[331] | 1 | /*
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| 2 | * Copyright 2011-2016 The OpenSSL Project Authors. All Rights Reserved.
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| 3 | *
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| 4 | * Licensed under the OpenSSL license (the "License"). You may not use
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| 5 | * this file except in compliance with the License. You can obtain a copy
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| 6 | * in the file LICENSE in the source distribution or at
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| 7 | * https://www.openssl.org/source/license.html
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| 8 | */
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| 9 |
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| 10 | /* Copyright 2011 Google Inc.
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| 11 | *
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| 12 | * Licensed under the Apache License, Version 2.0 (the "License");
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| 13 | *
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| 14 | * you may not use this file except in compliance with the License.
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| 15 | * You may obtain a copy of the License at
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| 16 | *
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| 17 | * http://www.apache.org/licenses/LICENSE-2.0
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| 18 | *
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| 19 | * Unless required by applicable law or agreed to in writing, software
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| 20 | * distributed under the License is distributed on an "AS IS" BASIS,
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| 21 | * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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| 22 | * See the License for the specific language governing permissions and
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| 23 | * limitations under the License.
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| 24 | */
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| 25 |
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| 26 | /*
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| 27 | * A 64-bit implementation of the NIST P-256 elliptic curve point multiplication
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| 28 | *
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| 29 | * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
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| 30 | * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
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| 31 | * work which got its smarts from Daniel J. Bernstein's work on the same.
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| 32 | */
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| 33 |
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| 34 | #include <openssl/opensslconf.h>
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| 35 | #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
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| 36 | NON_EMPTY_TRANSLATION_UNIT
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| 37 | #else
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| 38 |
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| 39 | # include <stdint.h>
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| 40 | # include <string.h>
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| 41 | # include <openssl/err.h>
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| 42 | # include "ec_lcl.h"
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| 43 |
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| 44 | # if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
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| 45 | /* even with gcc, the typedef won't work for 32-bit platforms */
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| 46 | typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
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| 47 | * platforms */
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| 48 | typedef __int128_t int128_t;
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| 49 | # else
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| 50 | # error "Need GCC 3.1 or later to define type uint128_t"
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| 51 | # endif
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| 52 |
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| 53 | typedef uint8_t u8;
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| 54 | typedef uint32_t u32;
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| 55 | typedef uint64_t u64;
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| 56 | typedef int64_t s64;
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| 57 |
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| 58 | /*
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| 59 | * The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We
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| 60 | * can serialise an element of this field into 32 bytes. We call this an
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| 61 | * felem_bytearray.
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| 62 | */
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| 63 |
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| 64 | typedef u8 felem_bytearray[32];
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| 65 |
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| 66 | /*
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| 67 | * These are the parameters of P256, taken from FIPS 186-3, page 86. These
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| 68 | * values are big-endian.
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| 69 | */
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| 70 | static const felem_bytearray nistp256_curve_params[5] = {
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| 71 | {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */
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| 72 | 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
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| 73 | 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
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| 74 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
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| 75 | {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */
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| 76 | 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
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| 77 | 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
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| 78 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc}, /* b */
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| 79 | {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
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| 80 | 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
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| 81 | 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
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| 82 | 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
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| 83 | {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */
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| 84 | 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
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| 85 | 0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
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| 86 | 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
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| 87 | {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */
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| 88 | 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
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| 89 | 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
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| 90 | 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
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| 91 | };
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| 92 |
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| 93 | /*-
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| 94 | * The representation of field elements.
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| 95 | * ------------------------------------
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| 96 | *
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| 97 | * We represent field elements with either four 128-bit values, eight 128-bit
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| 98 | * values, or four 64-bit values. The field element represented is:
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| 99 | * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p)
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| 100 | * or:
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| 101 | * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p)
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| 102 | *
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| 103 | * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
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| 104 | * apart, but are 128-bits wide, the most significant bits of each limb overlap
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| 105 | * with the least significant bits of the next.
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| 106 | *
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| 107 | * A field element with four limbs is an 'felem'. One with eight limbs is a
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| 108 | * 'longfelem'
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| 109 | *
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| 110 | * A field element with four, 64-bit values is called a 'smallfelem'. Small
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| 111 | * values are used as intermediate values before multiplication.
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| 112 | */
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| 113 |
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| 114 | # define NLIMBS 4
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| 115 |
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| 116 | typedef uint128_t limb;
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| 117 | typedef limb felem[NLIMBS];
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| 118 | typedef limb longfelem[NLIMBS * 2];
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| 119 | typedef u64 smallfelem[NLIMBS];
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| 120 |
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| 121 | /* This is the value of the prime as four 64-bit words, little-endian. */
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| 122 | static const u64 kPrime[4] =
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| 123 | { 0xfffffffffffffffful, 0xffffffff, 0, 0xffffffff00000001ul };
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| 124 | static const u64 bottom63bits = 0x7ffffffffffffffful;
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| 125 |
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| 126 | /*
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| 127 | * bin32_to_felem takes a little-endian byte array and converts it into felem
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| 128 | * form. This assumes that the CPU is little-endian.
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| 129 | */
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| 130 | static void bin32_to_felem(felem out, const u8 in[32])
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| 131 | {
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| 132 | out[0] = *((u64 *)&in[0]);
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| 133 | out[1] = *((u64 *)&in[8]);
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| 134 | out[2] = *((u64 *)&in[16]);
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| 135 | out[3] = *((u64 *)&in[24]);
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| 136 | }
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| 137 |
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| 138 | /*
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| 139 | * smallfelem_to_bin32 takes a smallfelem and serialises into a little
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| 140 | * endian, 32 byte array. This assumes that the CPU is little-endian.
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| 141 | */
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| 142 | static void smallfelem_to_bin32(u8 out[32], const smallfelem in)
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| 143 | {
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| 144 | *((u64 *)&out[0]) = in[0];
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| 145 | *((u64 *)&out[8]) = in[1];
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| 146 | *((u64 *)&out[16]) = in[2];
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| 147 | *((u64 *)&out[24]) = in[3];
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| 148 | }
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| 149 |
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| 150 | /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
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| 151 | static void flip_endian(u8 *out, const u8 *in, unsigned len)
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| 152 | {
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| 153 | unsigned i;
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| 154 | for (i = 0; i < len; ++i)
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| 155 | out[i] = in[len - 1 - i];
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| 156 | }
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| 157 |
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| 158 | /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
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| 159 | static int BN_to_felem(felem out, const BIGNUM *bn)
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| 160 | {
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| 161 | felem_bytearray b_in;
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| 162 | felem_bytearray b_out;
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| 163 | unsigned num_bytes;
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| 164 |
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| 165 | /* BN_bn2bin eats leading zeroes */
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| 166 | memset(b_out, 0, sizeof(b_out));
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| 167 | num_bytes = BN_num_bytes(bn);
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| 168 | if (num_bytes > sizeof b_out) {
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| 169 | ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
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| 170 | return 0;
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| 171 | }
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| 172 | if (BN_is_negative(bn)) {
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| 173 | ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
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| 174 | return 0;
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| 175 | }
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| 176 | num_bytes = BN_bn2bin(bn, b_in);
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| 177 | flip_endian(b_out, b_in, num_bytes);
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| 178 | bin32_to_felem(out, b_out);
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| 179 | return 1;
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| 180 | }
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| 181 |
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| 182 | /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
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| 183 | static BIGNUM *smallfelem_to_BN(BIGNUM *out, const smallfelem in)
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| 184 | {
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| 185 | felem_bytearray b_in, b_out;
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| 186 | smallfelem_to_bin32(b_in, in);
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| 187 | flip_endian(b_out, b_in, sizeof b_out);
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| 188 | return BN_bin2bn(b_out, sizeof b_out, out);
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| 189 | }
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| 190 |
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| 191 | /*-
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| 192 | * Field operations
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| 193 | * ----------------
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| 194 | */
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| 195 |
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| 196 | static void smallfelem_one(smallfelem out)
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| 197 | {
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| 198 | out[0] = 1;
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| 199 | out[1] = 0;
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| 200 | out[2] = 0;
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| 201 | out[3] = 0;
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| 202 | }
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| 203 |
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| 204 | static void smallfelem_assign(smallfelem out, const smallfelem in)
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| 205 | {
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| 206 | out[0] = in[0];
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| 207 | out[1] = in[1];
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| 208 | out[2] = in[2];
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| 209 | out[3] = in[3];
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| 210 | }
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| 211 |
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| 212 | static void felem_assign(felem out, const felem in)
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| 213 | {
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| 214 | out[0] = in[0];
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| 215 | out[1] = in[1];
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| 216 | out[2] = in[2];
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| 217 | out[3] = in[3];
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| 218 | }
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| 219 |
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| 220 | /* felem_sum sets out = out + in. */
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| 221 | static void felem_sum(felem out, const felem in)
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| 222 | {
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| 223 | out[0] += in[0];
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| 224 | out[1] += in[1];
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| 225 | out[2] += in[2];
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| 226 | out[3] += in[3];
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| 227 | }
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| 228 |
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| 229 | /* felem_small_sum sets out = out + in. */
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| 230 | static void felem_small_sum(felem out, const smallfelem in)
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| 231 | {
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| 232 | out[0] += in[0];
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| 233 | out[1] += in[1];
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| 234 | out[2] += in[2];
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| 235 | out[3] += in[3];
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| 236 | }
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| 237 |
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| 238 | /* felem_scalar sets out = out * scalar */
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| 239 | static void felem_scalar(felem out, const u64 scalar)
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| 240 | {
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| 241 | out[0] *= scalar;
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| 242 | out[1] *= scalar;
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| 243 | out[2] *= scalar;
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| 244 | out[3] *= scalar;
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| 245 | }
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| 246 |
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| 247 | /* longfelem_scalar sets out = out * scalar */
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| 248 | static void longfelem_scalar(longfelem out, const u64 scalar)
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| 249 | {
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| 250 | out[0] *= scalar;
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| 251 | out[1] *= scalar;
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| 252 | out[2] *= scalar;
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| 253 | out[3] *= scalar;
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| 254 | out[4] *= scalar;
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| 255 | out[5] *= scalar;
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| 256 | out[6] *= scalar;
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| 257 | out[7] *= scalar;
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| 258 | }
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| 259 |
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| 260 | # define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
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| 261 | # define two105 (((limb)1) << 105)
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| 262 | # define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
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| 263 |
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| 264 | /* zero105 is 0 mod p */
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| 265 | static const felem zero105 =
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| 266 | { two105m41m9, two105, two105m41p9, two105m41p9 };
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| 267 |
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| 268 | /*-
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| 269 | * smallfelem_neg sets |out| to |-small|
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| 270 | * On exit:
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| 271 | * out[i] < out[i] + 2^105
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| 272 | */
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| 273 | static void smallfelem_neg(felem out, const smallfelem small)
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| 274 | {
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| 275 | /* In order to prevent underflow, we subtract from 0 mod p. */
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| 276 | out[0] = zero105[0] - small[0];
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| 277 | out[1] = zero105[1] - small[1];
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| 278 | out[2] = zero105[2] - small[2];
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| 279 | out[3] = zero105[3] - small[3];
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| 280 | }
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| 281 |
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| 282 | /*-
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| 283 | * felem_diff subtracts |in| from |out|
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| 284 | * On entry:
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| 285 | * in[i] < 2^104
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| 286 | * On exit:
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| 287 | * out[i] < out[i] + 2^105
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| 288 | */
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| 289 | static void felem_diff(felem out, const felem in)
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| 290 | {
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| 291 | /*
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| 292 | * In order to prevent underflow, we add 0 mod p before subtracting.
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| 293 | */
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| 294 | out[0] += zero105[0];
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| 295 | out[1] += zero105[1];
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| 296 | out[2] += zero105[2];
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| 297 | out[3] += zero105[3];
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| 298 |
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| 299 | out[0] -= in[0];
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| 300 | out[1] -= in[1];
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| 301 | out[2] -= in[2];
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| 302 | out[3] -= in[3];
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| 303 | }
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| 304 |
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| 305 | # define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
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| 306 | # define two107 (((limb)1) << 107)
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| 307 | # define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
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| 308 |
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| 309 | /* zero107 is 0 mod p */
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| 310 | static const felem zero107 =
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| 311 | { two107m43m11, two107, two107m43p11, two107m43p11 };
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| 312 |
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| 313 | /*-
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| 314 | * An alternative felem_diff for larger inputs |in|
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| 315 | * felem_diff_zero107 subtracts |in| from |out|
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| 316 | * On entry:
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| 317 | * in[i] < 2^106
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| 318 | * On exit:
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| 319 | * out[i] < out[i] + 2^107
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| 320 | */
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| 321 | static void felem_diff_zero107(felem out, const felem in)
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| 322 | {
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| 323 | /*
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| 324 | * In order to prevent underflow, we add 0 mod p before subtracting.
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| 325 | */
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| 326 | out[0] += zero107[0];
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| 327 | out[1] += zero107[1];
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| 328 | out[2] += zero107[2];
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| 329 | out[3] += zero107[3];
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| 330 |
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| 331 | out[0] -= in[0];
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| 332 | out[1] -= in[1];
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| 333 | out[2] -= in[2];
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| 334 | out[3] -= in[3];
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| 335 | }
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| 336 |
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| 337 | /*-
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| 338 | * longfelem_diff subtracts |in| from |out|
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| 339 | * On entry:
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| 340 | * in[i] < 7*2^67
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| 341 | * On exit:
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| 342 | * out[i] < out[i] + 2^70 + 2^40
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| 343 | */
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| 344 | static void longfelem_diff(longfelem out, const longfelem in)
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| 345 | {
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| 346 | static const limb two70m8p6 =
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| 347 | (((limb) 1) << 70) - (((limb) 1) << 8) + (((limb) 1) << 6);
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| 348 | static const limb two70p40 = (((limb) 1) << 70) + (((limb) 1) << 40);
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| 349 | static const limb two70 = (((limb) 1) << 70);
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| 350 | static const limb two70m40m38p6 =
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| 351 | (((limb) 1) << 70) - (((limb) 1) << 40) - (((limb) 1) << 38) +
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| 352 | (((limb) 1) << 6);
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| 353 | static const limb two70m6 = (((limb) 1) << 70) - (((limb) 1) << 6);
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| 354 |
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| 355 | /* add 0 mod p to avoid underflow */
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| 356 | out[0] += two70m8p6;
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| 357 | out[1] += two70p40;
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| 358 | out[2] += two70;
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| 359 | out[3] += two70m40m38p6;
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| 360 | out[4] += two70m6;
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| 361 | out[5] += two70m6;
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| 362 | out[6] += two70m6;
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| 363 | out[7] += two70m6;
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| 364 |
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| 365 | /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
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| 366 | out[0] -= in[0];
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| 367 | out[1] -= in[1];
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| 368 | out[2] -= in[2];
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| 369 | out[3] -= in[3];
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| 370 | out[4] -= in[4];
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| 371 | out[5] -= in[5];
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| 372 | out[6] -= in[6];
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| 373 | out[7] -= in[7];
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| 374 | }
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| 375 |
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| 376 | # define two64m0 (((limb)1) << 64) - 1
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| 377 | # define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
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| 378 | # define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
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| 379 | # define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
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| 380 |
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| 381 | /* zero110 is 0 mod p */
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| 382 | static const felem zero110 = { two64m0, two110p32m0, two64m46, two64m32 };
|
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| 383 |
|
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| 384 | /*-
|
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| 385 | * felem_shrink converts an felem into a smallfelem. The result isn't quite
|
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| 386 | * minimal as the value may be greater than p.
