/* crypto/ec/ecp_nistp224.c */ /* * Written by Emilia Kasper (Google) for the OpenSSL project. */ /* ==================================================================== * Copyright (c) 2000-2010 The OpenSSL Project. All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in * the documentation and/or other materials provided with the * distribution. * * 3. All advertising materials mentioning features or use of this * software must display the following acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit. (http://www.OpenSSL.org/)" * * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to * endorse or promote products derived from this software without * prior written permission. For written permission, please contact * licensing@OpenSSL.org. * * 5. Products derived from this software may not be called "OpenSSL" * nor may "OpenSSL" appear in their names without prior written * permission of the OpenSSL Project. * * 6. Redistributions of any form whatsoever must retain the following * acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit (http://www.OpenSSL.org/)" * * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED * OF THE POSSIBILITY OF SUCH DAMAGE. * ==================================================================== * * This product includes cryptographic software written by Eric Young * (eay@cryptsoft.com). This product includes software written by Tim * Hudson (tjh@cryptsoft.com). * */ /* * 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 */ #ifdef EC_NISTP224_64_GCC_128 #include #include #include #include "ec_lcl.h" typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */ typedef uint8_t u8; static const u8 nistp224_curve_params[5*28] = { 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 }; /******************************************************************************/ /* INTERNAL REPRESENTATION OF FIELD ELEMENTS * * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 * where each slice a_i is a 64-bit word, i.e., a field element is an fslice * array a with 4 elements, where a[i] = a_i. * Outputs from multiplications are represented as unreduced polynomials * 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 * where each b_i is a 128-bit word. We ensure that inputs to each field * 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 a_i < 2^57. We only reduce to the unique minimal * representation at the end of the computation. * */ typedef uint64_t fslice; /* Field element size (and group order size), in bytes: 28*8 = 224 */ static const unsigned fElemSize = 28; /* Precomputed multiples of the standard generator * b_0*G + b_1*2^56*G + b_2*2^112*G + b_3*2^168*G for * (b_3, b_2, b_1, b_0) in [0,15], i.e., gmul[0] = point_at_infinity, * gmul[1] = G, gmul[2] = 2^56*G, gmul[3] = 2^56*G + G, etc. * Points are given in Jacobian projective coordinates: words 0-3 represent the * X-coordinate (slice a_0 is word 0, etc.), words 4-7 represent the * Y-coordinate and words 8-11 represent the Z-coordinate. */ static const fslice gmul[16][3][4] = { {{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}, {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}, {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}} }; /* Precomputation for the group generator. */ typedef struct { fslice g_pre_comp[16][3][4]; int references; } NISTP224_PRE_COMP; const EC_METHOD *EC_GFp_nistp224_method(void) { static const EC_METHOD ret = { 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, ec_GFp_simple_set_compressed_coordinates, ec_GFp_simple_point2oct, ec_GFp_simple_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(fslice out[4], 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 fslice in[4]) { 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(fslice out[4], const BIGNUM *bn) { u8 b_in[fElemSize]; u8 b_out[fElemSize]; unsigned num_bytes; /* BN_bn2bin eats leading zeroes */ memset(b_out, 0, fElemSize); num_bytes = BN_num_bytes(bn); if (num_bytes > fElemSize) { 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 fslice in[4]) { u8 b_in[fElemSize], b_out[fElemSize]; felem_to_bin28(b_in, in); flip_endian(b_out, b_in, fElemSize); return BN_bin2bn(b_out, fElemSize, 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. * */ /* Sum two field elements: out += in */ static void felem_sum64(fslice out[4], const fslice in[4]) { out[0] += in[0]; out[1] += in[1]; out[2] += in[2]; out[3] += in[3]; } /* Subtract field elements: out -= in */ /* Assumes in[i] < 2^57 */ static void felem_diff64(fslice out[4], const fslice in[4]) { static const uint64_t two58p2 = (((uint64_t) 1) << 58) + (((uint64_t) 1) << 2); static const uint64_t two58m2 = (((uint64_t) 1) << 58) - (((uint64_t) 1) << 2); static const uint64_t two58m42m2 = (((uint64_t) 1) << 58) - (((uint64_t) 1) << 42) - (((uint64_t) 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: out128 -= in128 */ /* Assumes in[i] < 2^119 */ static void felem_diff128(uint128_t out[7], const uint128_t in[4]) { static const uint128_t two120 = ((uint128_t) 1) << 120; static const uint128_t two120m64 = (((uint128_t) 1) << 120) - (((uint128_t) 1) << 64); static const uint128_t two120m104m64 = (((uint128_t) 1) << 120) - (((uint128_t) 1) << 104) - (((uint128_t) 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(uint128_t out[7], const fslice in[4]) { static const uint128_t two64p8 = (((uint128_t) 1) << 64) + (((uint128_t) 1) << 8); static const uint128_t two64m8 = (((uint128_t) 1) << 64) - (((uint128_t) 1) << 8); static const uint128_t two64m48m8 = (((uint128_t) 1) << 64) - (((uint128_t) 1) << 48) - (((uint128_t) 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: out64 = out64 * scalar * The scalars we actually use are small, so results fit without overflow */ static void felem_scalar64(fslice out[4], const fslice scalar) { out[0] *= scalar; out[1] *= scalar; out[2] *= scalar; out[3] *= scalar; } /* Multiply an unreduced field element by a scalar: out128 = out128 * scalar * The scalars we actually use are small, so results fit without overflow */ static void felem_scalar128(uint128_t out[7], const uint128_t 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(uint128_t out[7], const fslice in[4]) { out[0] = ((uint128_t) in[0]) * in[0]; out[1] = ((uint128_t) in[0]) * in[1] * 2; out[2] = ((uint128_t) in[0]) * in[2] * 2 + ((uint128_t) in[1]) * in[1]; out[3] = ((uint128_t) in[0]) * in[3] * 2 + ((uint128_t) in[1]) * in[2] * 2; out[4] = ((uint128_t) in[1]) * in[3] * 2 + ((uint128_t) in[2]) * in[2]; out[5] = ((uint128_t) in[2]) * in[3] * 2; out[6] = ((uint128_t) in[3]) * in[3]; } /* Multiply two field elements: out = in1 * in2 */ static void felem_mul(uint128_t out[7], const fslice in1[4], const fslice in2[4]) { out[0] = ((uint128_t) in1[0]) * in2[0]; out[1] = ((uint128_t) in1[0]) * in2[1] + ((uint128_t) in1[1]) * in2[0]; out[2] = ((uint128_t) in1[0]) * in2[2] + ((uint128_t) in1[1]) * in2[1] + ((uint128_t) in1[2]) * in2[0]; out[3] = ((uint128_t) in1[0]) * in2[3] + ((uint128_t) in1[1]) * in2[2] + ((uint128_t) in1[2]) * in2[1] + ((uint128_t) in1[3]) * in2[0]; out[4] = ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] + ((uint128_t) in1[3]) * in2[1]; out[5] = ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2]; out[6] = ((uint128_t) in1[3]) * in2[3]; } /* Reduce 128-bit coefficients to 64-bit coefficients. Requires in[i] < 2^126, * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] < 2^57 */ static void felem_reduce(fslice out[4], const uint128_t in[7]) { static const uint128_t two127p15 = (((uint128_t) 1) << 127) + (((uint128_t) 1) << 15); static const uint128_t two127m71 = (((uint128_t) 1) << 127) - (((uint128_t) 1) << 71); static const uint128_t two127m71m55 = (((uint128_t) 1) << 127) - (((uint128_t) 1) << 71) - (((uint128_t) 1) << 55); uint128_t 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]; output[4] = 0; /* 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 */ /* Eliminate