/* crypto/ec/ecp_nistp521.c */ /* * Written by Adam Langley (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-521 elliptic curve point multiplication * * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c. * Otherwise based on Emilia's P224 work, which was inspired by my curve25519 * work which got its smarts from Daniel J. Bernstein's work on the same. */ #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; /* * The underlying field. P521 operates over GF(2^521-1). We can serialise an * element of this field into 66 bytes where the most significant byte * contains only a single bit. We call this an felem_bytearray. */ typedef u8 felem_bytearray[66]; /* * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5. * These values are big-endian. */ static const felem_bytearray nistp521_curve_params[5] = { {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff}, {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */ 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc}, {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */ 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85, 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3, 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1, 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e, 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50, 0x3f, 0x00}, {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */ 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95, 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f, 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d, 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7, 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff, 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a, 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5, 0xbd, 0x66}, {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */ 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d, 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b, 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e, 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4, 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad, 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72, 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, 0x66, 0x50} }; /*- * The representation of field elements. * ------------------------------------ * * We represent field elements with nine values. These values are either 64 or * 128 bits and the field element represented is: * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p) * Each of the nine values is called a 'limb'. Since the limbs are spaced only * 58 bits apart, but are greater than 58 bits in length, the most significant * bits of each limb overlap with the least significant bits of the next. * * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a * 'largefelem' */ # define NLIMBS 9 typedef uint64_t limb; typedef limb felem[NLIMBS]; typedef uint128_t largefelem[NLIMBS]; static const limb bottom57bits = 0x1ffffffffffffff; static const limb bottom58bits = 0x3ffffffffffffff; /* * bin66_to_felem takes a little-endian byte array and converts it into felem * form. This assumes that the CPU is little-endian. */ static void bin66_to_felem(felem out, const u8 in[66]) { out[0] = (*((limb *) & in[0])) & bottom58bits; out[1] = (*((limb *) & in[7]) >> 2) & bottom58bits; out[2] = (*((limb *) & in[14]) >> 4) & bottom58bits; out[3] = (*((limb *) & in[21]) >> 6) & bottom58bits; out[4] = (*((limb *) & in[29])) & bottom58bits; out[5] = (*((limb *) & in[36]) >> 2) & bottom58bits; out[6] = (*((limb *) & in[43]) >> 4) & bottom58bits; out[7] = (*((limb *) & in[50]) >> 6) & bottom58bits; out[8] = (*((limb *) & in[58])) & bottom57bits; } /* * felem_to_bin66 takes an felem and serialises into a little endian, 66 byte * array. This assumes that the CPU is little-endian. */ static void felem_to_bin66(u8 out[66], const felem in) { memset(out, 0, 66); (*((limb *) & out[0])) = in[0]; (*((limb *) & out[7])) |= in[1] << 2; (*((limb *) & out[14])) |= in[2] << 4; (*((limb *) & out[21])) |= in[3] << 6; (*((limb *) & out[29])) = in[4]; (*((limb *) & out[36])) |= in[5] << 2; (*((limb *) & out[43])) |= in[6] << 4; (*((limb *) & out[50])) |= in[7] << 6; (*((limb *) & out[58])) = in[8]; } /* 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]; } /* BN_to_felem converts an OpenSSL BIGNUM into an felem */ 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); bin66_to_felem(out, b_out); return 1; } /* felem_to_BN converts an felem into an OpenSSL BIGNUM */ static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) { felem_bytearray b_in, b_out; felem_to_bin66(b_in, in); flip_endian(b_out, b_in, sizeof b_out); return BN_bin2bn(b_out, sizeof b_out, out); } /*- * Field operations * ---------------- */ static void felem_one(felem out) { out[0] = 1; out[1] = 0; out[2] = 0; out[3] = 0; out[4] = 0; out[5] = 0; out[6] = 0; out[7] = 0; out[8] = 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]; out[4] = in[4]; out[5] = in[5]; out[6] = in[6]; out[7] = in[7]; out[8] = in[8]; } /* felem_sum64 sets out = out + in. */ static void felem_sum64(felem out, const felem in) { 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]; out[7] += in[7]; out[8] += in[8]; } /* felem_scalar sets out = in * scalar */ static void felem_scalar(felem out, const felem in, limb scalar) { out[0] = in[0] * scalar; out[1] = in[1] * scalar; out[2] = in[2] * scalar; out[3] = in[3] * scalar; out[4] = in[4] * scalar; out[5] = in[5] * scalar; out[6] = in[6] * scalar; out[7] = in[7] * scalar; out[8] = in[8] * scalar; } /* felem_scalar64 