|
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| 387 | *
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| 388 | * On entry:
|
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| 389 | * in[i] < 2^109
|
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| 390 | * On exit:
|
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| 391 | * out[i] < 2^64
|
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| 392 | */
|
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| 393 | static void felem_shrink(smallfelem out, const felem in)
|
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| 394 | {
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| 395 | felem tmp;
|
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| 396 | u64 a, b, mask;
|
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| 397 | s64 high, low;
|
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| 398 | static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */
|
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| 399 |
|
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| 400 | /* Carry 2->3 */
|
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| 401 | tmp[3] = zero110[3] + in[3] + ((u64)(in[2] >> 64));
|
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| 402 | /* tmp[3] < 2^110 */
|
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| 403 |
|
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| 404 | tmp[2] = zero110[2] + (u64)in[2];
|
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| 405 | tmp[0] = zero110[0] + in[0];
|
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| 406 | tmp[1] = zero110[1] + in[1];
|
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| 407 | /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
|
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| 408 |
|
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| 409 | /*
|
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| 410 | * We perform two partial reductions where we eliminate the high-word of
|
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| 411 | * tmp[3]. We don't update the other words till the end.
|
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| 412 | */
|
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| 413 | a = tmp[3] >> 64; /* a < 2^46 */
|
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| 414 | tmp[3] = (u64)tmp[3];
|
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| 415 | tmp[3] -= a;
|
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| 416 | tmp[3] += ((limb) a) << 32;
|
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| 417 | /* tmp[3] < 2^79 */
|
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| 418 |
|
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| 419 | b = a;
|
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| 420 | a = tmp[3] >> 64; /* a < 2^15 */
|
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| 421 | b += a; /* b < 2^46 + 2^15 < 2^47 */
|
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| 422 | tmp[3] = (u64)tmp[3];
|
---|
| 423 | tmp[3] -= a;
|
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| 424 | tmp[3] += ((limb) a) << 32;
|
---|
| 425 | /* tmp[3] < 2^64 + 2^47 */
|
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| 426 |
|
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| 427 | /*
|
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| 428 | * This adjusts the other two words to complete the two partial
|
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| 429 | * reductions.
|
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| 430 | */
|
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| 431 | tmp[0] += b;
|
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| 432 | tmp[1] -= (((limb) b) << 32);
|
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| 433 |
|
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| 434 | /*
|
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| 435 | * In order to make space in tmp[3] for the carry from 2 -> 3, we
|
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| 436 | * conditionally subtract kPrime if tmp[3] is large enough.
|
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| 437 | */
|
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| 438 | high = tmp[3] >> 64;
|
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| 439 | /* As tmp[3] < 2^65, high is either 1 or 0 */
|
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| 440 | high <<= 63;
|
---|
| 441 | high >>= 63;
|
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| 442 | /*-
|
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| 443 | * high is:
|
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| 444 | * all ones if the high word of tmp[3] is 1
|
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| 445 | * all zeros if the high word of tmp[3] if 0 */
|
---|
| 446 | low = tmp[3];
|
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| 447 | mask = low >> 63;
|
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| 448 | /*-
|
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| 449 | * mask is:
|
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| 450 | * all ones if the MSB of low is 1
|
---|
| 451 | * all zeros if the MSB of low if 0 */
|
---|
| 452 | low &= bottom63bits;
|
---|
| 453 | low -= kPrime3Test;
|
---|
| 454 | /* if low was greater than kPrime3Test then the MSB is zero */
|
---|
| 455 | low = ~low;
|
---|
| 456 | low >>= 63;
|
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| 457 | /*-
|
---|
| 458 | * low is:
|
---|
| 459 | * all ones if low was > kPrime3Test
|
---|
| 460 | * all zeros if low was <= kPrime3Test */
|
---|
| 461 | mask = (mask & low) | high;
|
---|
| 462 | tmp[0] -= mask & kPrime[0];
|
---|
| 463 | tmp[1] -= mask & kPrime[1];
|
---|
| 464 | /* kPrime[2] is zero, so omitted */
|
---|
| 465 | tmp[3] -= mask & kPrime[3];
|
---|
| 466 | /* tmp[3] < 2**64 - 2**32 + 1 */
|
---|
| 467 |
|
---|
| 468 | tmp[1] += ((u64)(tmp[0] >> 64));
|
---|
| 469 | tmp[0] = (u64)tmp[0];
|
---|
| 470 | tmp[2] += ((u64)(tmp[1] >> 64));
|
---|
| 471 | tmp[1] = (u64)tmp[1];
|
---|
| 472 | tmp[3] += ((u64)(tmp[2] >> 64));
|
---|
| 473 | tmp[2] = (u64)tmp[2];
|
---|
| 474 | /* tmp[i] < 2^64 */
|
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| 475 |
|
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| 476 | out[0] = tmp[0];
|
---|
| 477 | out[1] = tmp[1];
|
---|
| 478 | out[2] = tmp[2];
|
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| 479 | out[3] = tmp[3];
|
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| 480 | }
|
---|
| 481 |
|
---|
| 482 | /* smallfelem_expand converts a smallfelem to an felem */
|
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| 483 | static void smallfelem_expand(felem out, const smallfelem in)
|
---|
| 484 | {
|
---|
| 485 | out[0] = in[0];
|
---|
| 486 | out[1] = in[1];
|
---|
| 487 | out[2] = in[2];
|
---|
| 488 | out[3] = in[3];
|
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| 489 | }
|
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| 490 |
|
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| 491 | /*-
|
---|
| 492 | * smallfelem_square sets |out| = |small|^2
|
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| 493 | * On entry:
|
---|
| 494 | * small[i] < 2^64
|
---|
| 495 | * On exit:
|
---|
| 496 | * out[i] < 7 * 2^64 < 2^67
|
---|
| 497 | */
|
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| 498 | static void smallfelem_square(longfelem out, const smallfelem small)
|
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| 499 | {
|
---|
| 500 | limb a;
|
---|
| 501 | u64 high, low;
|
---|
| 502 |
|
---|
| 503 | a = ((uint128_t) small[0]) * small[0];
|
---|
| 504 | low = a;
|
---|
| 505 | high = a >> 64;
|
---|
| 506 | out[0] = low;
|
---|
| 507 | out[1] = high;
|
---|
| 508 |
|
---|
| 509 | a = ((uint128_t) small[0]) * small[1];
|
---|
| 510 | low = a;
|
---|
| 511 | high = a >> 64;
|
---|
| 512 | out[1] += low;
|
---|
| 513 | out[1] += low;
|
---|
| 514 | out[2] = high;
|
---|
| 515 |
|
---|
| 516 | a = ((uint128_t) small[0]) * small[2];
|
---|
| 517 | low = a;
|
---|
| 518 | high = a >> 64;
|
---|
| 519 | out[2] += low;
|
---|
| 520 | out[2] *= 2;
|
---|
| 521 | out[3] = high;
|
---|
| 522 |
|
---|
| 523 | a = ((uint128_t) small[0]) * small[3];
|
---|
| 524 | low = a;
|
---|
| 525 | high = a >> 64;
|
---|
| 526 | out[3] += low;
|
---|
| 527 | out[4] = high;
|
---|
| 528 |
|
---|
| 529 | a = ((uint128_t) small[1]) * small[2];
|
---|
| 530 | low = a;
|
---|
| 531 | high = a >> 64;
|
---|
| 532 | out[3] += low;
|
---|
| 533 | out[3] *= 2;
|
---|
| 534 | out[4] += high;
|
---|
| 535 |
|
---|
| 536 | a = ((uint128_t) small[1]) * small[1];
|
---|
| 537 | low = a;
|
---|
| 538 | high = a >> 64;
|
---|
| 539 | out[2] += low;
|
---|
| 540 | out[3] += high;
|
---|
| 541 |
|
---|
| 542 | a = ((uint128_t) small[1]) * small[3];
|
---|
| 543 | low = a;
|
---|
| 544 | high = a >> 64;
|
---|
| 545 | out[4] += low;
|
---|
| 546 | out[4] *= 2;
|
---|
| 547 | out[5] = high;
|
---|
| 548 |
|
---|
| 549 | a = ((uint128_t) small[2]) * small[3];
|
---|
| 550 | low = a;
|
---|
| 551 | high = a >> 64;
|
---|
| 552 | out[5] += low;
|
---|
| 553 | out[5] *= 2;
|
---|
| 554 | out[6] = high;
|
---|
| 555 | out[6] += high;
|
---|
| 556 |
|
---|
| 557 | a = ((uint128_t) small[2]) * small[2];
|
---|
| 558 | low = a;
|
---|
| 559 | high = a >> 64;
|
---|
| 560 | out[4] += low;
|
---|
| 561 | out[5] += high;
|
---|
| 562 |
|
---|
| 563 | a = ((uint128_t) small[3]) * small[3];
|
---|
| 564 | low = a;
|
---|
| 565 | high = a >> 64;
|
---|
| 566 | out[6] += low;
|
---|
| 567 | out[7] = high;
|
---|
| 568 | }
|
---|
| 569 |
|
---|
| 570 | /*-
|
---|
| 571 | * felem_square sets |out| = |in|^2
|
---|
| 572 | * On entry:
|
---|
| 573 | * in[i] < 2^109
|
---|
| 574 | * On exit:
|
---|
| 575 | * out[i] < 7 * 2^64 < 2^67
|
---|
| 576 | */
|
---|
| 577 | static void felem_square(longfelem out, const felem in)
|
---|
| 578 | {
|
---|
| 579 | u64 small[4];
|
---|
| 580 | felem_shrink(small, in);
|
---|
| 581 | smallfelem_square(out, small);
|
---|
| 582 | }
|
---|
| 583 |
|
---|
| 584 | /*-
|
---|
| 585 | * smallfelem_mul sets |out| = |small1| * |small2|
|
---|
| 586 | * On entry:
|
---|
| 587 | * small1[i] < 2^64
|
---|
| 588 | * small2[i] < 2^64
|
---|
| 589 | * On exit:
|
---|
| 590 | * out[i] < 7 * 2^64 < 2^67
|
---|
| 591 | */
|
---|
| 592 | static void smallfelem_mul(longfelem out, const smallfelem small1,
|
---|
| 593 | const smallfelem small2)
|
---|
| 594 | {
|
---|
| 595 | limb a;
|
---|
| 596 | u64 high, low;
|
---|
| 597 |
|
---|
| 598 | a = ((uint128_t) small1[0]) * small2[0];
|
---|
| 599 | low = a;
|
---|
| 600 | high = a >> 64;
|
---|
| 601 | out[0] = low;
|
---|
| 602 | out[1] = high;
|
---|
| 603 |
|
---|
| 604 | a = ((uint128_t) small1[0]) * small2[1];
|
---|
| 605 | low = a;
|
---|
| 606 | high = a >> 64;
|
---|
| 607 | out[1] += low;
|
---|
| 608 | out[2] = high;
|
---|
| 609 |
|
---|
| 610 | a = ((uint128_t) small1[1]) * small2[0];
|
---|
| 611 | low = a;
|
---|
| 612 | high = a >> 64;
|
---|
| 613 | out[1] += low;
|
---|
| 614 | out[2] += high;
|
---|
| 615 |
|
---|
| 616 | a = ((uint128_t) small1[0]) * small2[2];
|
---|
| 617 | low = a;
|
---|
| 618 | high = a >> 64;
|
---|
| 619 | out[2] += low;
|
---|
| 620 | out[3] = high;
|
---|
| 621 |
|
---|
| 622 | a = ((uint128_t) small1[1]) * small2[1];
|
---|
| 623 | low = a;
|
---|
| 624 | high = a >> 64;
|
---|
| 625 | out[2] += low;
|
---|
| 626 | out[3] += high;
|
---|
| 627 |
|
---|
| 628 | a = ((uint128_t) small1[2]) * small2[0];
|
---|
| 629 | low = a;
|
---|
| 630 | high = a >> 64;
|
---|
| 631 | out[2] += low;
|
---|
| 632 | out[3] += high;
|
---|
| 633 |
|
---|
| 634 | a = ((uint128_t) small1[0]) * small2[3];
|
---|
| 635 | low = a;
|
---|
| 636 | high = a >> 64;
|
---|
| 637 | out[3] += low;
|
---|
| 638 | out[4] = high;
|
---|
| 639 |
|
---|
| 640 | a = ((uint128_t) small1[1]) * small2[2];
|
---|
| 641 | low = a;
|
---|
| 642 | high = a >> 64;
|
---|
| 643 | out[3] += low;
|
---|
| 644 | out[4] += high;
|
---|
| 645 |
|
---|
| 646 | a = ((uint128_t) small1[2]) * small2[1];
|
---|
| 647 | low = a;
|
---|
| 648 | high = a >> 64;
|
---|
| 649 | out[3] += low;
|
---|
| 650 | out[4] += high;
|
---|
| 651 |
|
---|
| 652 | a = ((uint128_t) small1[3]) * small2[0];
|
---|
| 653 | low = a;
|
---|
| 654 | high = a >> 64;
|
---|
| 655 | out[3] += low;
|
---|
| 656 | out[4] += high;
|
---|
| 657 |
|
---|
| 658 | a = ((uint128_t) small1[1]) * small2[3];
|
---|
| 659 | low = a;
|
---|
| 660 | high = a >> 64;
|
---|
| 661 | out[4] += low;
|
---|
| 662 | out[5] = high;
|
---|
| 663 |
|
---|
| 664 | a = ((uint128_t) small1[2]) * small2[2];
|
---|
| 665 | low = a;
|
---|
| 666 | high = a >> 64;
|
---|
| 667 | out[4] += low;
|
---|
| 668 | out[5] += high;
|
---|
| 669 |
|
---|
| 670 | a = ((uint128_t) small1[3]) * small2[1];
|
---|
| 671 | low = a;
|
---|
| 672 | high = a >> 64;
|
---|
| 673 | out[4] += low;
|
---|
| 674 | out[5] += high;
|
---|
| 675 |
|
---|
| 676 | a = ((uint128_t) small1[2]) * small2[3];
|
---|
| 677 | low = a;
|
---|
| 678 | high = a >> 64;
|
---|
| 679 | out[5] += low;
|
---|
| 680 | out[6] = high;
|
---|
| 681 |
|
---|
| 682 | a = ((uint128_t) small1[3]) * small2[2];
|
---|
| 683 | low = a;
|
---|
| 684 | high = a >> 64;
|
---|
| 685 | out[5] += low;
|
---|
| 686 | out[6] += high;
|
---|
| 687 |
|
---|
| 688 | a = ((uint128_t) small1[3]) * small2[3];
|
---|
| 689 | low = a;
|
---|
| 690 | high = a >> 64;
|
---|
| 691 | out[6] += low;
|
---|
| 692 | out[7] = high;
|
---|
| 693 | }
|
---|
| 694 |
|
---|
| 695 | /*-
|
---|
| 696 | * felem_mul sets |out| = |in1| * |in2|
|
---|
| 697 | * On entry:
|
---|
| 698 | * in1[i] < 2^109
|
---|
| 699 | * in2[i] < 2^109
|
---|
| 700 | * On exit:
|
---|
| 701 | * out[i] < 7 * 2^64 < 2^67
|
---|
| 702 | */
|
---|
| 703 | static void felem_mul(longfelem out, const felem in1, const felem in2)
|
---|
| 704 | {
|
---|
| 705 | smallfelem small1, small2;
|
---|
| 706 | felem_shrink(small1, in1);
|
---|
| 707 | felem_shrink(small2, in2);
|
---|
| 708 | smallfelem_mul(out, small1, small2);
|
---|
| 709 | }
|
---|
| 710 |
|
---|
| 711 | /*-
|
---|
| 712 | * felem_small_mul sets |out| = |small1| * |in2|
|
---|
| 713 | * On entry:
|
---|
| 714 | * small1[i] < 2^64
|
---|
| 715 | * in2[i] < 2^109
|
---|
| 716 | * On exit:
|
---|
| 717 | * out[i] < 7 * 2^64 < 2^67
|
---|
| 718 | */
|
---|
| 719 | static void felem_small_mul(longfelem out, const smallfelem small1,
|
---|
| 720 | const felem in2)
|
---|
| 721 | {
|
---|
| 722 | smallfelem small2;
|
---|
| 723 | felem_shrink(small2, in2);
|
---|
| 724 | smallfelem_mul(out, small1, small2);
|
---|
| 725 | }
|
---|
| 726 |
|
---|
| 727 | # define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
|
---|
| 728 | # define two100 (((limb)1) << 100)
|
---|
| 729 | # define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
|
---|
| 730 | /* zero100 is 0 mod p */
|
---|
| 731 | static const felem zero100 =
|
---|
| 732 | { two100m36m4, two100, two100m36p4, two100m36p4 };
|
---|
| 733 |
|
---|
| 734 | /*-
|
---|
| 735 | * Internal function for the different flavours of felem_reduce.
