in2); felem_reduce(out, tmp); } /* 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, two224m96p1, two225m97p2; zero = in[0] | in[1] | in[2] | in[3]; zero = (((int64_t)(zero) - 1) >> 63) & 1; two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff); two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1; two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff); two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1; return (zero | two224m96p1 | two225m97p2); } static limb felem_is_zero_int(const felem in) { return (int) (felem_is_zero(in) & ((limb)1)); } /* 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; unsigned 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) { unsigned 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, unsigned int size, const felem pre_comp[/*size*/][3], felem out[3]) { unsigned i, j; limb *outlimbs = &out[0][0]; memset(outlimbs, 0, 3 * sizeof(felem)); 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--; 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, unsigned 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 unsigned num_points, const u8 *g_scalar, const int mixed, const felem pre_comp[][17][3], const felem g_pre_comp[2][16][3]) { int i, skip; unsigned num; unsigned gen_mul = (g_scalar != NULL); 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). */ skip = 1; /* save two point operations in the first round */ for (i = (num_points ? 220 : 27); i >= 0; --i) { /* double */ if (!skip) point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); /* add multiples of the generator */ if (gen_mul && (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 && (i % 5 == 0)) { /* loop over all scalars */ 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], mixed, tmp[0], tmp[1], tmp[2]); } else { memcpy(nq, tmp, 3 * sizeof(felem)); skip = 0; } } } } felem_assign(x_out, nq[0]); felem_assign(y_out, nq[1]); felem_assign(z_out, nq[2]); } /******************************************************************************/ /* FUNCTIONS TO MANAGE PRECOMPUTATION */ static NISTP224_PRE_COMP *nistp224_pre_comp_new() { NISTP224_PRE_COMP *ret = NULL; ret = (NISTP224_PRE_COMP *) OPENSSL_malloc(sizeof *ret); if (!ret) { ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); return ret; } memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp)); ret->references = 1; return ret; } static void *nistp224_pre_comp_dup(void *src_) { NISTP224_PRE_COMP *src = src_; /* no need to actually copy, these objects never change! */ CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP); return src_; } static void nistp224_pre_comp_free(void *pre_) { int i; NISTP224_PRE_COMP *pre = pre_; if (!pre) return; i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP); if (i > 0) return; OPENSSL_free(pre); } static void nistp224_pre_comp_clear_free(void *pre_) { int i; NISTP224_PRE_COMP *pre = pre_; if (!pre) return; i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP); if (i > 0) return; OPENSSL_cleanse(pre, sizeof *pre); OPENSSL_free(pre); } /******************************************************************************/ /* OPENSSL EC_METHOD FUNCTIONS */ int ec_GFp_nistp224_group_init(EC_GROUP *group) { int ret; ret = ec_GFp_simple_group_init(group); group->a_is_minus3 = 1; return ret; } int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { int ret = 0; BN_CTX *new_ctx = NULL; BIGNUM *curve_p, *curve_a, *curve_b; if (ctx == NULL) if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0; BN_CTX_start(ctx); if (((curve_p = BN_CTX_get(ctx)) == NULL) || ((curve_a = BN_CTX_get(ctx)) == NULL) || ((curve_b = BN_CTX_get(ctx)) == NULL)) goto err; BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p); BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a); BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b); if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) { ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE, EC_R_WRONG_CURVE_PARAMETERS); goto err; } group->field_mod_func = BN_nist_mod_224; ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); err: BN_CTX_end(ctx); if (new_ctx != NULL) BN_CTX_free(new_ctx); return ret; } /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns * (X', Y') = (X/Z^2, Y/Z^3) */ 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)) { ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES, 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) { if (!felem_to_BN(x, x_out)) { ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES, 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) { if (!felem_to_BN(y, y_out)) { ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB); return 0; } } return 1; } static void make_points_affine(size_t num, felem points[/*num*/][3], felem tmp_felems[/*num+1*/]) { /* Runs in constant time, unless an input is the point at infinity * (which normally shouldn't happen). */ ec_GFp_nistp_points_make_affine_internal( num, points, sizeof(felem), tmp_felems, (void (*)(void *)) felem_one, (int (*)(const void *)) felem_is_zero_int, (void (*)(void *, const void *)) felem_assign, (void (*)(void *, const void *)) felem_square_reduce, (void (*)(void *, const void *, const void *)) felem_mul_reduce, (void (*)(void *, const void *)) felem_inv, (void (*)(void *, const void *)) felem_contract); } /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values * Result is stored in r (r can equal one of the inputs). */ int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) { int ret = 0; int j; unsigned i; int mixed = 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 *tmp_felems = NULL; felem_bytearray tmp; unsigned num_bytes; int have_pre_comp = 0; size_t num_points = num; felem x_in, y_in, z_in, x_out, y_out, z_out; NISTP224_PRE_COMP *pre = NULL; const felem (*g_pre_comp)[16][3] = NULL; EC_POINT *generator = NULL; const EC_POINT *p = NULL; const BIGNUM *p_scalar = NULL; if (ctx == NULL) if ((ctx = new_ctx = BN_CTX_new()) == 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 (scalar != NULL) { pre = EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup, nistp224_pre_comp_free, nistp224_pre_comp_clear_free); if (pre) /* we have precomputation, try to use it */ g_pre_comp = (const felem (*)[16][3]) pre->g_pre_comp; else /* try to use the standard precomputation */ g_pre_comp = &gmul[0]; generator = EC_POINT_new(group); if (generator == NULL) goto err; /* get the generator from precomputation */ if (!felem_to_BN(x, g_pre_comp[0][1][0]) || !felem_to_BN(y, g_pre_comp[0][1][1]) || !felem_to_BN(z, g_pre_comp[0][1][2])) { ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB); goto err; } if (!EC_POINT_set_Jprojective_coordinates_GFp(group, generator, x, y, z, ctx)) goto err; if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) /* precomputation matches generator */ have_pre_comp = 1; else /* we don't have