sets out = out * scalar */ static void felem_scalar64(felem out, limb scalar) { out[0] *= scalar; out[1] *= scalar; out[2] *= scalar; out[3] *= scalar; out[4] *= scalar; out[5] *= scalar; out[6] *= scalar; out[7] *= scalar; out[8] *= scalar; } /* felem_scalar128 sets out = out * scalar */ static void felem_scalar128(largefelem out, limb scalar) { out[0] *= scalar; out[1] *= scalar; out[2] *= scalar; out[3] *= scalar; out[4] *= scalar; out[5] *= scalar; out[6] *= scalar; out[7] *= scalar; out[8] *= scalar; } /*- * felem_neg sets |out| to |-in| * On entry: * in[i] < 2^59 + 2^14 * On exit: * out[i] < 2^62 */ static void felem_neg(felem out, const felem in) { /* In order to prevent underflow, we subtract from 0 mod p. */ static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5); static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4); out[0] = two62m3 - in[0]; out[1] = two62m2 - in[1]; out[2] = two62m2 - in[2]; out[3] = two62m2 - in[3]; out[4] = two62m2 - in[4]; out[5] = two62m2 - in[5]; out[6] = two62m2 - in[6]; out[7] = two62m2 - in[7]; out[8] = two62m2 - in[8]; } /*- * felem_diff64 subtracts |in| from |out| * On entry: * in[i] < 2^59 + 2^14 * On exit: * out[i] < out[i] + 2^62 */ static void felem_diff64(felem out, const felem in) { /* * In order to prevent underflow, we add 0 mod p before subtracting. */ static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5); static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4); out[0] += two62m3 - in[0]; out[1] += two62m2 - in[1]; out[2] += two62m2 - in[2]; out[3] += two62m2 - in[3]; out[4] += two62m2 - in[4]; out[5] += two62m2 - in[5]; out[6] += two62m2 - in[6]; out[7] += two62m2 - in[7]; out[8] += two62m2 - in[8]; } /*- * felem_diff_128_64 subtracts |in| from |out| * On entry: * in[i] < 2^62 + 2^17 * On exit: * out[i] < out[i] + 2^63 */ static void felem_diff_128_64(largefelem out, const felem in) { /* * In order to prevent underflow, we add 0 mod p before subtracting. */ static const limb two63m6 = (((limb) 1) << 62) - (((limb) 1) << 5); static const limb two63m5 = (((limb) 1) << 62) - (((limb) 1) << 4); out[0] += two63m6 - in[0]; out[1] += two63m5 - in[1]; out[2] += two63m5 - in[2]; out[3] += two63m5 - in[3]; out[4] += two63m5 - in[4]; out[5] += two63m5 - in[5]; out[6] += two63m5 - in[6]; out[7] += two63m5 - in[7]; out[8] += two63m5 - in[8]; } /*- * felem_diff_128_64 subtracts |in| from |out| * On entry: * in[i] < 2^126 * On exit: * out[i] < out[i] + 2^127 - 2^69 */ static void felem_diff128(largefelem out, const largefelem in) { /* * In order to prevent underflow, we add 0 mod p before subtracting. */ static const uint128_t two127m70 = (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70); static const uint128_t two127m69 = (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69); out[0] += (two127m70 - in[0]); out[1] += (two127m69 - in[1]); out[2] += (two127m69 - in[2]); out[3] += (two127m69 - in[3]); out[4] += (two127m69 - in[4]); out[5] += (two127m69 - in[5]); out[6] += (two127m69 - in[6]); out[7] += (two127m69 - in[7]); out[8] += (two127m69 - in[8]); } /*- * felem_square sets |out| = |in|^2 * On entry: * in[i] < 2^62 * On exit: * out[i] < 17 * max(in[i]) * max(in[i]) */ static void felem_square(largefelem out, const felem in) { felem inx2, inx4; felem_scalar(inx2, in, 2); felem_scalar(inx4, in, 4); /*- * We have many cases were we want to do * in[x] * in[y] + * in[y] * in[x] * This is obviously just * 2 * in[x] * in[y] * However, rather than do the doubling on the 128 bit result, we * double one of the inputs to the multiplication by reading from * |inx2| */ out[0] = ((uint128_t) in[0]) * in[0]; out[1] = ((uint128_t) in[0]) * inx2[1]; out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1]; out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2]; out[4] = ((uint128_t) in[0]) * inx2[4] + ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2]; out[5] = ((uint128_t) in[0]) * inx2[5] + ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3]; out[6] = ((uint128_t) in[0]) * inx2[6] + ((uint128_t) in[1]) * inx2[5] + ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3]; out[7] = ((uint128_t) in[0]) * inx2[7] + ((uint128_t) in[1]) * inx2[6] + ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4]; out[8] = ((uint128_t) in[0]) * inx2[8] + ((uint128_t) in[1]) * inx2[7] + ((uint128_t) in[2]) * inx2[6] + ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4]; /* * The remaining limbs fall above 2^521, with the first falling at 2^522. * They correspond to locations one bit up from the limbs produced above * so we would have to multiply by two to align them. Again, rather than * operate on the 128-bit result, we double one of the inputs to the * multiplication. If we want to double for both this reason, and the * reason above, then we end up multiplying by four. */ /* 9 */ out[0] += ((uint128_t) in[1]) * inx4[8] + ((uint128_t) in[2]) * inx4[7] + ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5]; /* 10 */ out[1] += ((uint128_t) in[2]) * inx4[8] + ((uint128_t) in[3]) * inx4[7] + ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5]; /* 11 */ out[2] += ((uint128_t) in[3]) * inx4[8] + ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6]; /* 12 */ out[3] += ((uint128_t) in[4]) * inx4[8] + ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6]; /* 13 */ out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7]; /* 14 */ out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7]; /* 