|
---|
| 736 | * felem_reduce_ reduces the higher coefficients in[4]-in[7].
|
---|
| 737 | * On entry:
|
---|
| 738 | * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
|
---|
| 739 | * out[1] >= in[7] + 2^32*in[4]
|
---|
| 740 | * out[2] >= in[5] + 2^32*in[5]
|
---|
| 741 | * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
|
---|
| 742 | * On exit:
|
---|
| 743 | * out[0] <= out[0] + in[4] + 2^32*in[5]
|
---|
| 744 | * out[1] <= out[1] + in[5] + 2^33*in[6]
|
---|
| 745 | * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
|
---|
| 746 | * out[3] <= out[3] + 2^32*in[4] + 3*in[7]
|
---|
| 747 | */
|
---|
| 748 | static void felem_reduce_(felem out, const longfelem in)
|
---|
| 749 | {
|
---|
| 750 | int128_t c;
|
---|
| 751 | /* combine common terms from below */
|
---|
| 752 | c = in[4] + (in[5] << 32);
|
---|
| 753 | out[0] += c;
|
---|
| 754 | out[3] -= c;
|
---|
| 755 |
|
---|
| 756 | c = in[5] - in[7];
|
---|
| 757 | out[1] += c;
|
---|
| 758 | out[2] -= c;
|
---|
| 759 |
|
---|
| 760 | /* the remaining terms */
|
---|
| 761 | /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
|
---|
| 762 | out[1] -= (in[4] << 32);
|
---|
| 763 | out[3] += (in[4] << 32);
|
---|
| 764 |
|
---|
| 765 | /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
|
---|
| 766 | out[2] -= (in[5] << 32);
|
---|
| 767 |
|
---|
| 768 | /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
|
---|
| 769 | out[0] -= in[6];
|
---|
| 770 | out[0] -= (in[6] << 32);
|
---|
| 771 | out[1] += (in[6] << 33);
|
---|
| 772 | out[2] += (in[6] * 2);
|
---|
| 773 | out[3] -= (in[6] << 32);
|
---|
| 774 |
|
---|
| 775 | /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
|
---|
| 776 | out[0] -= in[7];
|
---|
| 777 | out[0] -= (in[7] << 32);
|
---|
| 778 | out[2] += (in[7] << 33);
|
---|
| 779 | out[3] += (in[7] * 3);
|
---|
| 780 | }
|
---|
| 781 |
|
---|
| 782 | /*-
|
---|
| 783 | * felem_reduce converts a longfelem into an felem.
|
---|
| 784 | * To be called directly after felem_square or felem_mul.
|
---|
| 785 | * On entry:
|
---|
| 786 | * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
|
---|
| 787 | * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
|
---|
| 788 | * On exit:
|
---|
| 789 | * out[i] < 2^101
|
---|
| 790 | */
|
---|
| 791 | static void felem_reduce(felem out, const longfelem in)
|
---|
| 792 | {
|
---|
| 793 | out[0] = zero100[0] + in[0];
|
---|
| 794 | out[1] = zero100[1] + in[1];
|
---|
| 795 | out[2] = zero100[2] + in[2];
|
---|
| 796 | out[3] = zero100[3] + in[3];
|
---|
| 797 |
|
---|
| 798 | felem_reduce_(out, in);
|
---|
| 799 |
|
---|
| 800 | /*-
|
---|
| 801 | * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
|
---|
| 802 | * out[1] > 2^100 - 2^64 - 7*2^96 > 0
|
---|
| 803 | * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
|
---|
| 804 | * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
|
---|
| 805 | *
|
---|
| 806 | * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
|
---|
| 807 | * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
|
---|
| 808 | * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
|
---|
| 809 | * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101
|
---|
| 810 | */
|
---|
| 811 | }
|
---|
| 812 |
|
---|
| 813 | /*-
|
---|
| 814 | * felem_reduce_zero105 converts a larger longfelem into an felem.
|
---|
| 815 | * On entry:
|
---|
| 816 | * in[0] < 2^71
|
---|
| 817 | * On exit:
|
---|
| 818 | * out[i] < 2^106
|
---|
| 819 | */
|
---|
| 820 | static void felem_reduce_zero105(felem out, const longfelem in)
|
---|
| 821 | {
|
---|
| 822 | out[0] = zero105[0] + in[0];
|
---|
| 823 | out[1] = zero105[1] + in[1];
|
---|
| 824 | out[2] = zero105[2] + in[2];
|
---|
| 825 | out[3] = zero105[3] + in[3];
|
---|
| 826 |
|
---|
| 827 | felem_reduce_(out, in);
|
---|
| 828 |
|
---|
| 829 | /*-
|
---|
| 830 | * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
|
---|
| 831 | * out[1] > 2^105 - 2^71 - 2^103 > 0
|
---|
| 832 | * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
|
---|
| 833 | * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
|
---|
| 834 | *
|
---|
| 835 | * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
|
---|
| 836 | * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
|
---|
| 837 | * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
|
---|
| 838 | * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
|
---|
| 839 | */
|
---|
| 840 | }
|
---|
| 841 |
|
---|
| 842 | /*
|
---|
| 843 | * subtract_u64 sets *result = *result - v and *carry to one if the
|
---|
| 844 | * subtraction underflowed.
|
---|
| 845 | */
|
---|
| 846 | static void subtract_u64(u64 *result, u64 *carry, u64 v)
|
---|
| 847 | {
|
---|
| 848 | uint128_t r = *result;
|
---|
| 849 | r -= v;
|
---|
| 850 | *carry = (r >> 64) & 1;
|
---|
| 851 | *result = (u64)r;
|
---|
| 852 | }
|
---|
| 853 |
|
---|
| 854 | /*
|
---|
| 855 | * felem_contract converts |in| to its unique, minimal representation. On
|
---|
| 856 | * entry: in[i] < 2^109
|
---|
| 857 | */
|
---|
| 858 | static void felem_contract(smallfelem out, const felem in)
|
---|
| 859 | {
|
---|
| 860 | unsigned i;
|
---|
| 861 | u64 all_equal_so_far = 0, result = 0, carry;
|
---|
| 862 |
|
---|
| 863 | felem_shrink(out, in);
|
---|
| 864 | /* small is minimal except that the value might be > p */
|
---|
| 865 |
|
---|
| 866 | all_equal_so_far--;
|
---|
| 867 | /*
|
---|
| 868 | * We are doing a constant time test if out >= kPrime. We need to compare
|
---|
| 869 | * each u64, from most-significant to least significant. For each one, if
|
---|
| 870 | * all words so far have been equal (m is all ones) then a non-equal
|
---|
| 871 | * result is the answer. Otherwise we continue.
|
---|
| 872 | */
|
---|
| 873 | for (i = 3; i < 4; i--) {
|
---|
| 874 | u64 equal;
|
---|
| 875 | uint128_t a = ((uint128_t) kPrime[i]) - out[i];
|
---|
| 876 | /*
|
---|
| 877 | * if out[i] > kPrime[i] then a will underflow and the high 64-bits
|
---|
| 878 | * will all be set.
|
---|
| 879 | */
|
---|
| 880 | result |= all_equal_so_far & ((u64)(a >> 64));
|
---|
| 881 |
|
---|
| 882 | /*
|
---|
| 883 | * if kPrime[i] == out[i] then |equal| will be all zeros and the
|
---|
| 884 | * decrement will make it all ones.
|
---|
| 885 | */
|
---|
| 886 | equal = kPrime[i] ^ out[i];
|
---|
| 887 | equal--;
|
---|
| 888 | equal &= equal << 32;
|
---|
| 889 | equal &= equal << 16;
|
---|
| 890 | equal &= equal << 8;
|
---|
| 891 | equal &= equal << 4;
|
---|
| 892 | equal &= equal << 2;
|
---|
| 893 | equal &= equal << 1;
|
---|
| 894 | equal = ((s64) equal) >> 63;
|
---|
| 895 |
|
---|
| 896 | all_equal_so_far &= equal;
|
---|
| 897 | }
|
---|
| 898 |
|
---|
| 899 | /*
|
---|
| 900 | * if all_equal_so_far is still all ones then the two values are equal
|
---|
| 901 | * and so out >= kPrime is true.
|
---|
| 902 | */
|
---|
| 903 | result |= all_equal_so_far;
|
---|
| 904 |
|
---|
| 905 | /* if out >= kPrime then we subtract kPrime. */
|
---|
| 906 | subtract_u64(&out[0], &carry, result & kPrime[0]);
|
---|
| 907 | subtract_u64(&out[1], &carry, carry);
|
---|
| 908 | subtract_u64(&out[2], &carry, carry);
|
---|
| 909 | subtract_u64(&out[3], &carry, carry);
|
---|
| 910 |
|
---|
| 911 | subtract_u64(&out[1], &carry, result & kPrime[1]);
|
---|
| 912 | subtract_u64(&out[2], &carry, carry);
|
---|
| 913 | subtract_u64(&out[3], &carry, carry);
|
---|
| 914 |
|
---|
| 915 | subtract_u64(&out[2], &carry, result & kPrime[2]);
|
---|
| 916 | subtract_u64(&out[3], &carry, carry);
|
---|
| 917 |
|
---|
| 918 | subtract_u64(&out[3], &carry, result & kPrime[3]);
|
---|
| 919 | }
|
---|
| 920 |
|
---|
| 921 | static void smallfelem_square_contract(smallfelem out, const smallfelem in)
|
---|
| 922 | {
|
---|
| 923 | longfelem longtmp;
|
---|
| 924 | felem tmp;
|
---|
| 925 |
|
---|
| 926 | smallfelem_square(longtmp, in);
|
---|
| 927 | felem_reduce(tmp, longtmp);
|
---|
| 928 | felem_contract(out, tmp);
|
---|
| 929 | }
|
---|
| 930 |
|
---|
| 931 | static void smallfelem_mul_contract(smallfelem out, const smallfelem in1,
|
---|
| 932 | const smallfelem in2)
|
---|
| 933 | {
|
---|
| 934 | longfelem longtmp;
|
---|
| 935 | felem tmp;
|
---|
| 936 |
|
---|
| 937 | smallfelem_mul(longtmp, in1, in2);
|
---|
| 938 | felem_reduce(tmp, longtmp);
|
---|
| 939 | felem_contract(out, tmp);
|
---|
| 940 | }
|
---|
| 941 |
|
---|
| 942 | /*-
|
---|
| 943 | * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
|
---|
| 944 | * otherwise.