valid precomputation: * treat the generator as a random point */ num_points = num_points + 1; } if (num_points > 0) { if (num_points >= 3) { /* unless we precompute multiples for just one or two points, * converting those into affine form is time well spent */ mixed = 1; } secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray)); pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem)); if (mixed) tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem)); if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_felems == NULL))) { ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, 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)); for (i = 0; i < num_points; ++i) { if (i == num) /* the generator */ { p = EC_GROUP_get0_generator(group); p_scalar = scalar; } else /* the i^th point */ { p = points[i]; p_scalar = scalars[i]; } if ((p_scalar != NULL) && (p != NULL)) { /* reduce scalar to 0 <= 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)) { ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, 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); 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 (mixed) make_points_affine(num_points * 17, pre_comp[0], tmp_felems); } /* the scalar for the generator */ if ((scalar != NULL) && (have_pre_comp)) { memset(g_secret, 0, sizeof g_secret); /* reduce scalar to 0 <= scalar < 2^224 */ if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar))) { /* this is an unusual input, and we don't guarantee * constant-timeness */ if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx)) { ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB); goto err; } num_bytes = BN_bn2bin(tmp_scalar, tmp); } else num_bytes = BN_bn2bin(scalar, tmp); flip_endian(g_secret, tmp, num_bytes); /* do the multiplication with generator precomputation*/ batch_mul(x_out, y_out, z_out, (const felem_bytearray (*)) secrets, num_points, g_secret, mixed, (const felem (*)[17][3]) pre_comp, g_pre_comp); } else /* do the multiplication without generator precomputation */ batch_mul(x_out, y_out, z_out, (const felem_bytearray (*)) secrets, num_points, NULL, mixed, (const felem (*)[17][3]) pre_comp, NULL); /* 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))) { ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB); goto err; } ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); err: BN_CTX_end(ctx); if (generator != NULL) EC_POINT_free(generator); if (new_ctx != NULL) BN_CTX_free(new_ctx); if (secrets != NULL) OPENSSL_free(secrets); if (pre_comp != NULL) OPENSSL_free(pre_comp); if (tmp_felems != NULL) OPENSSL_free(tmp_felems); return ret; } int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx) { int ret = 0; NISTP224_PRE_COMP *pre = NULL; int i, j; BN_CTX *new_ctx = NULL; BIGNUM *x, *y; EC_POINT *generator = NULL; felem tmp_felems[32]; /* throw away old precomputation */ EC_EX_DATA_free_data(&group->extra_data, nistp224_pre_comp_dup, nistp224_pre_comp_free, nistp224_pre_comp_clear_free); if (ctx == NULL) if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0; BN_CTX_start(ctx); if (((x = BN_CTX_get(ctx)) == NULL) || ((y = BN_CTX_get(ctx)) == NULL)) goto err; /* get the generator */ if (group->generator == NULL) goto err; generator = EC_POINT_new(group); if (generator == NULL) goto err; BN_bin2bn(nistp224_curve_params[3], sizeof (felem_bytearray), x); BN_bin2bn(nistp224_curve_params[4], sizeof (felem_bytearray), y); if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx)) goto err; if ((pre = nistp224_pre_comp_new()) == NULL) goto err; /* if the generator is the standard one, use built-in precomputation */ if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) { memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp)); ret = 1; goto err; } if ((!BN_to_felem(pre->g_pre_comp[0][1][0], &group->generator->X)) || (!BN_to_felem(pre->g_pre_comp[0][1][1], &group->generator->Y)) || (!BN_to_felem(pre->g_pre_comp[0][1][2], &group->generator->Z))) goto err; /* compute 2^56*G, 2^112*G, 2^168*G for the first table, * 2^28*G, 2^84*G, 2^140*G, 2^196*G for the second one */ for (i = 1; i <= 8; i <<= 1) { point_double( pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0], pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]); for (j = 0; j < 27; ++j) { point_double( pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]); } if (i == 8) break; point_double( pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2], pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]); for (j = 0; j < 27; ++j) { point_double( pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2], pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2]); } } for (i = 0; i < 2; i++) { /* g_pre_comp[i][0] is the point at infinity */ memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0])); /* the remaining multiples */ /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */ point_add( pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1], pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2], 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]); /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */ point_add( pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1], pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0], pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2], 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]); /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */ point_add( pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0], pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2], 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2]); /* 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G */ point_add( pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1], pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2], 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]); for (j = 1; j < 8; ++j) { /* odd multiples: add G resp. 2^28*G */ point_add( pre->g_pre_comp[i][2*j+1][0], pre->g_pre_comp[i][2*j+1][1], pre->g_pre_comp[i][2*j+1][2], pre->g_pre_comp[i][2*j][0], pre->g_pre_comp[i][2*j][1], pre->g_pre_comp[i][2*j][2], 0, pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1], pre->g_pre_comp[i][1][2]); } } make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems); if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp224_pre_comp_dup, nistp224_pre_comp_free, nistp224_pre_comp_clear_free)) goto err; ret = 1; pre = NULL; err: BN_CTX_end(ctx); if (generator != NULL) EC_POINT_free(generator); if (new_ctx != NULL) BN_CTX_free(new_ctx); if (pre) nistp224_pre_comp_free(pre); return ret; } int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group) { if (EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup, nistp224_pre_comp_free, nistp224_pre_comp_clear_free) != NULL) return 1; else return 0; } #else static void *dummy=&dummy; #endif