output[4] */ output[2] += output[4] >> 16; 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; out[1] = output[1] & 0x00ffffffffffffff; output[3] += output[2] >> 56; out[2] = output[2] & 0x00ffffffffffffff; /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, * out[3] < 2^57 (due to final carry) */ out[3] = output[3]; } /* Reduce to unique minimal representation */ static void felem_contract(fslice out[4], const fslice in[4]) { static const int64_t two56 = ((uint64_t) 1) << 56; /* 0 <= in < 2^225 */ /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */ int64_t tmp[4], a; tmp[0] = (int64_t) in[0] - (in[3] >> 56); tmp[1] = (int64_t) in[1] + ((in[3] >> 16) & 0x0000010000000000); tmp[2] = (int64_t) in[2]; tmp[3] = (int64_t) in[3] & 0x00ffffffffffffff; /* eliminate negative coefficients */ a = tmp[0] >> 63; tmp[0] += two56 & a; tmp[1] -= 1 & a; a = tmp[1] >> 63; tmp[1] += two56 & a; tmp[2] -= 1 & a; a = tmp[2] >> 63; tmp[2] += two56 & a; tmp[3] -= 1 & a; a = tmp[3] >> 63; tmp[3] += two56 & a; tmp[0] += 1 & a; tmp[1] -= (1 & a) << 40; /* carry 1 -> 2 -> 3 */ tmp[2] += tmp[1] >> 56; tmp[1] &= 0x00ffffffffffffff; tmp[3] += tmp[2] >> 56; tmp[2] &= 0x00ffffffffffffff; /* 0 <= in < 2^224 + 2^96 - 1 */ /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */ tmp[0] -= (tmp[3] >> 56); tmp[1] += ((tmp[3] >> 16) & 0x0000010000000000); tmp[3] &= 0x00ffffffffffffff; /* eliminate negative coefficients */ a = tmp[0] >> 63; tmp[0] += two56 & a; tmp[1] -= 1 & a; a = tmp[1] >> 63; tmp[1] += two56 & a; tmp[2] -= 1 & a; a = tmp[2] >> 63; tmp[2] += two56 & a; tmp[3] -= 1 & a; a = tmp[3] >> 63; tmp[3] += two56 & a; tmp[0] += 1 & a; tmp[1] -= (1 & a) << 40; /* carry 1 -> 2 -> 3 */ tmp[2] += tmp[1] >> 56; tmp[1] &= 0x00ffffffffffffff; tmp[3] += tmp[2] >> 56; tmp[2] &= 0x00ffffffffffffff; /* Now 0 <= in < 2^224 */ /* if in > 2^224 - 2^96, reduce */ /* a = 0 iff in > 2^224 - 2^96, i.e., * the high 128 bits are all 1 and the lower part is non-zero */ a = (tmp[3] + 1) | (tmp[2] + 1) | ((tmp[1] | 0x000000ffffffffff) + 1) | ((((tmp[1] & 0xffff) - 1) >> 63) & ((tmp[0] - 1) >> 63)); /* turn a into an all-one mask (if a = 0) or an all-zero mask */ a = ((a & 0x00ffffffffffffff) - 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; 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 fslice felem_is_zero(const fslice in[4]) { fslice 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); } /* Invert a field element */ /* Computation chain copied from djb's code */ static void felem_inv(fslice out[4], const fslice in[4]) { fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4]; uint128_t tmp[7]; 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(fslice *out, const fslice *in, unsigned len, fslice icopy) { unsigned i; /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */ const fslice copy = -icopy; for (i = 0; i < len; ++i) { const fslice tmp = copy & (in[i] ^ out[i]); out[i] ^= tmp; } } /* Copy in constant time: * if isel == 1, copy in2 to out, * if isel == 0, copy in1 to out. */ static void select_conditional(fslice *out, const fslice *in1, const fslice *in2, unsigned len, fslice isel) { unsigned i; /* isel is a (64-bit) 0 or 1, so sel is either all-zero or all-one */ const fslice sel = -isel; for (i = 0; i < len; ++i) { const fslice tmp = sel & (in1[i] ^ in2[i]); out[i] = in1[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(fslice x_out[4], fslice y_out[4], fslice z_out[4], const fslice x_in[4], const fslice y_in[4], const fslice z_in[4]) { uint128_t tmp[7], tmp2[7]; fslice delta[4]; fslice gamma[4]; fslice beta[4]; fslice alpha[4]; fslice ftmp[4], ftmp2[4]; memcpy(ftmp, x_in, 4 * sizeof(fslice)); memcpy(ftmp2, x_in, 4 * sizeof(fslice)); /* 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_diff64(ftmp, delta); /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */ felem_sum64(ftmp2, delta); /* ftmp2[i] < 2^57 + 2^57 = 2^58 */ felem_scalar64(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 */ memcpy(ftmp, beta, 4 * sizeof(fslice)); felem_scalar64(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_sum64(delta, gamma); /* delta[i] < 2^57 + 2^57 = 2^58 */ memcpy(ftmp, y_in, 4 * sizeof(fslice)); felem_sum64(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_scalar64(beta, 4); /* beta[i] < 4 * 2^57 = 2^59 */ felem_diff64(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 */ felem_scalar128(tmp2, 8); /* tmp2[i] < 8 * 2^116 = 2^119 */ felem_diff128(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 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(fslice x3[4], fslice y3[4], fslice z3[4], const fslice x1[4], const fslice y1[4], const fslice z1[4], const fslice x2[4], const fslice y2[4], const fslice z2[4]) { fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4], ftmp5[4]; uint128_t tmp[7], tmp2[7]; fslice z1_is_zero, z2_is_zero, x_equal, y_equal; /* ftmp = z1^2 */ felem_square(tmp, z1); felem_reduce(ftmp, tmp); /* ftmp2 = z2^2 */ felem_square(tmp, z2); felem_reduce(ftmp2, tmp); /* ftmp3 = z1^3 */ felem_mul(tmp, ftmp, z1); felem_reduce(ftmp3, tmp); /* ftmp4 = z2^3 */ felem_mul(tmp, ftmp2, z2); felem_reduce(ftmp4, tmp); /* ftmp3 = z1^3*y2 */ felem_mul(tmp, ftmp3, y2); /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ /* ftmp4 = z2^3*y1 */ felem_mul(tmp2, ftmp4, y1); felem_reduce(ftmp4, tmp2); /* 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); /* ftmp = z1^2*x2 */ felem_mul(tmp, ftmp, x2); /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ /* ftmp2 =z2^2*x1 */ felem_mul(tmp2, ftmp2, x1); felem_reduce(ftmp2, tmp2); /* ftmp = z1^2*x2 - z2^2*x1 */ felem_diff128(tmp, tmp2); /* 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 */ felem_mul(tmp, z1, z2); felem_reduce(ftmp5, tmp); /* z3 = (z1^2*x2 - z2^2*x1)*(z1*z2) */ felem_mul(tmp, ftmp, ftmp5); felem_reduce(z3, tmp); /* ftmp = (z1^2*x2 - z2^2*x1)^2 */ memcpy(ftmp5, ftmp, 4 * sizeof(fslice)); 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); /* ftmp4 = 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 */ memcpy(ftmp5, ftmp2, 4 * sizeof(fslice)); felem_scalar64(ftmp5, 2); /* ftmp5[i] < 2 * 2^57 = 2^58 */ /* x3 = (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(x3, tmp2); /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3 */ felem_diff64(ftmp2, x3); /* 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 - x3) */ felem_mul(tmp2, ftmp3, ftmp2); /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */ /* y3 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) - z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ felem_diff128(tmp2, tmp); /* tmp2[i] < 2^118 + 2^120 < 2^121 */ felem_reduce(y3, tmp2); /* the result (x3, y3, z3) 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(x3, x2, 4, z1_is_zero); copy_conditional(x3, x1, 4, z2_is_zero); copy_conditional(y3, y2, 4, z1_is_zero); copy_conditional(y3, y1, 4, z2_is_zero); copy_conditional(z3, z2, 4, z1_is_zero); copy_conditional(z3, z1, 4, z2_is_zero); } /* Select a point from an array of 16 precomputed point multiples, * in constant time: for bits = {b_0, b_1, b_2, b_3}, return the point * pre_comp[8*b_3 + 4*b_2 + 2*b_1 + b_0] */ static void select_point(const fslice bits[4], const fslice pre_comp[16][3][4], fslice out[12]) { fslice tmp[5][12]; select_conditional(tmp[0], pre_comp[7][0], pre_comp[15][0], 12, bits[3]); select_conditional(tmp[1], pre_comp[3][0], pre_comp[11][0], 12, bits[3]); select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]); select_conditional(tmp[0], pre_comp[5][0], pre_comp[13][0], 12, bits[3]); select_conditional(tmp[1], pre_comp[1][0], pre_comp[9][0], 12, bits[3]); select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]); select_conditional(tmp[4], tmp[3], tmp[2], 12, bits[1]); select_conditional(tmp[0], pre_comp[6][0], pre_comp[14][0], 12, bits[3]); select_conditional(tmp[1], pre_comp[2][0], pre_comp[10][0], 