|
---|
| 945 | * On entry:
|
---|
| 946 | * small[i] < 2^64
|
---|
| 947 | */
|
---|
| 948 | static limb smallfelem_is_zero(const smallfelem small)
|
---|
| 949 | {
|
---|
| 950 | limb result;
|
---|
| 951 | u64 is_p;
|
---|
| 952 |
|
---|
| 953 | u64 is_zero = small[0] | small[1] | small[2] | small[3];
|
---|
| 954 | is_zero--;
|
---|
| 955 | is_zero &= is_zero << 32;
|
---|
| 956 | is_zero &= is_zero << 16;
|
---|
| 957 | is_zero &= is_zero << 8;
|
---|
| 958 | is_zero &= is_zero << 4;
|
---|
| 959 | is_zero &= is_zero << 2;
|
---|
| 960 | is_zero &= is_zero << 1;
|
---|
| 961 | is_zero = ((s64) is_zero) >> 63;
|
---|
| 962 |
|
---|
| 963 | is_p = (small[0] ^ kPrime[0]) |
|
---|
| 964 | (small[1] ^ kPrime[1]) |
|
---|
| 965 | (small[2] ^ kPrime[2]) | (small[3] ^ kPrime[3]);
|
---|
| 966 | is_p--;
|
---|
| 967 | is_p &= is_p << 32;
|
---|
| 968 | is_p &= is_p << 16;
|
---|
| 969 | is_p &= is_p << 8;
|
---|
| 970 | is_p &= is_p << 4;
|
---|
| 971 | is_p &= is_p << 2;
|
---|
| 972 | is_p &= is_p << 1;
|
---|
| 973 | is_p = ((s64) is_p) >> 63;
|
---|
| 974 |
|
---|
| 975 | is_zero |= is_p;
|
---|
| 976 |
|
---|
| 977 | result = is_zero;
|
---|
| 978 | result |= ((limb) is_zero) << 64;
|
---|
| 979 | return result;
|
---|
| 980 | }
|
---|
| 981 |
|
---|
| 982 | static int smallfelem_is_zero_int(const smallfelem small)
|
---|
| 983 | {
|
---|
| 984 | return (int)(smallfelem_is_zero(small) & ((limb) 1));
|
---|
| 985 | }
|
---|
| 986 |
|
---|
| 987 | /*-
|
---|
| 988 | * felem_inv calculates |out| = |in|^{-1}
|
---|
| 989 | *
|
---|
| 990 | * Based on Fermat's Little Theorem:
|
---|
| 991 | * a^p = a (mod p)
|
---|
| 992 | * a^{p-1} = 1 (mod p)
|
---|
| 993 | * a^{p-2} = a^{-1} (mod p)
|
---|
| 994 | */
|
---|
| 995 | static void felem_inv(felem out, const felem in)
|
---|
| 996 | {
|
---|
| 997 | felem ftmp, ftmp2;
|
---|
| 998 | /* each e_I will hold |in|^{2^I - 1} */
|
---|
| 999 | felem e2, e4, e8, e16, e32, e64;
|
---|
| 1000 | longfelem tmp;
|
---|
| 1001 | unsigned i;
|
---|
| 1002 |
|
---|
| 1003 | felem_square(tmp, in);
|
---|
| 1004 | felem_reduce(ftmp, tmp); /* 2^1 */
|
---|
| 1005 | felem_mul(tmp, in, ftmp);
|
---|
| 1006 | felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
|
---|
| 1007 | felem_assign(e2, ftmp);
|
---|
| 1008 | felem_square(tmp, ftmp);
|
---|
| 1009 | felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
|
---|
| 1010 | felem_square(tmp, ftmp);
|
---|
| 1011 | felem_reduce(ftmp, tmp); /* 2^4 - 2^2 */
|
---|
| 1012 | felem_mul(tmp, ftmp, e2);
|
---|
| 1013 | felem_reduce(ftmp, tmp); /* 2^4 - 2^0 */
|
---|
| 1014 | felem_assign(e4, ftmp);
|
---|
| 1015 | felem_square(tmp, ftmp);
|
---|
| 1016 | felem_reduce(ftmp, tmp); /* 2^5 - 2^1 */
|
---|
| 1017 | felem_square(tmp, ftmp);
|
---|
| 1018 | felem_reduce(ftmp, tmp); /* 2^6 - 2^2 */
|
---|
| 1019 | felem_square(tmp, ftmp);
|
---|
| 1020 | felem_reduce(ftmp, tmp); /* 2^7 - 2^3 */
|
---|
| 1021 | felem_square(tmp, ftmp);
|
---|
| 1022 | felem_reduce(ftmp, tmp); /* 2^8 - 2^4 */
|
---|
| 1023 | felem_mul(tmp, ftmp, e4);
|
---|
| 1024 | felem_reduce(ftmp, tmp); /* 2^8 - 2^0 */
|
---|
| 1025 | felem_assign(e8, ftmp);
|
---|
| 1026 | for (i = 0; i < 8; i++) {
|
---|
| 1027 | felem_square(tmp, ftmp);
|
---|
| 1028 | felem_reduce(ftmp, tmp);
|
---|
| 1029 | } /* 2^16 - 2^8 */
|
---|
| 1030 | felem_mul(tmp, ftmp, e8);
|
---|
| 1031 | felem_reduce(ftmp, tmp); /* 2^16 - 2^0 */
|
---|
| 1032 | felem_assign(e16, ftmp);
|
---|
| 1033 | for (i = 0; i < 16; i++) {
|
---|
| 1034 | felem_square(tmp, ftmp);
|
---|
| 1035 | felem_reduce(ftmp, tmp);
|
---|
| 1036 | } /* 2^32 - 2^16 */
|
---|
| 1037 | felem_mul(tmp, ftmp, e16);
|
---|
| 1038 | felem_reduce(ftmp, tmp); /* 2^32 - 2^0 */
|
---|
| 1039 | felem_assign(e32, ftmp);
|
---|
| 1040 | for (i = 0; i < 32; i++) {
|
---|
| 1041 | felem_square(tmp, ftmp);
|
---|
| 1042 | felem_reduce(ftmp, tmp);
|
---|
| 1043 | } /* 2^64 - 2^32 */
|
---|
| 1044 | felem_assign(e64, ftmp);
|
---|
| 1045 | felem_mul(tmp, ftmp, in);
|
---|
| 1046 | felem_reduce(ftmp, tmp); /* 2^64 - 2^32 + 2^0 */
|
---|
| 1047 | for (i = 0; i < 192; i++) {
|
---|
| 1048 | felem_square(tmp, ftmp);
|
---|
| 1049 | felem_reduce(ftmp, tmp);
|
---|
| 1050 | } /* 2^256 - 2^224 + 2^192 */
|
---|
| 1051 |
|
---|
| 1052 | felem_mul(tmp, e64, e32);
|
---|
| 1053 | felem_reduce(ftmp2, tmp); /* 2^64 - 2^0 */
|
---|
| 1054 | for (i = 0; i < 16; i++) {
|
---|
| 1055 | felem_square(tmp, ftmp2);
|
---|
| 1056 | felem_reduce(ftmp2, tmp);
|
---|
| 1057 | } /* 2^80 - 2^16 */
|
---|
| 1058 | felem_mul(tmp, ftmp2, e16);
|
---|
| 1059 | felem_reduce(ftmp2, tmp); /* 2^80 - 2^0 */
|
---|
| 1060 | for (i = 0; i < 8; i++) {
|
---|
| 1061 | felem_square(tmp, ftmp2);
|
---|
| 1062 | felem_reduce(ftmp2, tmp);
|
---|
| 1063 | } /* 2^88 - 2^8 */
|
---|
| 1064 | felem_mul(tmp, ftmp2, e8);
|
---|
| 1065 | felem_reduce(ftmp2, tmp); /* 2^88 - 2^0 */
|
---|
| 1066 | for (i = 0; i < 4; i++) {
|
---|
| 1067 | felem_square(tmp, ftmp2);
|
---|
| 1068 | felem_reduce(ftmp2, tmp);
|
---|
| 1069 | } /* 2^92 - 2^4 */
|
---|
| 1070 | felem_mul(tmp, ftmp2, e4);
|
---|
| 1071 | felem_reduce(ftmp2, tmp); /* 2^92 - 2^0 */
|
---|
| 1072 | felem_square(tmp, ftmp2);
|
---|
| 1073 | felem_reduce(ftmp2, tmp); /* 2^93 - 2^1 */
|
---|
| 1074 | felem_square(tmp, ftmp2);
|
---|
| 1075 | felem_reduce(ftmp2, tmp); /* 2^94 - 2^2 */
|
---|
| 1076 | felem_mul(tmp, ftmp2, e2);
|
---|
| 1077 | felem_reduce(ftmp2, tmp); /* 2^94 - 2^0 */
|
---|
| 1078 | felem_square(tmp, ftmp2);
|
---|
| 1079 | felem_reduce(ftmp2, tmp); /* 2^95 - 2^1 */
|
---|
| 1080 | felem_square(tmp, ftmp2);
|
---|
| 1081 | felem_reduce(ftmp2, tmp); /* 2^96 - 2^2 */
|
---|
| 1082 | felem_mul(tmp, ftmp2, in);
|
---|
| 1083 | felem_reduce(ftmp2, tmp); /* 2^96 - 3 */
|
---|
| 1084 |
|
---|
| 1085 | felem_mul(tmp, ftmp2, ftmp);
|
---|
| 1086 | felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
|
---|
| 1087 | }
|
---|
| 1088 |
|
---|
| 1089 | static void smallfelem_inv_contract(smallfelem out, const smallfelem in)
|
---|
| 1090 | {
|
---|
| 1091 | felem tmp;
|
---|
| 1092 |
|
---|
| 1093 | smallfelem_expand(tmp, in);
|
---|
| 1094 | felem_inv(tmp, tmp);
|
---|
| 1095 | felem_contract(out, tmp);
|
---|
| 1096 | }
|
---|
| 1097 |
|
---|
| 1098 | /*-
|
---|
| 1099 | * Group operations
|
---|
| 1100 | * ----------------
|
---|
| 1101 | *
|
---|
| 1102 | * Building on top of the field operations we have the operations on the
|
---|
| 1103 | * elliptic curve group itself. Points on the curve are represented in Jacobian
|
---|
| 1104 | * coordinates
|
---|
| 1105 | */
|
---|
| 1106 |
|
---|
| 1107 | /*-
|
---|
| 1108 | * point_double calculates 2*(x_in, y_in, z_in)
|
---|
| 1109 | *
|
---|
| 1110 | * The method is taken from:
|
---|
| 1111 | * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
|
---|
| 1112 | *
|
---|
| 1113 | * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
|
---|
| 1114 | * while x_out == y_in is not (maybe this works, but it's not tested).
|
---|
| 1115 | */
|
---|
| 1116 | static void
|
---|
| 1117 | point_double(felem x_out, felem y_out, felem z_out,
|
---|
| 1118 | const felem x_in, const felem y_in, const felem z_in)
|
---|
| 1119 | {
|
---|
| 1120 | longfelem tmp, tmp2;
|
---|
| 1121 | felem delta, gamma, beta, alpha, ftmp, ftmp2;
|
---|
| 1122 | smallfelem small1, small2;
|
---|
| 1123 |
|
---|
| 1124 | felem_assign(ftmp, x_in);
|
---|
| 1125 | /* ftmp[i] < 2^106 */
|
---|
| 1126 | felem_assign(ftmp2, x_in);
|
---|
| 1127 | /* ftmp2[i] < 2^106 */
|
---|
| 1128 |
|
---|
| 1129 | /* delta = z^2 */
|
---|
| 1130 | felem_square(tmp, z_in);
|
---|
| 1131 | felem_reduce(delta, tmp);
|
---|
| 1132 | /* delta[i] < 2^101 */
|
---|
| 1133 |
|
---|
| 1134 | /* gamma = y^2 */
|
---|
| 1135 | felem_square(tmp, y_in);
|
---|
| 1136 | felem_reduce(gamma, tmp);
|
---|
| 1137 | /* gamma[i] < 2^101 */
|
---|
| 1138 | felem_shrink(small1, gamma);
|
---|
| 1139 |
|
---|
| 1140 | /* beta = x*gamma */
|
---|
| 1141 | felem_small_mul(tmp, small1, x_in);
|
---|
| 1142 | felem_reduce(beta, tmp);
|
---|
| 1143 | /* beta[i] < 2^101 */
|
---|
| 1144 |
|
---|
| 1145 | /* alpha = 3*(x-delta)*(x+delta) */
|
---|
| 1146 | felem_diff(ftmp, delta);
|
---|
| 1147 | /* ftmp[i] < 2^105 + 2^106 < 2^107 */
|
---|
| 1148 | felem_sum(ftmp2, delta);
|
---|
| 1149 | /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
|
---|
| 1150 | felem_scalar(ftmp2, 3);
|
---|
| 1151 | /* ftmp2[i] < 3 * 2^107 < 2^109 */
|
---|
| 1152 | felem_mul(tmp, ftmp, ftmp2);
|
---|
| 1153 | felem_reduce(alpha, tmp);
|
---|
| 1154 | /* alpha[i] < 2^101 */
|
---|
| 1155 | felem_shrink(small2, alpha);
|
---|
| 1156 |
|
---|
| 1157 | /* x' = alpha^2 - 8*beta */
|
---|
| 1158 | smallfelem_square(tmp, small2);
|
---|
| 1159 | felem_reduce(x_out, tmp);
|
---|
| 1160 | felem_assign(ftmp, beta);
|
---|
| 1161 | felem_scalar(ftmp, 8);
|
---|
| 1162 | /* ftmp[i] < 8 * 2^101 = 2^104 */
|
---|
| 1163 | felem_diff(x_out, ftmp);
|
---|
| 1164 | /* x_out[i] < 2^105 + 2^101 < 2^106 */
|
---|
| 1165 |
|
---|
| 1166 | /* z' = (y + z)^2 - gamma - delta */
|
---|
| 1167 | felem_sum(delta, gamma);
|
---|
| 1168 | /* delta[i] < 2^101 + 2^101 = 2^102 */
|
---|
| 1169 | felem_assign(ftmp, y_in);
|
---|
| 1170 | felem_sum(ftmp, z_in);
|
---|
| 1171 | /* ftmp[i] < 2^106 + 2^106 = 2^107 */
|
---|
| 1172 | felem_square(tmp, ftmp);
|
---|
| 1173 | felem_reduce(z_out, tmp);
|
---|
| 1174 | felem_diff(z_out, delta);
|
---|
| 1175 | /* z_out[i] < 2^105 + 2^101 < 2^106 */
|
---|
| 1176 |
|
---|
| 1177 | /* y' = alpha*(4*beta - x') - 8*gamma^2 */
|
---|
| 1178 | felem_scalar(beta, 4);
|
---|
| 1179 | /* beta[i] < 4 * 2^101 = 2^103 */
|
---|
| 1180 | felem_diff_zero107(beta, x_out);
|
---|
| 1181 | /* beta[i] < 2^107 + 2^103 < 2^108 */
|
---|
| 1182 | felem_small_mul(tmp, small2, beta);
|
---|
| 1183 | /* tmp[i] < 7 * 2^64 < 2^67 */
|
---|
| 1184 | smallfelem_square(tmp2, small1);
|
---|
| 1185 | /* tmp2[i] < 7 * 2^64 */
|
---|
| 1186 | longfelem_scalar(tmp2, 8);
|
---|
| 1187 | /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
|
---|
| 1188 | longfelem_diff(tmp, tmp2);
|
---|
| 1189 | /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
|
---|
| 1190 | felem_reduce_zero105(y_out, tmp);
|
---|
| 1191 | /* y_out[i] < 2^106 */
|
---|
| 1192 | }
|
---|
| 1193 |
|
---|
| 1194 | /*
|
---|
| 1195 | * point_double_small is the same as point_double, except that it operates on
|
---|
| 1196 | * smallfelems
|
---|
| 1197 | */
|
---|
| 1198 | static void
|
---|
| 1199 | point_double_small(smallfelem x_out, smallfelem y_out, smallfelem z_out,
|
---|
| 1200 | const smallfelem x_in, const smallfelem y_in,
|
---|
| 1201 | const smallfelem z_in)
|
---|
| 1202 | {
|
---|
| 1203 | felem felem_x_out, felem_y_out, felem_z_out;
|
---|
| 1204 | felem felem_x_in, felem_y_in, felem_z_in;
|
---|
| 1205 |
|
---|
| 1206 | smallfelem_expand(felem_x_in, x_in);
|
---|
| 1207 | smallfelem_expand(felem_y_in, y_in);
|
---|
| 1208 | smallfelem_expand(felem_z_in, z_in);
|
---|
| 1209 | point_double(felem_x_out, felem_y_out, felem_z_out,
|
---|
| 1210 | felem_x_in, felem_y_in, felem_z_in);
|
---|
| 1211 | felem_shrink(x_out, felem_x_out);
|
---|
| 1212 | felem_shrink(y_out, felem_y_out);
|
---|
| 1213 | felem_shrink(z_out, felem_z_out);
|
---|
| 1214 | }
|
---|
| 1215 |
|
---|
| 1216 | /* copy_conditional copies in to out iff mask is all ones. */
|
---|
| 1217 | static void copy_conditional(felem out, const felem in, limb mask)
|
---|
| 1218 | {
|
---|
| 1219 | unsigned i;
|
---|
| 1220 | for (i = 0; i < NLIMBS; ++i) {
|
---|
| 1221 | const limb tmp = mask & (in[i] ^ out[i]);
|
---|
| 1222 | out[i] ^= tmp;
|
---|
| 1223 | }
|
---|
| 1224 | }
|
---|
| 1225 |
|
---|
| 1226 | /* copy_small_conditional copies in to out iff mask is all ones. */
|
---|
| 1227 | static void copy_small_conditional(felem out, const smallfelem in, limb mask)
|
---|
| 1228 | {
|
---|
| 1229 | unsigned i;
|
---|
| 1230 | const u64 mask64 = mask;
|
---|
| 1231 | for (i = 0; i < NLIMBS; ++i) {
|
---|
| 1232 | out[i] = ((limb) (in[i] & mask64)) | (out[i] & ~mask);
|
---|
| 1233 | }
|
---|
| 1234 | }
|
---|
| 1235 |
|
---|
| 1236 | /*-
|
---|
| 1237 | * point_add calculates (x1, y1, z1) + (x2, y2, z2)
|
---|
| 1238 | *
|
---|
| 1239 | * The method is taken from:
|
---|
| 1240 | * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
|
---|
| 1241 | * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
|
---|
| 1242 | *
|
---|
| 1243 | * This function includes a branch for checking whether the two input points
|
---|
| 1244 | * are equal, (while not equal to the point at infinity). This case never
|
---|
| 1245 | * happens during single point multiplication, so there is no timing leak for
|
---|
| 1246 | * ECDH or ECDSA signing.