15 */ out[6] += ((uint128_t) in[7]) * inx4[8]; /* 16 */ out[7] += ((uint128_t) in[8]) * inx2[8]; } /*- * felem_mul sets |out| = |in1| * |in2| * On entry: * in1[i] < 2^64 * in2[i] < 2^63 * On exit: * out[i] < 17 * max(in1[i]) * max(in2[i]) */ static void felem_mul(largefelem out, const felem in1, const felem in2) { felem in2x2; felem_scalar(in2x2, in2, 2); 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[0]) * in2[4] + ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] + ((uint128_t) in1[3]) * in2[1] + ((uint128_t) in1[4]) * in2[0]; out[5] = ((uint128_t) in1[0]) * in2[5] + ((uint128_t) in1[1]) * in2[4] + ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2] + ((uint128_t) in1[4]) * in2[1] + ((uint128_t) in1[5]) * in2[0]; out[6] = ((uint128_t) in1[0]) * in2[6] + ((uint128_t) in1[1]) * in2[5] + ((uint128_t) in1[2]) * in2[4] + ((uint128_t) in1[3]) * in2[3] + ((uint128_t) in1[4]) * in2[2] + ((uint128_t) in1[5]) * in2[1] + ((uint128_t) in1[6]) * in2[0]; out[7] = ((uint128_t) in1[0]) * in2[7] + ((uint128_t) in1[1]) * in2[6] + ((uint128_t) in1[2]) * in2[5] + ((uint128_t) in1[3]) * in2[4] + ((uint128_t) in1[4]) * in2[3] + ((uint128_t) in1[5]) * in2[2] + ((uint128_t) in1[6]) * in2[1] + ((uint128_t) in1[7]) * in2[0]; out[8] = ((uint128_t) in1[0]) * in2[8] + ((uint128_t) in1[1]) * in2[7] + ((uint128_t) in1[2]) * in2[6] + ((uint128_t) in1[3]) * in2[5] + ((uint128_t) in1[4]) * in2[4] + ((uint128_t) in1[5]) * in2[3] + ((uint128_t) in1[6]) * in2[2] + ((uint128_t) in1[7]) * in2[1] + ((uint128_t) in1[8]) * in2[0]; /* See comment in felem_square about the use of in2x2 here */ out[0] += ((uint128_t) in1[1]) * in2x2[8] + ((uint128_t) in1[2]) * in2x2[7] + ((uint128_t) in1[3]) * in2x2[6] + ((uint128_t) in1[4]) * in2x2[5] + ((uint128_t) in1[5]) * in2x2[4] + ((uint128_t) in1[6]) * in2x2[3] + ((uint128_t) in1[7]) * in2x2[2] + ((uint128_t) in1[8]) * in2x2[1]; out[1] += ((uint128_t) in1[2]) * in2x2[8] + ((uint128_t) in1[3]) * in2x2[7] + ((uint128_t) in1[4]) * in2x2[6] + ((uint128_t) in1[5]) * in2x2[5] + ((uint128_t) in1[6]) * in2x2[4] + ((uint128_t) in1[7]) * in2x2[3] + ((uint128_t) in1[8]) * in2x2[2]; out[2] += ((uint128_t) in1[3]) * in2x2[8] + ((uint128_t) in1[4]) * in2x2[7] + ((uint128_t) in1[5]) * in2x2[6] + ((uint128_t) in1[6]) * in2x2[5] + ((uint128_t) in1[7]) * in2x2[4] + ((uint128_t) in1[8]) * in2x2[3]; out[3] += ((uint128_t) in1[4]) * in2x2[8] + ((uint128_t) in1[5]) * in2x2[7] + ((uint128_t) in1[6]) * in2x2[6] + ((uint128_t) in1[7]) * in2x2[5] + ((uint128_t) in1[8]) * in2x2[4]; out[4] += ((uint128_t) in1[5]) * in2x2[8] + ((uint128_t) in1[6]) * in2x2[7] + ((uint128_t) in1[7]) * in2x2[6] + ((uint128_t) in1[8]) * in2x2[5]; out[5] += ((uint128_t) in1[6]) * in2x2[8] + ((uint128_t) in1[7]) * in2x2[7] + ((uint128_t) in1[8]) * in2x2[6]; out[6] += ((uint128_t) in1[7]) * in2x2[8] + ((uint128_t) in1[8]) * in2x2[7]; out[7] += ((uint128_t) in1[8]) * in2x2[8]; } static const limb bottom52bits = 0xfffffffffffff; /*- * felem_reduce converts a largefelem to an felem. * On entry: * in[i] < 2^128 * On exit: * out[i] < 2^59 + 2^14 */ static void felem_reduce(felem out, const largefelem in) { u64 overflow1, overflow2; out[0] = ((limb) in[0]) & bottom58bits; out[1] = ((limb) in[1]) & bottom58bits; out[2] = ((limb) in[2]) & bottom58bits; out[3] = ((limb) in[3]) & bottom58bits; out[4] = ((limb) in[4]) & bottom58bits; out[5] = ((limb) in[5]) & bottom58bits; out[6] = ((limb) in[6]) & bottom58bits; out[7] = ((limb) in[7]) & bottom58bits; out[8] = ((limb) in[8]) & bottom58bits; /* out[i] < 2^58 */ out[1] += ((limb) in[0]) >> 58; out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6; /*- * out[1] < 2^58 + 2^6 + 2^58 * = 2^59 + 2^6 */ out[2] += ((limb) (in[0] >> 64)) >> 52; out[2] += ((limb) in[1]) >> 58; out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6; out[3] += ((limb) (in[1] >> 64)) >> 52; out[3] += ((limb) in[2]) >> 58; out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6; out[4] += ((limb) (in[2] >> 64)) >> 52; out[4] += ((limb) in[3]) >> 58; out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6; out[5] += ((limb) (in[3] >> 64)) >> 52; out[5] += ((limb) in[4]) >> 58; out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6; out[6] += ((limb) (in[4] >> 64)) >> 52; out[6] += ((limb) in[5]) >> 58; out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6; out[7] += ((limb) (in[5] >> 64)) >> 52; out[7] += ((limb) in[6]) >> 58; out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6; out[8] += ((limb) (in[6] >> 64)) >> 52; out[8] += ((limb) in[7]) >> 58; out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6; /*- * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12 * < 2^59 + 2^13 */ overflow1 = ((limb) (in[7] >> 64)) >> 52; overflow1 += ((limb) in[8]) >> 58; overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6; overflow2 = ((limb) (in[8] >> 64)) >> 52; overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */ overflow2 <<= 1; /* overflow2 < 2^13 */ out[0] += overflow1; /* out[0] < 2^60 */ out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */ out[1] += out[0] >> 58; out[0] &= bottom58bits; /*- * out[0] < 2^58 * out[1] < 2^59 + 2^6 + 2^13 + 2^2 * < 2^59 + 2^14 */ } static void felem_square_reduce(felem out, const felem in) { largefelem tmp; felem_square(tmp, in); felem_reduce(out, tmp); } static void felem_mul_reduce(felem out, const felem in1, const felem in2) { largefelem tmp; felem_mul(tmp, in1, in2); felem_reduce(out, tmp); } /*- * felem_inv calculates |out| = |in|^{-1} * * Based on Fermat's Little Theorem: * a^p = a (mod p) * a^{p-1} = 1 (mod p) * a^{p-2} = a^{-1} (mod p) */ static void felem_inv(felem out, const felem in) { felem ftmp, ftmp2, ftmp3, ftmp4; largefelem tmp; unsigned i; felem_square(tmp, in); felem_reduce(ftmp, tmp); /* 2^1 */ felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */ felem_assign(ftmp2, ftmp); felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */ felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */ felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */ felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */ felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */ felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */ felem_assign(ftmp2, ftmp3); felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */ felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */ felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */ felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */ felem_assign(ftmp4, ftmp3); felem_mul(tmp, ftmp3, ftmp); felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */ felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */ felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */ felem_assign(ftmp2, ftmp3); for (i = 0; i < 8; i++) { felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */ } felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */ felem_assign(ftmp2, ftmp3); for (i = 0; i < 16; i++) { felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */ } felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */ felem_assign(ftmp2, ftmp3); for (i = 0; i < 32; i++) { felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */ } felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */ felem_assign(ftmp2, ftmp3); for (i = 0; i < 64; i++) { felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */ } felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */ felem_assign(ftmp2, ftmp3); for (i = 0; i < 128; i++) { felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */ } felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */ felem_assign(ftmp2, ftmp3); for (i = 0; i < 256; i++) { felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */ } felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */ for (i = 0; i < 9; i++) { felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */ } felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */ felem_mul(tmp, ftmp3, in); felem_reduce(out, tmp); /* 2^512 - 3 */ } /* This is 2^521-1, expressed as an felem */ static const felem kPrime = { 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff }; /*- * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0 * otherwise. * On entry: * in[i] < 2^59 + 2^14 */ static limb felem_is_zero(const felem in) { felem ftmp; limb is_zero, is_p; felem_assign(ftmp, in); ftmp[0] += ftmp[8] >> 57; ftmp[8] &= bottom57bits; /* ftmp[8] < 2^57 */ ftmp[1] += ftmp[0] >> 58; ftmp[0] &= bottom58bits; ftmp[2] += ftmp[1] >> 58; ftmp[1] &= bottom58bits; ftmp[3] += ftmp[2] >> 58; ftmp[2] &= bottom58bits; ftmp[4] += ftmp[3] >> 58; ftmp[3] &= bottom58bits; ftmp[5] += ftmp[4] >> 58; ftmp[4] &= bottom58bits; ftmp[6] += ftmp[5] >> 58; ftmp[5] &= bottom58bits; ftmp[7] += ftmp[6] >> 58; ftmp[6] &= bottom58bits; ftmp[8] += ftmp[7] >> 58; ftmp[7] &= bottom58bits; /* ftmp[8] < 2^57 + 4 */ /* * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater * than our bound for ftmp[8]. Therefore we only have to check if the * zero is zero or 2^521-1. */ is_zero = 0; is_zero |= ftmp[0]; is_zero |= ftmp[1]; is_zero |= ftmp[2]; is_zero |= ftmp[3]; is_zero |= ftmp[4]; is_zero |= ftmp[5]; is_zero |= ftmp[6]; is_zero |= ftmp[7]; is_zero |= ftmp[8]; is_zero--; /* * We know that ftmp[i] < 2^63, therefore the only way that the top bit * can be set is if is_zero was 0 before the decrement. */ is_zero = ((s64) is_zero) >> 63; is_p = ftmp[0] ^ kPrime[0]; is_p |= ftmp[1] ^ kPrime[1]; is_p |= ftmp[2] ^ kPrime[2]; is_p |= ftmp[3] ^ kPrime[3]; is_p |= ftmp[4] ^ kPrime[4]; is_p |= ftmp[5] ^ kPrime[5]; is_p |= ftmp[6] ^ kPrime[6]; is_p |= ftmp[7] ^ kPrime[7]; is_p |= ftmp[8] ^ kPrime[8]; is_p--; is_p = ((s64) is_p) >> 63; is_zero |= is_p; return is_zero; } static int felem_is_zero_int(const felem in) { return (int)(felem_is_zero(in) & ((limb) 1)); } /*- * felem_contract converts |in| to its unique, minimal representation. * On entry: * in[i] < 2^59 + 2^14 */ static void felem_contract(felem out, const felem in) { limb is_p, is_greater, sign; static const limb two58 = ((limb) 1) << 58; felem_assign(out, in); out[0] += out[8] >> 57; out[8] &= bottom57bits; /* out[8] < 2^57 */ out[1] += out[0] >> 58; out[0] &= bottom58bits; out[2] += out[1] >> 58; out[1] &= bottom58bits; out[3] += out[2] >> 58; out[2] &= bottom58bits; out[4] += out[3] >> 58; out[3] &= bottom58bits; out[5] += out[4] >> 58; out[4] &= bottom58bits; out[6] += out[5] >> 58; out[5] &= bottom58bits; out[7] += out[6] >> 58; out[6] &= bottom58bits; out[8] += out[7] >> 58; out[7] &= bottom58bits; /* out[8] < 2^57 + 4 */ /* * If the value is greater than 2^521-1 then we have to subtract 2^521-1 * out. See the comments in felem_is_zero regarding why we don't test for * other multiples of the prime. */ /* * First, if |out| is equal to 2^521-1, we subtract it out to get zero. */ is_p = out[0] ^ kPrime[0]; is_p |= out[1] ^ kPrime[1]; is_p |= out[2] ^ kPrime[2]; is_p |= out[3] ^ kPrime[3]; is_p |= out[4] ^ kPrime[4]; is_p |= out[5] ^ kPrime[5]; is_p |= out[6] ^ kPrime[6]; is_p |= out[7] ^ kPrime[7]; is_p |= out[8] ^ kPrime[8]; is_p--; is_p &= is_p << 32; is_p &= is_p << 16; is_p &= is_p << 8; is_p &= is_p << 4; is_p &= is_p << 2; is_p &= is_p << 1; is_p = ((s64) is_p) >> 63; is_p = ~is_p; /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */ out[0] &= is_p; out[1] &= is_p; out[2] &= is_p; out[3] &= is_p; out[4] &= is_p; out[5] &= is_p; out[6] &= is_p; out[7] &= is_p; out[8] &= is_p; /* * In order to test that |out| >= 2^521-1 we need only test if out[8] >> * 57 is greater than zero as (2^521-1) + x >= 2^522 */ is_greater = out[8] >> 57; is_greater |= is_greater << 32; is_greater |= is_greater << 16; is_greater |= is_greater << 8; is_greater |= is_greater << 4; is_greater |= is_greater << 2; is_greater |= is_greater << 1; is_greater = ((s64) is_greater) >> 63; out[0] -= kPrime[0] & is_greater; out[1] -= kPrime[1] & is_greater; out[2] -= kPrime[2] & is_greater; out[3] -= kPrime[3] & is_greater; out[4] -= kPrime[4] & is_greater; out[5] -= kPrime[5] & is_greater; out[6] -= kPrime[6] & is_greater; out[7] -= kPrime[7] & is_greater; out[8] -= kPrime[8] & is_greater; /* Eliminate negative coefficients */ sign = -(out[0] >> 63); out[0] += (two58 & sign); out[1] -= (1 & sign); sign = -(out[1] >> 63); out[1] += (two58 & sign); out[2] -= (1 & sign); sign = -(out[2] >> 63); out[2] += (two58 & sign); out[3] -= (1 & sign); sign = -(out[3] >> 63); out[3] += (two58 & sign); out[4] -= (1 & sign); sign = -(out[4] >> 63); out[4] += (two58 & sign); out[5] -= (1 & sign); sign = -(out[0] >> 63); out[5] += (two58 & sign); out[6] -= (1 & sign); sign = -(out[6] >> 63); out[6] += (two58 & sign); out[7] -= (1 & sign); sign = -(out[7] >> 63); out[7] += (two58 & sign); out[8] -= (1 & sign); sign = -(out[5] >> 63); out[5] += (two58 & sign); out[6] -= (1 & sign); sign = -(out[6] >> 63); out[6] += (two58 & sign); out[7] -= (1 & sign); sign = -(out[7] >> 63); out[7] += (two58 & sign); out[8] -= (1 & sign); } /*- * Group operations * ---------------- * * Building on top of the field operations we have the operations on the * elliptic curve group itself. Points on the curve are represented in Jacobian * coordinates */ /*- * point_double calcuates 2*(x_in, y_in, z_in) * * The method is taken from: * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b * * 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) { largefelem 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); /* delta[i] < 2^59 + 2^14 */ /* gamma = y^2 */ felem_square(tmp, y_in); felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */ /* beta = x*gamma */ felem_mul(tmp, x_in, gamma); felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */ /* alpha = 3*(x-delta)*(x+delta) */ felem_diff64(ftmp, delta); /* ftmp[i] < 2^61 */ felem_sum64(ftmp2, delta); /* ftmp2[i] < 2^60 + 2^15 */ felem_scalar64(ftmp2, 3); /* ftmp2[i] < 3*2^60 + 3*2^15 */ felem_mul(tmp, ftmp, ftmp2); /*- * tmp[i] < 17(3*2^121 + 3*2^76) * = 61*2^121 + 61*2^76 * < 64*2^121 + 64*2^76 * = 2^127 + 2^82 * < 2^128 */ felem_reduce(alpha, tmp); /* x' = alpha^2 - 8*beta */ felem_square(tmp, alpha); /* * tmp[i] < 17*2^120 < 2^125 */ felem_assign(ftmp, beta); felem_scalar64(ftmp, 8); /* ftmp[i] < 2^62 + 2^17 */ felem_diff_128_64(tmp, ftmp); /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */ felem_reduce(x_out, tmp); /* z' = (y + z)^2 - gamma - delta */ felem_sum64(delta, gamma); /* delta[i] < 2^60 + 2^15 */ felem_assign(ftmp, y_in); felem_sum64(ftmp, z_in); /* ftmp[i] < 2^60 + 2^15 */ felem_square(tmp, ftmp); /* * tmp[i] < 17(2^122) < 2^127 */ felem_diff_128_64(tmp, delta); /* tmp[i] < 2^127 + 2^63 */ felem_reduce(z_out, tmp); /* y' = alpha*(4*beta - x') - 8*gamma^2 */ felem_scalar64(beta, 4); /* beta[i] < 2^61 + 2^16 */ felem_diff64(beta, x_out); /* beta[i] < 2^61 + 2^60 + 2^16 */ felem_mul(tmp, alpha, beta); /*- * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16)) * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30) * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30) * < 2^128 */ felem_square(tmp2, gamma); /*- * tmp2[i] < 17*(2^59 + 2^14)^2 * = 17*(2^118 + 2^74 + 2^28) */ felem_scalar128(tmp2, 8); /*- * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28) * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31 * < 2^126 */ felem_diff128(tmp, tmp2); /*- * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30) * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 + * 2^74 + 2^69 + 2^34 + 2^30 * < 2^128 */ felem_reduce(y_out, tmp); } /* copy_conditional copies in to out iff mask is all ones. */ static void copy_conditional(felem out, const felem in, limb mask) { unsigned i; for (i = 0; i < NLIMBS; ++i) { const limb tmp = mask & (in[i] ^ out[i]); out[i] ^= tmp; } } /*- * point_add calcuates (x1, y1, z1) + (x2, y2, z2) * * The method is taken from * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). * * This function 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, ftmp6, x_out, y_out, z_out; largefelem tmp, tmp2; limb x_equal, y_equal, z1_is_zero, z2_is_zero; z1_is_zero = felem_is_zero(z1); z2_is_zero = felem_is_zero(z2); /* ftmp = z1z1 = z1**2 */ felem_square(tmp, z1); felem_reduce(ftmp, tmp); if (!mixed) { /* ftmp2 = z2z2 = z2**2 */ felem_square(tmp, z2); felem_reduce(ftmp2, tmp); /* u1 = ftmp3 = x1*z2z2 */ felem_mul(tmp, x1, ftmp2); felem_reduce(ftmp3, tmp); /* ftmp5 = z1 + z2 */ felem_assign(ftmp5, z1); felem_sum64(ftmp5, z2); /* ftmp5[i] < 2^61 */ /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */ felem_square(tmp, ftmp5); /* tmp[i] < 17*2^122 */ felem_diff_128_64(tmp, ftmp); /* tmp[i] < 17*2^122 + 2^63 */ felem_diff_128_64(tmp, ftmp2); /* tmp[i] < 