12, bits[3]); select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]); select_conditional(tmp[0], pre_comp[4][0], pre_comp[12][0], 12, bits[3]); select_conditional(tmp[1], pre_comp[0][0], pre_comp[8][0], 12, bits[3]); select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]); select_conditional(tmp[1], tmp[3], tmp[2], 12, bits[1]); select_conditional(out, tmp[1], tmp[4], 12, bits[0]); } /* Interleaved point multiplication using precomputed point multiples: * The small point multiples 0*P, 1*P, ..., 15*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(fslice x_out[4], fslice y_out[4], fslice z_out[4], const u8 scalars[][fElemSize], const unsigned num_points, const u8 *g_scalar, const fslice pre_comp[][16][3][4], const fslice g_pre_comp[16][3][4]) { unsigned i, j, num; unsigned gen_mul = (g_scalar != NULL); fslice nq[12], nqt[12], tmp[12]; fslice bits[4]; u8 byte; /* set nq to the point at infinity */ memset(nq, 0, 12 * sizeof(fslice)); /* Loop over all scalars msb-to-lsb, 4 bits at a time: for each nibble, * double 4 times, then add the precomputed point multiples. * If we are also adding multiples of the generator, then interleave * these additions with the last 56 doublings. */ for (i = (num_points ? 28 : 7); i > 0; --i) { for (j = 0; j < 8; ++j) { /* double once */ point_double(nq, nq+4, nq+8, nq, nq+4, nq+8); /* add multiples of the generator */ if ((gen_mul) && (i <= 7)) { bits[3] = (g_scalar[i+20] >> (7-j)) & 1; bits[2] = (g_scalar[i+13] >> (7-j)) & 1; bits[1] = (g_scalar[i+6] >> (7-j)) & 1; bits[0] = (g_scalar[i-1] >> (7-j)) & 1; /* select the point to add, in constant time */ select_point(bits, g_pre_comp, tmp); memcpy(nqt, nq, 12 * sizeof(fslice)); point_add(nq, nq+4, nq+8, nqt, nqt+4, nqt+8, tmp, tmp+4, tmp+8); } /* do an addition after every 4 doublings */ if (j % 4 == 3) { /* loop over all scalars */ for (num = 0; num < num_points; ++num) { byte = scalars[num][i-1]; bits[3] = (byte >> (10-j)) & 1; bits[2] = (byte >> (9-j)) & 1; bits[1] = (byte >> (8-j)) & 1; bits[0] = (byte >> (7-j)) & 1; /* select the point to add */ select_point(bits, pre_comp[num], tmp); memcpy(nqt, nq, 12 * sizeof(fslice)); point_add(nq, nq+4, nq+8, nqt, nqt+4, nqt+8, tmp, tmp+4, tmp+8); } } } } memcpy(x_out, nq, 4 * sizeof(fslice)); memcpy(y_out, nq+4, 4 * sizeof(fslice)); memcpy(z_out, nq+8, 4 * sizeof(fslice)); } /******************************************************************************/ /* FUNCTIONS TO MANAGE PRECOMPUTATION */ static NISTP224_PRE_COMP *nistp224_pre_comp_new() { NISTP224_PRE_COMP *ret = NULL; ret = (NISTP224_PRE_COMP *)OPENSSL_malloc(sizeof(NISTP224_PRE_COMP)); 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, fElemSize, curve_p); BN_bin2bn(nistp224_curve_params + 28, fElemSize, curve_a); BN_bin2bn(nistp224_curve_params + 56, fElemSize, 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) { fslice z1[4], z2[4], x_in[4], y_in[4], x_out[4], y_out[4]; uint128_t tmp[7]; 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; } /* 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 i, j; BN_CTX *new_ctx = NULL; BIGNUM *x, *y, *z, *tmp_scalar; u8 g_secret[fElemSize]; u8 (*secrets)[fElemSize] = NULL; fslice (*pre_comp)[16][3][4] = NULL; u8 tmp[fElemSize]; unsigned num_bytes; int have_pre_comp = 0; size_t num_points = num; fslice x_in[4], y_in[4], z_in[4], x_out[4], y_out[4], z_out[4]; NISTP224_PRE_COMP *pre = NULL; fslice (*g_pre_comp)[3][4] = 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 = pre->g_pre_comp; else /* try to use the standard precomputation */ g_pre_comp = (fslice (*)[3][4]) gmul; generator = EC_POINT_new(group); if (generator == NULL) goto err; /* get the generator from precomputation */ if (!felem_to_BN(x, g_pre_comp[1][0]) || !felem_to_BN(y, g_pre_comp[1][1]) || !felem_to_BN(z, g_pre_comp[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; } secrets = OPENSSL_malloc(num_points * fElemSize); pre_comp = OPENSSL_malloc(num_points * 16 * 3 * 4 * sizeof(fslice)); if ((num_points) && ((secrets == NULL) || (pre_comp == 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 * fElemSize); memset(pre_comp, 0, num_points * 16 * 3 * 4 * sizeof(fslice)); 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)) { num_bytes = BN_num_bytes(p_scalar); /* reduce scalar to 0 <= scalar < 2^224 */ if ((num_bytes > fElemSize) || (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 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; memcpy(pre_comp[i][1][0], x_out, 4 * sizeof(fslice)); memcpy(pre_comp[i][1][1], y_out, 4 * sizeof(fslice)); memcpy(pre_comp[i][1][2], z_out, 4 * sizeof(fslice)); for (j = 1; j < 8; ++j) { point_double(pre_comp[i][2*j][0], pre_comp[i][2*j][1], pre_comp[i][2*j][2], pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2]); point_add(pre_comp[i][2*j+1][0], pre_comp[i][2*j+1][1], pre_comp[i][2*j+1][2], pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2], pre_comp[i][2*j][0], pre_comp[i][2*j][1], pre_comp[i][2*j][2]); } } } /* the scalar for the generator */ if ((scalar != NULL) && (have_pre_comp)) { memset(g_secret, 0, fElemSize); num_bytes = BN_num_bytes(scalar); /* reduce scalar to 0 <= scalar < 2^224 */ if ((num_bytes > fElemSize) || (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 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 u8 (*)[fElemSize]) secrets, num_points, g_secret, (const fslice (*)[16][3][4]) pre_comp, (const fslice (*)[3][4]) g_pre_comp); } else /* do the multiplication without generator precomputation */ batch_mul(x_out, y_out, z_out, (const u8 (*)[fElemSize]) secrets, num_points, NULL, (const fslice (*)[16][3][4]) 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); 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; /* 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 + 84, fElemSize, x); BN_bin2bn(nistp224_curve_params + 112, fElemSize, 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[1][0], &group->generator->X)) || (!BN_to_felem(pre->g_pre_comp[1][1], &group->generator->Y)) || (!BN_to_felem(pre->g_pre_comp[1][2], &group->generator->Z))) goto err; /* compute 2^56*G, 2^112*G, 2^168*G */ for (i = 1; i < 5; ++i) { point_double(pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2], pre->g_pre_comp[i][0], pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]); for (j = 0; j < 55; ++j) { point_double(pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2], pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2]); } } /* g_pre_comp[0] is the point at infinity */ memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0])); /* the remaining multiples */ /* 2^56*G + 2^112*G */ point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1], pre->g_pre_comp[6][2], pre->g_pre_comp[4][0], pre->g_pre_comp[4][1], pre->g_pre_comp[4][2], pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], pre->g_pre_comp[2][2]); /* 2^56*G + 2^168*G */ point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1], pre->g_pre_comp[10][2], pre->g_pre_comp[8][0], pre->g_pre_comp[8][1], pre->g_pre_comp[8][2], pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], pre->g_pre_comp[2][2]); /* 2^112*G + 2^168*G */ point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1], pre->g_pre_comp[12][2], pre->g_pre_comp[8][0], pre->g_pre_comp[8][1], pre->g_pre_comp[8][2], pre->g_pre_comp[4][0], pre->g_pre_comp[4][1], pre->g_pre_comp[4][2]); /* 2^56*G + 2^112*G + 2^168*G */ point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1], pre->g_pre_comp[14][2], pre->g_pre_comp[12][0], pre->g_pre_comp[12][1], pre->g_pre_comp[12][2], pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], pre->g_pre_comp[2][2]); for (i = 1; i < 8; ++i) { /* odd multiples: add G */ point_add(pre->g_pre_comp[2*i+1][0], pre->g_pre_comp[2*i+1][1], pre->g_pre_comp[2*i+1][2], pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2], pre->g_pre_comp[1][0], pre->g_pre_comp[1][1], pre->g_pre_comp[1][2]); } 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; } #endif