|
---|
| 1247 | */
|
---|
| 1248 | static void point_add(felem x3, felem y3, felem z3,
|
---|
| 1249 | const felem x1, const felem y1, const felem z1,
|
---|
| 1250 | const int mixed, const smallfelem x2,
|
---|
| 1251 | const smallfelem y2, const smallfelem z2)
|
---|
| 1252 | {
|
---|
| 1253 | felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
|
---|
| 1254 | longfelem tmp, tmp2;
|
---|
| 1255 | smallfelem small1, small2, small3, small4, small5;
|
---|
| 1256 | limb x_equal, y_equal, z1_is_zero, z2_is_zero;
|
---|
| 1257 |
|
---|
| 1258 | felem_shrink(small3, z1);
|
---|
| 1259 |
|
---|
| 1260 | z1_is_zero = smallfelem_is_zero(small3);
|
---|
| 1261 | z2_is_zero = smallfelem_is_zero(z2);
|
---|
| 1262 |
|
---|
| 1263 | /* ftmp = z1z1 = z1**2 */
|
---|
| 1264 | smallfelem_square(tmp, small3);
|
---|
| 1265 | felem_reduce(ftmp, tmp);
|
---|
| 1266 | /* ftmp[i] < 2^101 */
|
---|
| 1267 | felem_shrink(small1, ftmp);
|
---|
| 1268 |
|
---|
| 1269 | if (!mixed) {
|
---|
| 1270 | /* ftmp2 = z2z2 = z2**2 */
|
---|
| 1271 | smallfelem_square(tmp, z2);
|
---|
| 1272 | felem_reduce(ftmp2, tmp);
|
---|
| 1273 | /* ftmp2[i] < 2^101 */
|
---|
| 1274 | felem_shrink(small2, ftmp2);
|
---|
| 1275 |
|
---|
| 1276 | felem_shrink(small5, x1);
|
---|
| 1277 |
|
---|
| 1278 | /* u1 = ftmp3 = x1*z2z2 */
|
---|
| 1279 | smallfelem_mul(tmp, small5, small2);
|
---|
| 1280 | felem_reduce(ftmp3, tmp);
|
---|
| 1281 | /* ftmp3[i] < 2^101 */
|
---|
| 1282 |
|
---|
| 1283 | /* ftmp5 = z1 + z2 */
|
---|
| 1284 | felem_assign(ftmp5, z1);
|
---|
| 1285 | felem_small_sum(ftmp5, z2);
|
---|
| 1286 | /* ftmp5[i] < 2^107 */
|
---|
| 1287 |
|
---|
| 1288 | /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
|
---|
| 1289 | felem_square(tmp, ftmp5);
|
---|
| 1290 | felem_reduce(ftmp5, tmp);
|
---|
| 1291 | /* ftmp2 = z2z2 + z1z1 */
|
---|
| 1292 | felem_sum(ftmp2, ftmp);
|
---|
| 1293 | /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
|
---|
| 1294 | felem_diff(ftmp5, ftmp2);
|
---|
| 1295 | /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
|
---|
| 1296 |
|
---|
| 1297 | /* ftmp2 = z2 * z2z2 */
|
---|
| 1298 | smallfelem_mul(tmp, small2, z2);
|
---|
| 1299 | felem_reduce(ftmp2, tmp);
|
---|
| 1300 |
|
---|
| 1301 | /* s1 = ftmp2 = y1 * z2**3 */
|
---|
| 1302 | felem_mul(tmp, y1, ftmp2);
|
---|
| 1303 | felem_reduce(ftmp6, tmp);
|
---|
| 1304 | /* ftmp6[i] < 2^101 */
|
---|
| 1305 | } else {
|
---|
| 1306 | /*
|
---|
| 1307 | * We'll assume z2 = 1 (special case z2 = 0 is handled later)
|
---|
| 1308 | */
|
---|
| 1309 |
|
---|
| 1310 | /* u1 = ftmp3 = x1*z2z2 */
|
---|
| 1311 | felem_assign(ftmp3, x1);
|
---|
| 1312 | /* ftmp3[i] < 2^106 */
|
---|
| 1313 |
|
---|
| 1314 | /* ftmp5 = 2z1z2 */
|
---|
| 1315 | felem_assign(ftmp5, z1);
|
---|
| 1316 | felem_scalar(ftmp5, 2);
|
---|
| 1317 | /* ftmp5[i] < 2*2^106 = 2^107 */
|
---|
| 1318 |
|
---|
| 1319 | /* s1 = ftmp2 = y1 * z2**3 */
|
---|
| 1320 | felem_assign(ftmp6, y1);
|
---|
| 1321 | /* ftmp6[i] < 2^106 */
|
---|
| 1322 | }
|
---|
| 1323 |
|
---|
| 1324 | /* u2 = x2*z1z1 */
|
---|
| 1325 | smallfelem_mul(tmp, x2, small1);
|
---|
| 1326 | felem_reduce(ftmp4, tmp);
|
---|
| 1327 |
|
---|
| 1328 | /* h = ftmp4 = u2 - u1 */
|
---|
| 1329 | felem_diff_zero107(ftmp4, ftmp3);
|
---|
| 1330 | /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
|
---|
| 1331 | felem_shrink(small4, ftmp4);
|
---|
| 1332 |
|
---|
| 1333 | x_equal = smallfelem_is_zero(small4);
|
---|
| 1334 |
|
---|
| 1335 | /* z_out = ftmp5 * h */
|
---|
| 1336 | felem_small_mul(tmp, small4, ftmp5);
|
---|
| 1337 | felem_reduce(z_out, tmp);
|
---|
| 1338 | /* z_out[i] < 2^101 */
|
---|
| 1339 |
|
---|
| 1340 | /* ftmp = z1 * z1z1 */
|
---|
| 1341 | smallfelem_mul(tmp, small1, small3);
|
---|
| 1342 | felem_reduce(ftmp, tmp);
|
---|
| 1343 |
|
---|
| 1344 | /* s2 = tmp = y2 * z1**3 */
|
---|
| 1345 | felem_small_mul(tmp, y2, ftmp);
|
---|
| 1346 | felem_reduce(ftmp5, tmp);
|
---|
| 1347 |
|
---|
| 1348 | /* r = ftmp5 = (s2 - s1)*2 */
|
---|
| 1349 | felem_diff_zero107(ftmp5, ftmp6);
|
---|
| 1350 | /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
|
---|
| 1351 | felem_scalar(ftmp5, 2);
|
---|
| 1352 | /* ftmp5[i] < 2^109 */
|
---|
| 1353 | felem_shrink(small1, ftmp5);
|
---|
| 1354 | y_equal = smallfelem_is_zero(small1);
|
---|
| 1355 |
|
---|
| 1356 | if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
|
---|
| 1357 | point_double(x3, y3, z3, x1, y1, z1);
|
---|
| 1358 | return;
|
---|
| 1359 | }
|
---|
| 1360 |
|
---|
| 1361 | /* I = ftmp = (2h)**2 */
|
---|
| 1362 | felem_assign(ftmp, ftmp4);
|
---|
| 1363 | felem_scalar(ftmp, 2);
|
---|
| 1364 | /* ftmp[i] < 2*2^108 = 2^109 */
|
---|
| 1365 | felem_square(tmp, ftmp);
|
---|
| 1366 | felem_reduce(ftmp, tmp);
|
---|
| 1367 |
|
---|
| 1368 | /* J = ftmp2 = h * I */
|
---|
| 1369 | felem_mul(tmp, ftmp4, ftmp);
|
---|
| 1370 | felem_reduce(ftmp2, tmp);
|
---|
| 1371 |
|
---|
| 1372 | /* V = ftmp4 = U1 * I */
|
---|
| 1373 | felem_mul(tmp, ftmp3, ftmp);
|
---|
| 1374 | felem_reduce(ftmp4, tmp);
|
---|
| 1375 |
|
---|
| 1376 | /* x_out = r**2 - J - 2V */
|
---|
| 1377 | smallfelem_square(tmp, small1);
|
---|
| 1378 | felem_reduce(x_out, tmp);
|
---|
| 1379 | felem_assign(ftmp3, ftmp4);
|
---|
| 1380 | felem_scalar(ftmp4, 2);
|
---|
| 1381 | felem_sum(ftmp4, ftmp2);
|
---|
| 1382 | /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
|
---|
| 1383 | felem_diff(x_out, ftmp4);
|
---|
| 1384 | /* x_out[i] < 2^105 + 2^101 */
|
---|
| 1385 |
|
---|
| 1386 | /* y_out = r(V-x_out) - 2 * s1 * J */
|
---|
| 1387 | felem_diff_zero107(ftmp3, x_out);
|
---|
| 1388 | /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
|
---|
| 1389 | felem_small_mul(tmp, small1, ftmp3);
|
---|
| 1390 | felem_mul(tmp2, ftmp6, ftmp2);
|
---|
| 1391 | longfelem_scalar(tmp2, 2);
|
---|
| 1392 | /* tmp2[i] < 2*2^67 = 2^68 */
|
---|
| 1393 | longfelem_diff(tmp, tmp2);
|
---|
| 1394 | /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
|
---|
| 1395 | felem_reduce_zero105(y_out, tmp);
|
---|
| 1396 | /* y_out[i] < 2^106 */
|
---|
| 1397 |
|
---|
| 1398 | copy_small_conditional(x_out, x2, z1_is_zero);
|
---|
| 1399 | copy_conditional(x_out, x1, z2_is_zero);
|
---|
| 1400 | copy_small_conditional(y_out, y2, z1_is_zero);
|
---|
| 1401 | copy_conditional(y_out, y1, z2_is_zero);
|
---|
| 1402 | copy_small_conditional(z_out, z2, z1_is_zero);
|
---|
| 1403 | copy_conditional(z_out, z1, z2_is_zero);
|
---|
| 1404 | felem_assign(x3, x_out);
|
---|
| 1405 | felem_assign(y3, y_out);
|
---|
| 1406 | felem_assign(z3, z_out);
|
---|
| 1407 | }
|
---|
| 1408 |
|
---|
| 1409 | /*
|
---|
| 1410 | * point_add_small is the same as point_add, except that it operates on
|
---|
| 1411 | * smallfelems
|
---|
| 1412 | */
|
---|
| 1413 | static void point_add_small(smallfelem x3, smallfelem y3, smallfelem z3,
|
---|
| 1414 | smallfelem x1, smallfelem y1, smallfelem z1,
|
---|
| 1415 | smallfelem x2, smallfelem y2, smallfelem z2)
|
---|
| 1416 | {
|
---|
| 1417 | felem felem_x3, felem_y3, felem_z3;
|
---|
| 1418 | felem felem_x1, felem_y1, felem_z1;
|
---|
| 1419 | smallfelem_expand(felem_x1, x1);
|
---|
| 1420 | smallfelem_expand(felem_y1, y1);
|
---|
| 1421 | smallfelem_expand(felem_z1, z1);
|
---|
| 1422 | point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0,
|
---|
| 1423 | x2, y2, z2);
|
---|
| 1424 | felem_shrink(x3, felem_x3);
|
---|
| 1425 | felem_shrink(y3, felem_y3);
|
---|
| 1426 | felem_shrink(z3, felem_z3);
|
---|
| 1427 | }
|
---|
| 1428 |
|
---|
| 1429 | /*-
|
---|
| 1430 | * Base point pre computation
|
---|
| 1431 | * --------------------------
|
---|
| 1432 | *
|
---|
| 1433 | * Two different sorts of precomputed tables are used in the following code.
|
---|
| 1434 | * Each contain various points on the curve, where each point is three field
|
---|
| 1435 | * elements (x, y, z).
|
---|
| 1436 | *
|
---|
| 1437 | * For the base point table, z is usually 1 (0 for the point at infinity).
|
---|
| 1438 | * This table has 2 * 16 elements, starting with the following:
|
---|
| 1439 | * index | bits | point
|
---|
| 1440 | * ------+---------+------------------------------
|
---|
| 1441 | * 0 | 0 0 0 0 | 0G
|
---|
| 1442 | * 1 | 0 0 0 1 | 1G
|
---|
| 1443 | * 2 | 0 0 1 0 | 2^64G
|
---|
| 1444 | * 3 | 0 0 1 1 | (2^64 + 1)G
|
---|
| 1445 | * 4 | 0 1 0 0 | 2^128G
|
---|
| 1446 | * 5 | 0 1 0 1 | (2^128 + 1)G
|
---|
| 1447 | * 6 | 0 1 1 0 | (2^128 + 2^64)G
|
---|
| 1448 | * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
|
---|
| 1449 | * 8 | 1 0 0 0 | 2^192G
|
---|
| 1450 | * 9 | 1 0 0 1 | (2^192 + 1)G
|
---|
| 1451 | * 10 | 1 0 1 0 | (2^192 + 2^64)G
|
---|
| 1452 | * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
|
---|
| 1453 | * 12 | 1 1 0 0 | (2^192 + 2^128)G
|
---|
| 1454 | * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
|
---|
| 1455 | * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
|
---|
| 1456 | * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
|
---|
| 1457 | * followed by a copy of this with each element multiplied by 2^32.
|
---|
| 1458 | *
|
---|
| 1459 | * The reason for this is so that we can clock bits into four different
|
---|
| 1460 | * locations when doing simple scalar multiplies against the base point,
|
---|
| 1461 | * and then another four locations using the second 16 elements.