17*2^122 + 2^64 */ felem_reduce(ftmp5, tmp); /* ftmp2 = z2 * z2z2 */ felem_mul(tmp, ftmp2, z2); felem_reduce(ftmp2, tmp); /* s1 = ftmp6 = y1 * z2**3 */ felem_mul(tmp, y1, ftmp2); felem_reduce(ftmp6, tmp); } else { /* * We'll assume z2 = 1 (special case z2 = 0 is handled later) */ /* u1 = ftmp3 = x1*z2z2 */ felem_assign(ftmp3, x1); /* ftmp5 = 2*z1z2 */ felem_scalar(ftmp5, z1, 2); /* s1 = ftmp6 = y1 * z2**3 */ felem_assign(ftmp6, y1); } /* u2 = x2*z1z1 */ felem_mul(tmp, x2, ftmp); /* tmp[i] < 17*2^120 */ /* h = ftmp4 = u2 - u1 */ felem_diff_128_64(tmp, ftmp3); /* tmp[i] < 17*2^120 + 2^63 */ felem_reduce(ftmp4, tmp); x_equal = felem_is_zero(ftmp4); /* z_out = ftmp5 * h */ felem_mul(tmp, ftmp5, ftmp4); felem_reduce(z_out, tmp); /* ftmp = z1 * z1z1 */ felem_mul(tmp, ftmp, z1); felem_reduce(ftmp, tmp); /* s2 = tmp = y2 * z1**3 */ felem_mul(tmp, y2, ftmp); /* tmp[i] < 17*2^120 */ /* r = ftmp5 = (s2 - s1)*2 */ felem_diff_128_64(tmp, ftmp6); /* tmp[i] < 17*2^120 + 2^63 */ felem_reduce(ftmp5, tmp); y_equal = felem_is_zero(ftmp5); felem_scalar64(ftmp5, 2); /* ftmp5[i] < 2^61 */ if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { point_double(x3, y3, z3, x1, y1, z1); return; } /* I = ftmp = (2h)**2 */ felem_assign(ftmp, ftmp4); felem_scalar64(ftmp, 2); /* ftmp[i] < 2^61 */ felem_square(tmp, ftmp); /* tmp[i] < 17*2^122 */ felem_reduce(ftmp, tmp); /* J = ftmp2 = h * I */ felem_mul(tmp, ftmp4, ftmp); felem_reduce(ftmp2, tmp); /* V = ftmp4 = U1 * I */ felem_mul(tmp, ftmp3, ftmp); felem_reduce(ftmp4, tmp); /* x_out = r**2 - J - 2V */ felem_square(tmp, ftmp5); /* tmp[i] < 17*2^122 */ felem_diff_128_64(tmp, ftmp2); /* tmp[i] < 17*2^122 + 2^63 */ felem_assign(ftmp3, ftmp4); felem_scalar64(ftmp4, 2); /* ftmp4[i] < 2^61 */ felem_diff_128_64(tmp, ftmp4); /* tmp[i] < 17*2^122 + 2^64 */ felem_reduce(x_out, tmp); /* y_out = r(V-x_out) - 2 * s1 * J */ felem_diff64(ftmp3, x_out); /* * ftmp3[i] < 2^60 + 2^60 = 2^61 */ felem_mul(tmp, ftmp5, ftmp3); /* tmp[i] < 17*2^122 */ felem_mul(tmp2, ftmp6, ftmp2); /* tmp2[i] < 17*2^120 */ felem_scalar128(tmp2, 2); /* tmp2[i] < 17*2^121 */ felem_diff128(tmp, tmp2); /*- * tmp[i] < 2^127 - 2^69 + 17*2^122 * = 2^126 - 2^122 - 2^6 - 2^2 - 1 * < 2^127 */ felem_reduce(y_out, tmp); 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); } /*- * Base point pre computation * -------------------------- * * Two different sorts of precomputed tables are used in the following code. * Each contain various points on the curve, where each point is three field * elements (x, y, z). * * For the base point table, z is usually 1 (0 for the point at infinity). * This table has 16 elements: * index | bits | point * ------+---------+------------------------------ * 0 | 0 0 0 0 | 0G * 1 | 0 0 0 1 | 1G * 2 | 0 0 1 0 | 2^130G * 3 | 0 0 1 1 | (2^130 + 1)G * 4 | 0 1 0 0 | 2^260G * 5 | 0 1 0 1 | (2^260 + 1)G * 6 | 0 1 1 0 | (2^260 + 2^130)G * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G * 8 | 1 0 0 0 | 2^390G * 9 | 1 0 0 1 | (2^390 + 1)G * 10 | 1 0 1 0 | (2^390 + 2^130)G * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G * 12 | 1 1 0 0 | (2^390 + 2^260)G * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G * * The reason for this is so that we can clock bits into four different * locations when doing simple scalar multiplies against the base point. * * Tables for other points have table[i] = iG for i in 0 .. 16. */ /* gmul is the table of precomputed base points */ static const felem gmul[16][3] = { {{0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334, 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8, 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404}, {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353, 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45, 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad, 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e, 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5}, {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58, 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c, 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873, 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c, 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9}, {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52, 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e, 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2, 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561, 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065}, {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a, 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e, 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6, 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51, 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe}, {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d, 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c, 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27, 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f, 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256}, {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa, 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2, 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890, 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74, 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23}, {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516, 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1, 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce, 