|
---|
| 1462 | *
|
---|
| 1463 | * Tables for other points have table[i] = iG for i in 0 .. 16. */
|
---|
| 1464 |
|
---|
| 1465 | /* gmul is the table of precomputed base points */
|
---|
| 1466 | static const smallfelem gmul[2][16][3] = {
|
---|
| 1467 | {{{0, 0, 0, 0},
|
---|
| 1468 | {0, 0, 0, 0},
|
---|
| 1469 | {0, 0, 0, 0}},
|
---|
| 1470 | {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2,
|
---|
| 1471 | 0x6b17d1f2e12c4247},
|
---|
| 1472 | {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16,
|
---|
| 1473 | 0x4fe342e2fe1a7f9b},
|
---|
| 1474 | {1, 0, 0, 0}},
|
---|
| 1475 | {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de,
|
---|
| 1476 | 0x0fa822bc2811aaa5},
|
---|
| 1477 | {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b,
|
---|
| 1478 | 0xbff44ae8f5dba80d},
|
---|
| 1479 | {1, 0, 0, 0}},
|
---|
| 1480 | {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789,
|
---|
| 1481 | 0x300a4bbc89d6726f},
|
---|
| 1482 | {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f,
|
---|
| 1483 | 0x72aac7e0d09b4644},
|
---|
| 1484 | {1, 0, 0, 0}},
|
---|
| 1485 | {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e,
|
---|
| 1486 | 0x447d739beedb5e67},
|
---|
| 1487 | {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7,
|
---|
| 1488 | 0x2d4825ab834131ee},
|
---|
| 1489 | {1, 0, 0, 0}},
|
---|
| 1490 | {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60,
|
---|
| 1491 | 0xef9519328a9c72ff},
|
---|
| 1492 | {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c,
|
---|
| 1493 | 0x611e9fc37dbb2c9b},
|
---|
| 1494 | {1, 0, 0, 0}},
|
---|
| 1495 | {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf,
|
---|
| 1496 | 0x550663797b51f5d8},
|
---|
| 1497 | {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5,
|
---|
| 1498 | 0x157164848aecb851},
|
---|
| 1499 | {1, 0, 0, 0}},
|
---|
| 1500 | {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391,
|
---|
| 1501 | 0xeb5d7745b21141ea},
|
---|
| 1502 | {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee,
|
---|
| 1503 | 0xeafd72ebdbecc17b},
|
---|
| 1504 | {1, 0, 0, 0}},
|
---|
| 1505 | {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5,
|
---|
| 1506 | 0xa6d39677a7849276},
|
---|
| 1507 | {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf,
|
---|
| 1508 | 0x674f84749b0b8816},
|
---|
| 1509 | {1, 0, 0, 0}},
|
---|
| 1510 | {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb,
|
---|
| 1511 | 0x4e769e7672c9ddad},
|
---|
| 1512 | {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281,
|
---|
| 1513 | 0x42b99082de830663},
|
---|
| 1514 | {1, 0, 0, 0}},
|
---|
| 1515 | {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478,
|
---|
| 1516 | 0x78878ef61c6ce04d},
|
---|
| 1517 | {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def,
|
---|
| 1518 | 0xb6cb3f5d7b72c321},
|
---|
| 1519 | {1, 0, 0, 0}},
|
---|
| 1520 | {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae,
|
---|
| 1521 | 0x0c88bc4d716b1287},
|
---|
| 1522 | {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa,
|
---|
| 1523 | 0xdd5ddea3f3901dc6},
|
---|
| 1524 | {1, 0, 0, 0}},
|
---|
| 1525 | {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3,
|
---|
| 1526 | 0x68f344af6b317466},
|
---|
| 1527 | {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3,
|
---|
| 1528 | 0x31b9c405f8540a20},
|
---|
| 1529 | {1, 0, 0, 0}},
|
---|
| 1530 | {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0,
|
---|
| 1531 | 0x4052bf4b6f461db9},
|
---|
| 1532 | {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8,
|
---|
| 1533 | 0xfecf4d5190b0fc61},
|
---|
| 1534 | {1, 0, 0, 0}},
|
---|
| 1535 | {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a,
|
---|
| 1536 | 0x1eddbae2c802e41a},
|
---|
| 1537 | {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0,
|
---|
| 1538 | 0x43104d86560ebcfc},
|
---|
| 1539 | {1, 0, 0, 0}},
|
---|
| 1540 | {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a,
|
---|
| 1541 | 0xb48e26b484f7a21c},
|
---|
| 1542 | {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668,
|
---|
| 1543 | 0xfac015404d4d3dab},
|
---|
| 1544 | {1, 0, 0, 0}}},
|
---|
| 1545 | {{{0, 0, 0, 0},
|
---|
| 1546 | {0, 0, 0, 0},
|
---|
| 1547 | {0, 0, 0, 0}},
|
---|
| 1548 | {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da,
|
---|
| 1549 | 0x7fe36b40af22af89},
|
---|
| 1550 | {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1,
|
---|
| 1551 | 0xe697d45825b63624},
|
---|
| 1552 | {1, 0, 0, 0}},
|
---|
| 1553 | {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902,
|
---|
| 1554 | 0x4a5b506612a677a6},
|
---|
| 1555 | {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40,
|
---|
| 1556 | 0xeb13461ceac089f1},
|
---|
| 1557 | {1, 0, 0, 0}},
|
---|
| 1558 | {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857,
|
---|
| 1559 | 0x0781b8291c6a220a},
|
---|
| 1560 | {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434,
|
---|
| 1561 | 0x690cde8df0151593},
|
---|
| 1562 | {1, 0, 0, 0}},
|
---|
| 1563 | {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326,
|
---|
| 1564 | 0x8a535f566ec73617},
|
---|
| 1565 | {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf,
|
---|
| 1566 | 0x0455c08468b08bd7},
|
---|
| 1567 | {1, 0, 0, 0}},
|
---|
| 1568 | {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279,
|
---|
| 1569 | 0x06bada7ab77f8276},
|
---|
| 1570 | {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70,
|
---|
| 1571 | 0x5b476dfd0e6cb18a},
|
---|
| 1572 | {1, 0, 0, 0}},
|
---|
| 1573 | {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8,
|
---|
| 1574 | 0x3e29864e8a2ec908},
|
---|
| 1575 | {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed,
|
---|
| 1576 | 0x239b90ea3dc31e7e},
|
---|
| 1577 | {1, 0, 0, 0}},
|
---|
| 1578 | {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4,
|
---|
| 1579 | 0x820f4dd949f72ff7},
|
---|
| 1580 | {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3,
|
---|
| 1581 | 0x140406ec783a05ec},
|
---|
| 1582 | {1, 0, 0, 0}},
|
---|
| 1583 | {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe,
|
---|
| 1584 | 0x68f6b8542783dfee},
|
---|
| 1585 | {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028,
|
---|
| 1586 | 0xcbe1feba92e40ce6},
|
---|
| 1587 | {1, 0, 0, 0}},
|
---|
| 1588 | {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927,
|
---|
| 1589 | 0xd0b2f94d2f420109},
|
---|
| 1590 | {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a,
|
---|
| 1591 | 0x971459828b0719e5},
|
---|
| 1592 | {1, 0, 0, 0}},
|
---|
| 1593 | {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687,
|
---|
| 1594 | 0x961610004a866aba},
|
---|
| 1595 | {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c,
|
---|
| 1596 | 0x7acb9fadcee75e44},
|
---|
| 1597 | {1, 0, 0, 0}},
|
---|
| 1598 | {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea,
|
---|
| 1599 | 0x24eb9acca333bf5b},
|
---|
| 1600 | {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d,
|
---|
| 1601 | 0x69f891c5acd079cc},
|
---|
| 1602 | {1, 0, 0, 0}},
|
---|
| 1603 | {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514,
|
---|
| 1604 | 0xe51f547c5972a107},
|
---|
| 1605 | {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06,
|
---|
| 1606 | 0x1c309a2b25bb1387},
|
---|
| 1607 | {1, 0, 0, 0}},
|
---|
| 1608 | {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828,
|
---|
| 1609 | 0x20b87b8aa2c4e503},
|
---|
| 1610 | {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044,
|
---|
| 1611 | 0xf5c6fa49919776be},
|
---|
| 1612 | {1, 0, 0, 0}},
|
---|
| 1613 | {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56,
|
---|
| 1614 | 0x1ed7d1b9332010b9},
|
---|
| 1615 | {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24,
|
---|
| 1616 | 0x3a2b03f03217257a},
|
---|
| 1617 | {1, 0, 0, 0}},
|
---|
| 1618 | {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b,
|
---|
| 1619 | 0x15fee545c78dd9f6},
|
---|
| 1620 | {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb,
|
---|
| 1621 | 0x4ab5b6b2b8753f81},
|
---|
| 1622 | {1, 0, 0, 0}}}
|
---|
| 1623 | };
|
---|
| 1624 |
|
---|
| 1625 | /*
|
---|
| 1626 | * select_point selects the |idx|th point from a precomputation table and
|
---|
| 1627 | * copies it to out.
|
---|
| 1628 | */
|
---|
| 1629 | static void select_point(const u64 idx, unsigned int size,
|
---|
| 1630 | const smallfelem pre_comp[16][3], smallfelem out[3])
|
---|
| 1631 | {
|
---|
| 1632 | unsigned i, j;
|
---|
| 1633 | u64 *outlimbs = &out[0][0];
|
---|
| 1634 |
|
---|
| 1635 | memset(out, 0, sizeof(*out) * 3);
|
---|
| 1636 |
|
---|
| 1637 | for (i = 0; i < size; i++) {
|
---|
| 1638 | const u64 *inlimbs = (u64 *)&pre_comp[i][0][0];
|
---|
| 1639 | u64 mask = i ^ idx;
|
---|
| 1640 | mask |= mask >> 4;
|
---|
| 1641 | mask |= mask >> 2;
|
---|
| 1642 | mask |= mask >> 1;
|
---|
| 1643 | mask &= 1;
|
---|
| 1644 | mask--;
|
---|
| 1645 | for (j = 0; j < NLIMBS * 3; j++)
|
---|
| 1646 | outlimbs[j] |= inlimbs[j] & mask;
|
---|
| 1647 | }
|
---|
| 1648 | }
|
---|
| 1649 |
|
---|
| 1650 | /* get_bit returns the |i|th bit in |in| */
|
---|
| 1651 | static char get_bit(const felem_bytearray in, int i)
|
---|
| 1652 | {
|
---|
| 1653 | if ((i < 0) || (i >= 256))
|
---|
| 1654 | return 0;
|
---|
| 1655 | return (in[i >> 3] >> (i & 7)) & 1;
|
---|
| 1656 | }
|
---|
| 1657 |
|
---|
| 1658 | /*
|
---|
| 1659 | * Interleaved point multiplication using precomputed point multiples: The
|
---|
| 1660 | * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars
|
---|
| 1661 | * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
|
---|
| 1662 | * generator, using certain (large) precomputed multiples in g_pre_comp.
|
---|
| 1663 | * Output point (X, Y, Z) is stored in x_out, y_out, z_out
|
---|
| 1664 | */
|
---|
| 1665 | static void batch_mul(felem x_out, felem y_out, felem z_out,
|
---|
| 1666 | const felem_bytearray scalars[],
|
---|
| 1667 | const unsigned num_points, const u8 *g_scalar,
|
---|
| 1668 | const int mixed, const smallfelem pre_comp[][17][3],
|
---|
| 1669 | const smallfelem g_pre_comp[2][16][3])
|
---|
| 1670 | {
|
---|
| 1671 | int i, skip;
|
---|
| 1672 | unsigned num, gen_mul = (g_scalar != NULL);
|
---|
| 1673 | felem nq[3], ftmp;
|
---|
| 1674 | smallfelem tmp[3];
|
---|
| 1675 | u64 bits;
|
---|
| 1676 | u8 sign, digit;
|
---|
| 1677 |
|
---|
| 1678 | /* set nq to the point at infinity */
|
---|
| 1679 | memset(nq, 0, sizeof(nq));
|
---|
| 1680 |
|
---|
| 1681 | /*
|
---|
| 1682 | * Loop over all scalars msb-to-lsb, interleaving additions of multiples
|
---|
| 1683 | * of the generator (two in each of the last 32 rounds) and additions of
|
---|
| 1684 | * other points multiples (every 5th round).
|
---|
| 1685 | */
|
---|
| 1686 | skip = 1; /* save two point operations in the first
|
---|
| 1687 | * round */
|
---|
| 1688 | for (i = (num_points ? 255 : 31); i >= 0; --i) {
|
---|
| 1689 | /* double */
|
---|
| 1690 | if (!skip)
|
---|
| 1691 | point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
|
---|
| 1692 |
|
---|
| 1693 | /* add multiples of the generator */
|
---|
| 1694 | if (gen_mul && (i <= 31)) {
|
---|
| 1695 | /* first, look 32 bits upwards */
|
---|
| 1696 | bits = get_bit(g_scalar, i + 224) << 3;
|
---|
| 1697 | bits |= get_bit(g_scalar, i + 160) << 2;
|
---|
| 1698 | bits |= get_bit(g_scalar, i + 96) << 1;
|
---|
| 1699 | bits |= get_bit(g_scalar, i + 32);
|
---|
| 1700 | /* select the point to add, in constant time */
|
---|
| 1701 | select_point(bits, 16, g_pre_comp[1], tmp);
|
---|
| 1702 |
|
---|
| 1703 | if (!skip) {
|
---|
| 1704 | /* Arg 1 below is for "mixed" */
|
---|
| 1705 | point_add(nq[0], nq[1], nq[2],
|
---|
| 1706 | nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
|
---|
| 1707 | } else {
|
---|
| 1708 | smallfelem_expand(nq[0], tmp[0]);
|
---|
| 1709 | smallfelem_expand(nq[1], tmp[1]);
|
---|
| 1710 | smallfelem_expand(nq[2], tmp[2]);
|
---|
| 1711 | skip = 0;
|
---|
| 1712 | }
|
---|
| 1713 |
|
---|
| 1714 | /* second, look at the current position */
|
---|
| 1715 | bits = get_bit(g_scalar, i + 192) << 3;
|
---|
| 1716 | bits |= get_bit(g_scalar, i + 128) << 2;
|
---|
| 1717 | bits |= get_bit(g_scalar, i + 64) << 1;
|
---|
| 1718 | bits |= get_bit(g_scalar, i);
|
---|
| 1719 | /* select the point to add, in constant time */
|
---|
| 1720 | select_point(bits, 16, g_pre_comp[0], tmp);
|
---|
| 1721 | /* Arg 1 below is for "mixed" */
|
---|
| 1722 | point_add(nq[0], nq[1], nq[2],
|
---|
| 1723 | nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
|
---|
| 1724 | }
|
---|
| 1725 |
|
---|
| 1726 | /* do other additions every 5 doublings */
|
---|
| 1727 | if (num_points && (i % 5 == 0)) {
|
---|
| 1728 | /* loop over all scalars */
|
---|
| 1729 | for (num = 0; num < num_points; ++num) {
|
---|
| 1730 | bits = get_bit(scalars[num], i + 4) << 5;
|
---|
| 1731 | bits |= get_bit(scalars[num], i + 3) << 4;
|
---|
| 1732 | bits |= get_bit(scalars[num], i + 2) << 3;
|
---|
| 1733 | bits |= get_bit(scalars[num], i + 1) << 2;
|
---|
| 1734 | bits |= get_bit(scalars[num], i) << 1;
|
---|
| 1735 | bits |= get_bit(scalars[num], i - 1);
|
---|
| 1736 | ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
|
---|
| 1737 |
|
---|
| 1738 | /*
|
---|
| 1739 | * select the point to add or subtract, in constant time
|
---|
| 1740 | */
|
---|
| 1741 | select_point(digit, 17, pre_comp[num], tmp);
|
---|
| 1742 | smallfelem_neg(ftmp, tmp[1]); /* (X, -Y, Z) is the negative
|
---|
| 1743 | * point */
|
---|
| 1744 | copy_small_conditional(ftmp, tmp[1], (((limb) sign) - 1));
|
---|
| 1745 | felem_contract(tmp[1], ftmp);
|
---|
| 1746 |
|
---|
| 1747 | if (!skip) {
|
---|
| 1748 | point_add(nq[0], nq[1], nq[2],
|
---|
| 1749 | nq[0], nq[1], nq[2],
|
---|
| 1750 | mixed, tmp[0], tmp[1], tmp[2]);
|
---|
| 1751 | } else {
|
---|
| 1752 | smallfelem_expand(nq[0], tmp[0]);
|
---|
| 1753 | smallfelem_expand(nq[1], tmp[1]);
|
---|
| 1754 | smallfelem_expand(nq[2], tmp[2]);
|
---|
| 1755 | skip = 0;
|
---|
| 1756 | }
|
---|
| 1757 | }
|
---|
| 1758 | }
|
---|
| 1759 | }
|
---|
| 1760 | felem_assign(x_out, nq[0]);
|
---|
| 1761 | felem_assign(y_out, nq[1]);
|
---|
| 1762 | felem_assign(z_out, nq[2]);
|
---|
| 1763 | }
|
---|
| 1764 |
|
---|
| 1765 | /* Precomputation for the group generator. */
|
---|
| 1766 | struct nistp256_pre_comp_st {
|
---|
| 1767 | smallfelem g_pre_comp[2][16][3];
|
---|
| 1768 | int references;
|
---|
| 1769 | CRYPTO_RWLOCK *lock;
|
---|
| 1770 | };
|
---|
| 1771 |
|
---|
| 1772 | const EC_METHOD *EC_GFp_nistp256_method(void)
|
---|
| 1773 | {
|
---|
| 1774 | static const EC_METHOD ret = {
|
---|
| 1775 | EC_FLAGS_DEFAULT_OCT,
|
---|
| 1776 | NID_X9_62_prime_field,
|
---|
| 1777 | ec_GFp_nistp256_group_init,
|
---|
| 1778 | ec_GFp_simple_group_finish,
|
---|
| 1779 | ec_GFp_simple_group_clear_finish,
|
---|
| 1780 | ec_GFp_nist_group_copy,
|
---|
| 1781 | ec_GFp_nistp256_group_set_curve,
|
---|
| 1782 | ec_GFp_simple_group_get_curve,
|
---|
| 1783 | ec_GFp_simple_group_get_degree,
|
---|
| 1784 | ec_group_simple_order_bits,
|
---|
| 1785 | ec_GFp_simple_group_check_discriminant,
|
---|
| 1786 | ec_GFp_simple_point_init,
|
---|
| 1787 | ec_GFp_simple_point_finish,
|
---|
| 1788 | ec_GFp_simple_point_clear_finish,
|
---|
| 1789 | ec_GFp_simple_point_copy,
|
---|
| 1790 | ec_GFp_simple_point_set_to_infinity,
|
---|
| 1791 | ec_GFp_simple_set_Jprojective_coordinates_GFp,
|
---|
| 1792 | ec_GFp_simple_get_Jprojective_coordinates_GFp,
|
---|
| 1793 | ec_GFp_simple_point_set_affine_coordinates,
|
---|
| 1794 | ec_GFp_nistp256_point_get_affine_coordinates,
|
---|
| 1795 | 0 /* point_set_compressed_coordinates */ ,
|
---|
| 1796 | 0 /* point2oct */ ,
|
---|
| 1797 | 0 /* oct2point */ ,
|
---|
| 1798 | ec_GFp_simple_add,
|
---|
| 1799 | ec_GFp_simple_dbl,
|
---|
| 1800 | ec_GFp_simple_invert,
|
---|
| 1801 | ec_GFp_simple_is_at_infinity,
|
---|
| 1802 | ec_GFp_simple_is_on_curve,
|
---|
| 1803 | ec_GFp_simple_cmp,
|
---|
| 1804 | ec_GFp_simple_make_affine,
|
---|
| 1805 | ec_GFp_simple_points_make_affine,
|
---|
| 1806 | ec_GFp_nistp256_points_mul,
|
---|
| 1807 | ec_GFp_nistp256_precompute_mult,
|
---|
| 1808 | ec_GFp_nistp256_have_precompute_mult,
|
---|
| 1809 | ec_GFp_nist_field_mul,
|
---|
| 1810 | ec_GFp_nist_field_sqr,
|
---|
| 1811 | 0 /* field_div */ ,
|
---|
| 1812 | 0 /* field_encode */ ,
|
---|
| 1813 | 0 /* field_decode */ ,
|
---|
| 1814 | 0, /* field_set_to_one */
|
---|
| 1815 | ec_key_simple_priv2oct,
|
---|
| 1816 | ec_key_simple_oct2priv,
|
---|
| 1817 | 0, /* set private */
|
---|
| 1818 | ec_key_simple_generate_key,
|
---|
| 1819 | ec_key_simple_check_key,
|
---|
| 1820 | ec_key_simple_generate_public_key,
|
---|
| 1821 | 0, /* keycopy */
|
---|
| 1822 | 0, /* keyfinish */
|
---|
| 1823 | ecdh_simple_compute_key
|
---|
| 1824 | };
|
---|
| 1825 |
|
---|
| 1826 | return &ret;
|
---|
| 1827 | }
|
---|
| 1828 |
|
---|
| 1829 | /******************************************************************************/
|
---|
| 1830 | /*
|
---|
| 1831 | * FUNCTIONS TO MANAGE PRECOMPUTATION
|
---|
| 1832 | */
|
---|
| 1833 |
|
---|
| 1834 | static NISTP256_PRE_COMP *nistp256_pre_comp_new()
|
---|
| 1835 | {
|
---|
| 1836 | NISTP256_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
|
---|
| 1837 |
|
---|
| 1838 | if (ret == NULL) {
|
---|
| 1839 | ECerr(EC_F_NISTP256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
|
---|
| 1840 | return ret;
|
---|
| 1841 | }
|
---|
| 1842 |
|
---|
| 1843 | ret->references = 1;
|
---|
| 1844 |
|
---|
| 1845 | ret->lock = CRYPTO_THREAD_lock_new();
|
---|
| 1846 | if (ret->lock == NULL) {
|
---|
| 1847 | ECerr(EC_F_NISTP256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
|
---|
| 1848 | OPENSSL_free(ret);
|
---|
| 1849 | return NULL;
|
---|
| 1850 | }
|
---|
| 1851 | return ret;
|
---|
| 1852 | }
|
---|
| 1853 |
|
---|
| 1854 | NISTP256_PRE_COMP *EC_nistp256_pre_comp_dup(NISTP256_PRE_COMP *p)
|
---|
| 1855 | {
|
---|
| 1856 | int i;
|
---|
| 1857 | if (p != NULL)
|
---|
| 1858 | CRYPTO_atomic_add(&p->references, 1, &i, p->lock);
|
---|
| 1859 | return p;
|
---|
| 1860 | }
|
---|
| 1861 |
|
---|
| 1862 | void EC_nistp256_pre_comp_free(NISTP256_PRE_COMP *pre)
|
---|
| 1863 | {
|
---|
| 1864 | int i;
|
---|
| 1865 |
|
---|
| 1866 | if (pre == NULL)
|
---|
| 1867 | return;
|
---|
| 1868 |
|
---|
| 1869 | CRYPTO_atomic_add(&pre->references, -1, &i, pre->lock);
|
---|
| 1870 | REF_PRINT_COUNT("EC_nistp256", x);
|
---|
| 1871 | if (i > 0)
|
---|
| 1872 | return;
|
---|
| 1873 | REF_ASSERT_ISNT(i < 0);
|
---|
| 1874 |
|
---|
| 1875 | CRYPTO_THREAD_lock_free(pre->lock);
|
---|
| 1876 | OPENSSL_free(pre);
|
---|
| 1877 | }
|
---|
| 1878 |
|
---|
| 1879 | /******************************************************************************/
|
---|
| 1880 | /*
|
---|
| 1881 | * OPENSSL EC_METHOD FUNCTIONS
|
---|
| 1882 | */
|
---|
| 1883 |
|
---|
| 1884 | int ec_GFp_nistp256_group_init(EC_GROUP *group)
|
---|
| 1885 | {
|
---|
| 1886 | int ret;
|
---|
| 1887 | ret = ec_GFp_simple_group_init(group);
|
---|
| 1888 | group->a_is_minus3 = 1;
|
---|
| 1889 | return ret;
|
---|
| 1890 | }
|
---|
| 1891 |
|
---|
| 1892 | int ec_GFp_nistp256_group_set_curve(EC_GROUP *group, const BIGNUM *p,
|
---|
| 1893 | const BIGNUM *a, const BIGNUM *b,
|
---|
| 1894 | BN_CTX *ctx)
|
---|
| 1895 | {
|
---|
| 1896 | int ret = 0;
|
---|
| 1897 | BN_CTX *new_ctx = NULL;
|
---|
| 1898 | BIGNUM *curve_p, *curve_a, *curve_b;
|
---|
| 1899 |
|
---|
| 1900 | if (ctx == NULL)
|
---|
| 1901 | if ((ctx = new_ctx = BN_CTX_new()) == NULL)
|
---|
| 1902 | return 0;
|
---|
| 1903 | BN_CTX_start(ctx);
|
---|
| 1904 | if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
|
---|
| 1905 | ((curve_a = BN_CTX_get(ctx)) == NULL) ||
|
---|
| 1906 | ((curve_b = BN_CTX_get(ctx)) == NULL))
|
---|
| 1907 | goto err;
|
---|
| 1908 | BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p);
|
---|
| 1909 | BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a);
|
---|
| 1910 | BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b);
|
---|
| 1911 | if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
|
---|
| 1912 | ECerr(EC_F_EC_GFP_NISTP256_GROUP_SET_CURVE,
|
---|
| 1913 | EC_R_WRONG_CURVE_PARAMETERS);
|
---|
| 1914 | goto err;
|
---|
| 1915 | }
|
---|
| 1916 | group->field_mod_func = BN_nist_mod_256;
|
---|
| 1917 | ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
|
---|
| 1918 | err:
|
---|
| 1919 | BN_CTX_end(ctx);
|
---|
| 1920 | BN_CTX_free(new_ctx);
|
---|
| 1921 | return ret;
|
---|
| 1922 | }
|
---|
| 1923 |
|
---|
| 1924 | /*
|
---|
| 1925 | * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
|
---|
| 1926 | * (X/Z^2, Y/Z^3)
|
---|
| 1927 | */
|
---|
| 1928 | int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP *group,
|
---|
| 1929 | const EC_POINT *point,
|
---|
| 1930 | BIGNUM *x, BIGNUM *y,
|
---|
| 1931 | BN_CTX *ctx)
|
---|
| 1932 | {
|
---|
| 1933 | felem z1, z2, x_in, y_in;
|
---|
| 1934 | smallfelem x_out, y_out;
|
---|
| 1935 | longfelem tmp;
|
---|
| 1936 |
|
---|
| 1937 | if (EC_POINT_is_at_infinity(group, point)) {
|
---|
| 1938 | ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES,
|
---|
| 1939 | EC_R_POINT_AT_INFINITY);
|
---|
| 1940 | return 0;
|
---|
| 1941 | }
|
---|
| 1942 | if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
|
---|
| 1943 | (!BN_to_felem(z1, point->Z)))
|
---|
| 1944 | return 0;
|
---|
| 1945 | felem_inv(z2, z1);
|
---|
| 1946 | felem_square(tmp, z2);
|
---|
| 1947 | felem_reduce(z1, tmp);
|
---|
| 1948 | felem_mul(tmp, x_in, z1);
|
---|
| 1949 | felem_reduce(x_in, tmp);
|
---|
| 1950 | felem_contract(x_out, x_in);
|
---|
| 1951 | if (x != NULL) {
|
---|
| 1952 | if (!smallfelem_to_BN(x, x_out)) {
|
---|
| 1953 | ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES,
|
---|
| 1954 | ERR_R_BN_LIB);
|
---|
| 1955 | return 0;
|
---|
| 1956 | }
|
---|
| 1957 | }
|
---|
| 1958 | felem_mul(tmp, z1, z2);
|
---|
| 1959 | felem_reduce(z1, tmp);
|
---|
| 1960 | felem_mul(tmp, y_in, z1);
|
---|
| 1961 | felem_reduce(y_in, tmp);
|
---|
| 1962 | felem_contract(y_out, y_in);
|
---|
| 1963 | if (y != NULL) {
|
---|
| 1964 | if (!smallfelem_to_BN(y, y_out)) {
|
---|
| 1965 | ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES,
|
---|
| 1966 | ERR_R_BN_LIB);
|
---|
| 1967 | return 0;
|
---|
| 1968 | }
|
---|
| 1969 | }
|
---|
| 1970 | return 1;
|
---|
| 1971 | }
|
---|
| 1972 |
|
---|
| 1973 | /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */
|
---|
| 1974 | static void make_points_affine(size_t num, smallfelem points[][3],
|
---|
| 1975 | smallfelem tmp_smallfelems[])
|
---|
| 1976 | {
|
---|
| 1977 | /*
|
---|
| 1978 | * Runs in constant time, unless an input is the point at infinity (which
|
---|
| 1979 | * normally shouldn't happen).
|
---|
| 1980 | */
|
---|
| 1981 | ec_GFp_nistp_points_make_affine_internal(num,
|
---|
| 1982 | points,
|
---|
| 1983 | sizeof(smallfelem),
|
---|
| 1984 | tmp_smallfelems,
|
---|
| 1985 | (void (*)(void *))smallfelem_one,
|
---|
| 1986 | (int (*)(const void *))
|
---|
| 1987 | smallfelem_is_zero_int,
|
---|
| 1988 | (void (*)(void *, const void *))
|
---|
| 1989 | smallfelem_assign,
|
---|
| 1990 | (void (*)(void *, const void *))
|
---|
| 1991 | smallfelem_square_contract,
|
---|
| 1992 | (void (*)
|
---|
| 1993 | (void *, const void *,
|
---|
| 1994 | const void *))
|
---|
| 1995 | smallfelem_mul_contract,
|
---|
| 1996 | (void (*)(void *, const void *))
|
---|
| 1997 | smallfelem_inv_contract,
|
---|
| 1998 | /* nothing to contract */
|
---|
| 1999 | (void (*)(void *, const void *))
|
---|
| 2000 | smallfelem_assign);
|
---|
| 2001 | }
|
---|
| 2002 |
|
---|
| 2003 | /*
|
---|
| 2004 | * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
|
---|
| 2005 | * values Result is stored in r (r can equal one of the inputs).