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7, 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5}, {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318, 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83, 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae, 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef, 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203}, {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447, 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283, 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5, 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c, 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a}, {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df, 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645, 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292, 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422, 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b}, {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30, 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb, 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767, 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3, 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf}, {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2, 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692, 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3, 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade, 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684}, {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8, 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a, 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608, 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610, 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d}, {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006, 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86, 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42}, {1, 0, 0, 0, 0, 0, 0, 0, 0}}, {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c, 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9, 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f}, {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7, 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c, 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055}, {1, 0, 0, 0, 0, 0, 0, 0, 0}} }; /* * select_point selects the |idx|th point from a precomputation table and * copies it to out. */ /* pre_comp below is of the size provided in |size| */ static void select_point(const limb idx, unsigned int size, const felem pre_comp[][3], felem out[3]) { unsigned i, j; limb *outlimbs = &out[0][0]; memset(out, 0, sizeof(*out) * 3); for (i = 0; i < size; i++) { const limb *inlimbs = &pre_comp[i][0][0]; limb mask = i ^ idx; mask |= mask >> 4; mask |= mask >> 2; mask |= mask >> 1; mask &= 1; mask--; for (j = 0; j < NLIMBS * 3; j++) outlimbs[j] |= inlimbs[j] & mask; } } /* get_bit returns the |i|th bit in |in| */ static char get_bit(const felem_bytearray in, int i) { if (i < 0) 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[16][3]) { int i, skip; unsigned num, gen_mul = (g_scalar != NULL); felem nq[3], tmp[4]; limb bits; u8 sign, digit; /* set nq to the point at infinity */ memset(nq, 0, sizeof(nq)); /* * Loop over all scalars msb-to-lsb, interleaving additions of multiples * of the generator (last quarter of rounds) and additions of other * points multiples (every 5th round). */ skip = 1; /* save two point operations in the first * round */ for (i = (num_points ? 520 : 130); 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 <= 130)) { bits = get_bit(g_scalar, i + 390) << 3; if (i < 130) { bits |= get_bit(g_scalar, i + 260) << 2; bits |= get_bit(g_scalar, i + 130) << 1; bits |= get_bit(g_scalar, i); } /* select the point to add, in constant time */ select_point(bits, 16, g_pre_comp, tmp); if (!skip) { /* The 1 argument below is for "mixed" */ point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]); } else { memcpy(nq, tmp, 3 * sizeof(felem)); skip = 0; } } /* 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, in constant time */ 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], (-(limb) 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]); } /* Precomputation for the group generator. */ struct nistp521_pre_comp_st { felem g_pre_comp[16][3]; int references; }; const EC_METHOD *EC_GFp_nistp521_method(void) { static const EC_METHOD ret = { EC_FLAGS_DEFAULT_OCT, NID_X9_62_prime_field, ec_GFp_nistp521_group_init, ec_GFp_simple_group_finish, ec_GFp_simple_group_clear_finish, ec_GFp_nist_group_copy, ec_GFp_nistp521_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_nistp521_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_nistp521_points_mul, ec_GFp_nistp521_precompute_mult, ec_GFp_nistp521_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; } /******************************************************************************/ /* * FUNCTIONS TO MANAGE PRECOMPUTATION */ static NISTP521_PRE_COMP *nistp521_pre_comp_new() { NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret)); if (ret == NULL) { ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); return ret; } ret->references = 1; return ret; } NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p) { if (p != NULL) CRYPTO_add(&p->references, 1, CRYPTO_LOCK_EC_PRE_COMP); return p; } void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p) { if (p == NULL || CRYPTO_add(&p->references, -1, CRYPTO_LOCK_EC_PRE_COMP) > 0) return; OPENSSL_free(p); } /******************************************************************************/ /* * OPENSSL EC_METHOD FUNCTIONS */ int ec_GFp_nistp521_group_init(EC_GROUP *group) { int ret; ret = ec_GFp_simple_group_init(group); group->a_is_minus3 = 1; return ret; } int ec_GFp_nistp521_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(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p); BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a); BN_bin2bn(nistp521_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_NISTP521_GROUP_SET_CURVE, EC_R_WRONG_CURVE_PARAMETERS); goto err; } group->field_mod_func = BN_nist_mod_521; ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); err: BN_CTX_end(ctx); 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_nistp521_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; largefelem tmp; if (EC_POINT_is_at_infinity(group, point)) { ECerr(EC_F_EC_GFP_NISTP521_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_NISTP521_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_NISTP521_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB); return 0; } } return 1; } /* points below is of size |num|, and tmp_felems is of size |num+1/ */ static void make_points_affine(size_t num, felem points[][3], felem tmp_felems[]) { /* * 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_nistp521_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; 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 i, 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; NISTP521_PRE_COMP *pre = NULL; felem(*g_pre_comp)[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 = group->pre_comp.nistp521; if (pre) /* we have precomputation, try to use it */ g_pre_comp = &pre->g_pre_comp[0]; else /* try to use the standard precomputation */ g_pre_comp = (felem(*)[3]) 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_NISTP521_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++; } if (num_points > 0) { if (num_points >= 2) { /* * unless we precompute multiples for just one point, converting * those into affine form is time well spent */ mixed = 1; } secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points); pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points); if (mixed) tmp_felems = OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1)); if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_felems == NULL))) { ECerr(EC_F_EC_GFP_NISTP521_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 */ for (i = 0; i < num_points; ++i) { if (i == num) /* * we didn't have a valid precomputation, so we pick 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 < 2^521 */ if ((BN_num_bits(p_scalar) > 521) || (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_NISTP521_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; memcpy(pre_comp[i][1][0], x_out, sizeof(felem)); memcpy(pre_comp[i][1][1], y_out, sizeof(felem)); memcpy(pre_comp[i][1][2], z_out, sizeof(felem)); 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 < 2^521 */ if ((BN_num_bits(scalar) > 521) || (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_NISTP521_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, (const felem(*)[3])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_NISTP521_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); EC_POINT_free(generator); BN_CTX_free(new_ctx); OPENSSL_free(secrets); OPENSSL_free(pre_comp); OPENSSL_free(tmp_felems); return ret; } int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx) { int ret = 0; NISTP521_PRE_COMP *pre = NULL; int i, j; BN_CTX *new_ctx = NULL; BIGNUM *x, *y; EC_POINT *generator = NULL; felem tmp_felems[16]; /* throw away old precomputation */ EC_nistp521_pre_comp_free(group->pre_comp.nistp521); group->pre_comp.nistp521 = 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)) goto err; /* get the generator */ if (group->generator == NULL) goto err; generator = EC_POINT_new(group); if (generator == NULL) goto err; BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x); BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y); if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx)) goto err; if ((pre = nistp521_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 2^130*G, 2^260*G, 2^390*G */ for (i = 1; i <= 4; i <<= 1) { 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 < 129; ++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 */ /* 2^130*G + 2^260*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], 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], pre->g_pre_comp[2][2]); /* 2^130*G + 2^390*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], 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], pre->g_pre_comp[2][2]); /* 2^260*G + 2^390*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], 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1], pre->g_pre_comp[4][2]); /* 2^130*G + 2^260*G + 2^390*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], 0, 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], 0, pre->g_pre_comp[1][0], pre->g_pre_comp[1][1], pre->g_pre_comp[1][2]); } make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems); SETPRECOMP(group, nistp521, pre); ret = 1; pre = NULL; err: BN_CTX_end(ctx); EC_POINT_free(generator); BN_CTX_free(new_ctx); EC_nistp521_pre_comp_free(pre); return ret; } int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group) { return HAVEPRECOMP(group, nistp521); } #else static void *dummy = &dummy; #endif