|
---|
| 2006 | */
|
---|
| 2007 | int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r,
|
---|
| 2008 | const BIGNUM *scalar, size_t num,
|
---|
| 2009 | const EC_POINT *points[],
|
---|
| 2010 | const BIGNUM *scalars[], BN_CTX *ctx)
|
---|
| 2011 | {
|
---|
| 2012 | int ret = 0;
|
---|
| 2013 | int j;
|
---|
| 2014 | int mixed = 0;
|
---|
| 2015 | BN_CTX *new_ctx = NULL;
|
---|
| 2016 | BIGNUM *x, *y, *z, *tmp_scalar;
|
---|
| 2017 | felem_bytearray g_secret;
|
---|
| 2018 | felem_bytearray *secrets = NULL;
|
---|
| 2019 | smallfelem (*pre_comp)[17][3] = NULL;
|
---|
| 2020 | smallfelem *tmp_smallfelems = NULL;
|
---|
| 2021 | felem_bytearray tmp;
|
---|
| 2022 | unsigned i, num_bytes;
|
---|
| 2023 | int have_pre_comp = 0;
|
---|
| 2024 | size_t num_points = num;
|
---|
| 2025 | smallfelem x_in, y_in, z_in;
|
---|
| 2026 | felem x_out, y_out, z_out;
|
---|
| 2027 | NISTP256_PRE_COMP *pre = NULL;
|
---|
| 2028 | const smallfelem(*g_pre_comp)[16][3] = NULL;
|
---|
| 2029 | EC_POINT *generator = NULL;
|
---|
| 2030 | const EC_POINT *p = NULL;
|
---|
| 2031 | const BIGNUM *p_scalar = NULL;
|
---|
| 2032 |
|
---|
| 2033 | if (ctx == NULL)
|
---|
| 2034 | if ((ctx = new_ctx = BN_CTX_new()) == NULL)
|
---|
| 2035 | return 0;
|
---|
| 2036 | BN_CTX_start(ctx);
|
---|
| 2037 | if (((x = BN_CTX_get(ctx)) == NULL) ||
|
---|
| 2038 | ((y = BN_CTX_get(ctx)) == NULL) ||
|
---|
| 2039 | ((z = BN_CTX_get(ctx)) == NULL) ||
|
---|
| 2040 | ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
|
---|
| 2041 | goto err;
|
---|
| 2042 |
|
---|
| 2043 | if (scalar != NULL) {
|
---|
| 2044 | pre = group->pre_comp.nistp256;
|
---|
| 2045 | if (pre)
|
---|
| 2046 | /* we have precomputation, try to use it */
|
---|
| 2047 | g_pre_comp = (const smallfelem(*)[16][3])pre->g_pre_comp;
|
---|
| 2048 | else
|
---|
| 2049 | /* try to use the standard precomputation */
|
---|
| 2050 | g_pre_comp = &gmul[0];
|
---|
| 2051 | generator = EC_POINT_new(group);
|
---|
| 2052 | if (generator == NULL)
|
---|
| 2053 | goto err;
|
---|
| 2054 | /* get the generator from precomputation */
|
---|
| 2055 | if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) ||
|
---|
| 2056 | !smallfelem_to_BN(y, g_pre_comp[0][1][1]) ||
|
---|
| 2057 | !smallfelem_to_BN(z, g_pre_comp[0][1][2])) {
|
---|
| 2058 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
|
---|
| 2059 | goto err;
|
---|
| 2060 | }
|
---|
| 2061 | if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
|
---|
| 2062 | generator, x, y, z,
|
---|
| 2063 | ctx))
|
---|
| 2064 | goto err;
|
---|
| 2065 | if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
|
---|
| 2066 | /* precomputation matches generator */
|
---|
| 2067 | have_pre_comp = 1;
|
---|
| 2068 | else
|
---|
| 2069 | /*
|
---|
| 2070 | * we don't have valid precomputation: treat the generator as a
|
---|
| 2071 | * random point
|
---|
| 2072 | */
|
---|
| 2073 | num_points++;
|
---|
| 2074 | }
|
---|
| 2075 | if (num_points > 0) {
|
---|
| 2076 | if (num_points >= 3) {
|
---|
| 2077 | /*
|
---|
| 2078 | * unless we precompute multiples for just one or two points,
|
---|
| 2079 | * converting those into affine form is time well spent
|
---|
| 2080 | */
|
---|
| 2081 | mixed = 1;
|
---|
| 2082 | }
|
---|
| 2083 | secrets = OPENSSL_malloc(sizeof(*secrets) * num_points);
|
---|
| 2084 | pre_comp = OPENSSL_malloc(sizeof(*pre_comp) * num_points);
|
---|
| 2085 | if (mixed)
|
---|
| 2086 | tmp_smallfelems =
|
---|
| 2087 | OPENSSL_malloc(sizeof(*tmp_smallfelems) * (num_points * 17 + 1));
|
---|
| 2088 | if ((secrets == NULL) || (pre_comp == NULL)
|
---|
| 2089 | || (mixed && (tmp_smallfelems == NULL))) {
|
---|
| 2090 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
|
---|
| 2091 | goto err;
|
---|
| 2092 | }
|
---|
| 2093 |
|
---|
| 2094 | /*
|
---|
| 2095 | * we treat NULL scalars as 0, and NULL points as points at infinity,
|
---|
| 2096 | * i.e., they contribute nothing to the linear combination
|
---|
| 2097 | */
|
---|
| 2098 | memset(secrets, 0, sizeof(*secrets) * num_points);
|
---|
| 2099 | memset(pre_comp, 0, sizeof(*pre_comp) * num_points);
|
---|
| 2100 | for (i = 0; i < num_points; ++i) {
|
---|
| 2101 | if (i == num)
|
---|
| 2102 | /*
|
---|
| 2103 | * we didn't have a valid precomputation, so we pick the
|
---|
| 2104 | * generator
|
---|
| 2105 | */
|
---|
| 2106 | {
|
---|
| 2107 | p = EC_GROUP_get0_generator(group);
|
---|
| 2108 | p_scalar = scalar;
|
---|
| 2109 | } else
|
---|
| 2110 | /* the i^th point */
|
---|
| 2111 | {
|
---|
| 2112 | p = points[i];
|
---|
| 2113 | p_scalar = scalars[i];
|
---|
| 2114 | }
|
---|
| 2115 | if ((p_scalar != NULL) && (p != NULL)) {
|
---|
| 2116 | /* reduce scalar to 0 <= scalar < 2^256 */
|
---|
| 2117 | if ((BN_num_bits(p_scalar) > 256)
|
---|
| 2118 | || (BN_is_negative(p_scalar))) {
|
---|
| 2119 | /*
|
---|
| 2120 | * this is an unusual input, and we don't guarantee
|
---|
| 2121 | * constant-timeness
|
---|
| 2122 | */
|
---|
| 2123 | if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
|
---|
| 2124 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
|
---|
| 2125 | goto err;
|
---|
| 2126 | }
|
---|
| 2127 | num_bytes = BN_bn2bin(tmp_scalar, tmp);
|
---|
| 2128 | } else
|
---|
| 2129 | num_bytes = BN_bn2bin(p_scalar, tmp);
|
---|
| 2130 | flip_endian(secrets[i], tmp, num_bytes);
|
---|
| 2131 | /* precompute multiples */
|
---|
| 2132 | if ((!BN_to_felem(x_out, p->X)) ||
|
---|
| 2133 | (!BN_to_felem(y_out, p->Y)) ||
|
---|
| 2134 | (!BN_to_felem(z_out, p->Z)))
|
---|
| 2135 | goto err;
|
---|
| 2136 | felem_shrink(pre_comp[i][1][0], x_out);
|
---|
| 2137 | felem_shrink(pre_comp[i][1][1], y_out);
|
---|
| 2138 | felem_shrink(pre_comp[i][1][2], z_out);
|
---|
| 2139 | for (j = 2; j <= 16; ++j) {
|
---|
| 2140 | if (j & 1) {
|
---|
| 2141 | point_add_small(pre_comp[i][j][0], pre_comp[i][j][1],
|
---|
| 2142 | pre_comp[i][j][2], pre_comp[i][1][0],
|
---|
| 2143 | pre_comp[i][1][1], pre_comp[i][1][2],
|
---|
| 2144 | pre_comp[i][j - 1][0],
|
---|
| 2145 | pre_comp[i][j - 1][1],
|
---|
| 2146 | pre_comp[i][j - 1][2]);
|
---|
| 2147 | } else {
|
---|
| 2148 | point_double_small(pre_comp[i][j][0],
|
---|
| 2149 | pre_comp[i][j][1],
|
---|
| 2150 | pre_comp[i][j][2],
|
---|
| 2151 | pre_comp[i][j / 2][0],
|
---|
| 2152 | pre_comp[i][j / 2][1],
|
---|
| 2153 | pre_comp[i][j / 2][2]);
|
---|
| 2154 | }
|
---|
| 2155 | }
|
---|
| 2156 | }
|
---|
| 2157 | }
|
---|
| 2158 | if (mixed)
|
---|
| 2159 | make_points_affine(num_points * 17, pre_comp[0], tmp_smallfelems);
|
---|
| 2160 | }
|
---|
| 2161 |
|
---|
| 2162 | /* the scalar for the generator */
|
---|
| 2163 | if ((scalar != NULL) && (have_pre_comp)) {
|
---|
| 2164 | memset(g_secret, 0, sizeof(g_secret));
|
---|
| 2165 | /* reduce scalar to 0 <= scalar < 2^256 */
|
---|
| 2166 | if ((BN_num_bits(scalar) > 256) || (BN_is_negative(scalar))) {
|
---|
| 2167 | /*
|
---|
| 2168 | * this is an unusual input, and we don't guarantee
|
---|
| 2169 | * constant-timeness
|
---|
| 2170 | */
|
---|
| 2171 | if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
|
---|
| 2172 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
|
---|
| 2173 | goto err;
|
---|
| 2174 | }
|
---|
| 2175 | num_bytes = BN_bn2bin(tmp_scalar, tmp);
|
---|
| 2176 | } else
|
---|
| 2177 | num_bytes = BN_bn2bin(scalar, tmp);
|
---|
| 2178 | flip_endian(g_secret, tmp, num_bytes);
|
---|
| 2179 | /* do the multiplication with generator precomputation */
|
---|
| 2180 | batch_mul(x_out, y_out, z_out,
|
---|
| 2181 | (const felem_bytearray(*))secrets, num_points,
|
---|
| 2182 | g_secret,
|
---|
| 2183 | mixed, (const smallfelem(*)[17][3])pre_comp, g_pre_comp);
|
---|
| 2184 | } else
|
---|
| 2185 | /* do the multiplication without generator precomputation */
|
---|
| 2186 | batch_mul(x_out, y_out, z_out,
|
---|
| 2187 | (const felem_bytearray(*))secrets, num_points,
|
---|
| 2188 | NULL, mixed, (const smallfelem(*)[17][3])pre_comp, NULL);
|
---|
| 2189 | /* reduce the output to its unique minimal representation */
|
---|
| 2190 | felem_contract(x_in, x_out);
|
---|
| 2191 | felem_contract(y_in, y_out);
|
---|
| 2192 | felem_contract(z_in, z_out);
|
---|
| 2193 | if ((!smallfelem_to_BN(x, x_in)) || (!smallfelem_to_BN(y, y_in)) ||
|
---|
| 2194 | (!smallfelem_to_BN(z, z_in))) {
|
---|
| 2195 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB);
|
---|
| 2196 | goto err;
|
---|
| 2197 | }
|
---|
| 2198 | ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
|
---|
| 2199 |
|
---|
| 2200 | err:
|
---|
| 2201 | BN_CTX_end(ctx);
|
---|
| 2202 | EC_POINT_free(generator);
|
---|
| 2203 | BN_CTX_free(new_ctx);
|
---|
| 2204 | OPENSSL_free(secrets);
|
---|
| 2205 | OPENSSL_free(pre_comp);
|
---|
| 2206 | OPENSSL_free(tmp_smallfelems);
|
---|
| 2207 | return ret;
|
---|
| 2208 | }
|
---|
| 2209 |
|
---|
| 2210 | int ec_GFp_nistp256_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
|
---|
| 2211 | {
|
---|
| 2212 | int ret = 0;
|
---|
| 2213 | NISTP256_PRE_COMP *pre = NULL;
|
---|
| 2214 | int i, j;
|
---|
| 2215 | BN_CTX *new_ctx = NULL;
|
---|
| 2216 | BIGNUM *x, *y;
|
---|
| 2217 | EC_POINT *generator = NULL;
|
---|
| 2218 | smallfelem tmp_smallfelems[32];
|
---|
| 2219 | felem x_tmp, y_tmp, z_tmp;
|
---|
| 2220 |
|
---|
| 2221 | /* throw away old precomputation */
|
---|
| 2222 | EC_pre_comp_free(group);
|
---|
| 2223 | if (ctx == NULL)
|
---|
| 2224 | if ((ctx = new_ctx = BN_CTX_new()) == NULL)
|
---|
| 2225 | return 0;
|
---|
| 2226 | BN_CTX_start(ctx);
|
---|
| 2227 | if (((x = BN_CTX_get(ctx)) == NULL) || ((y = BN_CTX_get(ctx)) == NULL))
|
---|
| 2228 | goto err;
|
---|
| 2229 | /* get the generator */
|
---|
| 2230 | if (group->generator == NULL)
|
---|
| 2231 | goto err;
|
---|
| 2232 | generator = EC_POINT_new(group);
|
---|
| 2233 | if (generator == NULL)
|
---|
| 2234 | goto err;
|
---|
| 2235 | BN_bin2bn(nistp256_curve_params[3], sizeof(felem_bytearray), x);
|
---|
| 2236 | BN_bin2bn(nistp256_curve_params[4], sizeof(felem_bytearray), y);
|
---|
| 2237 | if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
|
---|
| 2238 | goto err;
|
---|
| 2239 | if ((pre = nistp256_pre_comp_new()) == NULL)
|
---|
| 2240 | goto err;
|
---|
| 2241 | /*
|
---|
| 2242 | * if the generator is the standard one, use built-in precomputation
|
---|
| 2243 | */
|
---|
| 2244 | if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
|
---|
| 2245 | memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
|
---|
| 2246 | goto done;
|
---|
| 2247 | }
|
---|
| 2248 | if ((!BN_to_felem(x_tmp, group->generator->X)) ||
|
---|
| 2249 | (!BN_to_felem(y_tmp, group->generator->Y)) ||
|
---|
| 2250 | (!BN_to_felem(z_tmp, group->generator->Z)))
|
---|
| 2251 | goto err;
|
---|
| 2252 | felem_shrink(pre->g_pre_comp[0][1][0], x_tmp);
|
---|
| 2253 | felem_shrink(pre->g_pre_comp[0][1][1], y_tmp);
|
---|
| 2254 | felem_shrink(pre->g_pre_comp[0][1][2], z_tmp);
|
---|
| 2255 | /*
|
---|
| 2256 | * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G, 2^96*G,
|
---|
| 2257 | * 2^160*G, 2^224*G for the second one
|
---|
| 2258 | */
|
---|
| 2259 | for (i = 1; i <= 8; i <<= 1) {
|
---|
| 2260 | point_double_small(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
|
---|
| 2261 | pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
|
---|
| 2262 | pre->g_pre_comp[0][i][1],
|
---|
| 2263 | pre->g_pre_comp[0][i][2]);
|
---|
| 2264 | for (j = 0; j < 31; ++j) {
|
---|
| 2265 | point_double_small(pre->g_pre_comp[1][i][0],
|
---|
| 2266 | pre->g_pre_comp[1][i][1],
|
---|
| 2267 | pre->g_pre_comp[1][i][2],
|
---|
| 2268 | pre->g_pre_comp[1][i][0],
|
---|
| 2269 | pre->g_pre_comp[1][i][1],
|
---|
| 2270 | pre->g_pre_comp[1][i][2]);
|
---|
| 2271 | }
|
---|
| 2272 | if (i == 8)
|
---|
| 2273 | break;
|
---|
| 2274 | point_double_small(pre->g_pre_comp[0][2 * i][0],
|
---|
| 2275 | pre->g_pre_comp[0][2 * i][1],
|
---|
| 2276 | pre->g_pre_comp[0][2 * i][2],
|
---|
| 2277 | pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
|
---|
| 2278 | pre->g_pre_comp[1][i][2]);
|
---|
| 2279 | for (j = 0; j < 31; ++j) {
|
---|
| 2280 | point_double_small(pre->g_pre_comp[0][2 * i][0],
|
---|
| 2281 | pre->g_pre_comp[0][2 * i][1],
|
---|
| 2282 | pre->g_pre_comp[0][2 * i][2],
|
---|
| 2283 | pre->g_pre_comp[0][2 * i][0],
|
---|
| 2284 | pre->g_pre_comp[0][2 * i][1],
|
---|
| 2285 | pre->g_pre_comp[0][2 * i][2]);
|
---|
| 2286 | }
|
---|
| 2287 | }
|
---|
| 2288 | for (i = 0; i < 2; i++) {
|
---|
| 2289 | /* g_pre_comp[i][0] is the point at infinity */
|
---|
| 2290 | memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
|
---|
| 2291 | /* the remaining multiples */
|
---|
| 2292 | /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
|
---|
| 2293 | point_add_small(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
|
---|
| 2294 | pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
|
---|
| 2295 | pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
|
---|
| 2296 | pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
|
---|
| 2297 | pre->g_pre_comp[i][2][2]);
|
---|
| 2298 | /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
|
---|
| 2299 | point_add_small(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
|
---|
| 2300 | pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
|
---|
| 2301 | pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
|
---|
| 2302 | pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
|
---|
| 2303 | pre->g_pre_comp[i][2][2]);
|
---|
| 2304 | /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
|
---|
| 2305 | point_add_small(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
|
---|
| 2306 | pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
|
---|
| 2307 | pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
|
---|
| 2308 | pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
|
---|
| 2309 | pre->g_pre_comp[i][4][2]);
|
---|
| 2310 | /*
|
---|
| 2311 | * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G + 2^224*G
|
---|
| 2312 | */
|
---|
| 2313 | point_add_small(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
|
---|
| 2314 | pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
|
---|
| 2315 | pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
|
---|
| 2316 | pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
|
---|
| 2317 | pre->g_pre_comp[i][2][2]);
|
---|
| 2318 | for (j = 1; j < 8; ++j) {
|
---|
| 2319 | /* odd multiples: add G resp. 2^32*G */
|
---|
| 2320 | point_add_small(pre->g_pre_comp[i][2 * j + 1][0],
|
---|
| 2321 | pre->g_pre_comp[i][2 * j + 1][1],
|
---|
| 2322 | pre->g_pre_comp[i][2 * j + 1][2],
|
---|
| 2323 | pre->g_pre_comp[i][2 * j][0],
|
---|
| 2324 | pre->g_pre_comp[i][2 * j][1],
|
---|
| 2325 | pre->g_pre_comp[i][2 * j][2],
|
---|
| 2326 | pre->g_pre_comp[i][1][0],
|
---|
| 2327 | pre->g_pre_comp[i][1][1],
|
---|
| 2328 | pre->g_pre_comp[i][1][2]);
|
---|
| 2329 | }
|
---|
| 2330 | }
|
---|
| 2331 | make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_smallfelems);
|
---|
| 2332 |
|
---|
| 2333 | done:
|
---|
| 2334 | SETPRECOMP(group, nistp256, pre);
|
---|
| 2335 | pre = NULL;
|
---|
| 2336 | ret = 1;
|
---|
| 2337 |
|
---|
| 2338 | err:
|
---|
| 2339 | BN_CTX_end(ctx);
|
---|
| 2340 | EC_POINT_free(generator);
|
---|
| 2341 | BN_CTX_free(new_ctx);
|
---|
| 2342 | EC_nistp256_pre_comp_free(pre);
|
---|
| 2343 | return ret;
|
---|
| 2344 | }
|
---|
| 2345 |
|
---|
| 2346 | int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP *group)
|
---|
| 2347 | {
|
---|
| 2348 | return HAVEPRECOMP(group, nistp256);
|
---|
| 2349 | }
|
---|
| 2350 | #endif
|
---|