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673 lines
24 KiB
673 lines
24 KiB
#ifndef ENABLE_MODULE_MUSIG
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/**********************************************************************
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* Copyright (c) 2014 Pieter Wuille *
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* Distributed under the MIT software license, see the accompanying *
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* file COPYING or https://www.gnu.org/licenses/gpl-3.0.en.html*
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**********************************************************************/
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#ifndef SECP256K1_SCALAR_IMPL_H
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#define SECP256K1_SCALAR_IMPL_H
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#include "group.h"
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#include "scalar.h"
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#if defined HAVE_CONFIG_H
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#include "libsecp256k1-config.h"
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#endif
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#if defined(EXHAUSTIVE_TEST_ORDER)
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#include "scalar_low_impl.h"
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#elif defined(USE_SCALAR_4X64)
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#include "scalar_4x64_impl.h"
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#elif defined(USE_SCALAR_8X32)
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#include "scalar_8x32_impl.h"
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#else
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#error "Please select scalar implementation"
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#endif
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#ifndef USE_NUM_NONE
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static void secp256k1_scalar_get_num(secp256k1_num *r, const secp256k1_scalar *a) {
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unsigned char c[32];
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secp256k1_scalar_get_b32(c, a);
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secp256k1_num_set_bin(r, c, 32);
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}
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/** secp256k1 curve order, see secp256k1_ecdsa_const_order_as_fe in ecdsa_impl.h */
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static void secp256k1_scalar_order_get_num(secp256k1_num *r) {
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#if defined(EXHAUSTIVE_TEST_ORDER)
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static const unsigned char order[32] = {
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0,0,0,0,0,0,0,0,
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0,0,0,0,0,0,0,0,
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0,0,0,0,0,0,0,0,
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0,0,0,0,0,0,0,EXHAUSTIVE_TEST_ORDER
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};
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#else
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static const unsigned char order[32] = {
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0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
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0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,
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0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B,
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0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41
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};
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#endif
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secp256k1_num_set_bin(r, order, 32);
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}
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#endif
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static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) {
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#if defined(EXHAUSTIVE_TEST_ORDER)
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int i;
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*r = 0;
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for (i = 0; i < EXHAUSTIVE_TEST_ORDER; i++)
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if ((i * *x) % EXHAUSTIVE_TEST_ORDER == 1)
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*r = i;
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/* If this VERIFY_CHECK triggers we were given a noninvertible scalar (and thus
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* have a composite group order; fix it in exhaustive_tests.c). */
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VERIFY_CHECK(*r != 0);
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}
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#else
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secp256k1_scalar *t;
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int i;
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/* First compute xN as x ^ (2^N - 1) for some values of N,
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* and uM as x ^ M for some values of M. */
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secp256k1_scalar x2, x3, x6, x8, x14, x28, x56, x112, x126;
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secp256k1_scalar u2, u5, u9, u11, u13;
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secp256k1_scalar_sqr(&u2, x);
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secp256k1_scalar_mul(&x2, &u2, x);
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secp256k1_scalar_mul(&u5, &u2, &x2);
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secp256k1_scalar_mul(&x3, &u5, &u2);
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secp256k1_scalar_mul(&u9, &x3, &u2);
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secp256k1_scalar_mul(&u11, &u9, &u2);
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secp256k1_scalar_mul(&u13, &u11, &u2);
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secp256k1_scalar_sqr(&x6, &u13);
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secp256k1_scalar_sqr(&x6, &x6);
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secp256k1_scalar_mul(&x6, &x6, &u11);
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secp256k1_scalar_sqr(&x8, &x6);
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secp256k1_scalar_sqr(&x8, &x8);
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secp256k1_scalar_mul(&x8, &x8, &x2);
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secp256k1_scalar_sqr(&x14, &x8);
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for (i = 0; i < 5; i++) {
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secp256k1_scalar_sqr(&x14, &x14);
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}
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secp256k1_scalar_mul(&x14, &x14, &x6);
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secp256k1_scalar_sqr(&x28, &x14);
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for (i = 0; i < 13; i++) {
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secp256k1_scalar_sqr(&x28, &x28);
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}
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secp256k1_scalar_mul(&x28, &x28, &x14);
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secp256k1_scalar_sqr(&x56, &x28);
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for (i = 0; i < 27; i++) {
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secp256k1_scalar_sqr(&x56, &x56);
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}
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secp256k1_scalar_mul(&x56, &x56, &x28);
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secp256k1_scalar_sqr(&x112, &x56);
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for (i = 0; i < 55; i++) {
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secp256k1_scalar_sqr(&x112, &x112);
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}
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secp256k1_scalar_mul(&x112, &x112, &x56);
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secp256k1_scalar_sqr(&x126, &x112);
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for (i = 0; i < 13; i++) {
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secp256k1_scalar_sqr(&x126, &x126);
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}
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secp256k1_scalar_mul(&x126, &x126, &x14);
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/* Then accumulate the final result (t starts at x126). */
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t = &x126;
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for (i = 0; i < 3; i++) {
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u5); /* 101 */
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for (i = 0; i < 4; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x3); /* 111 */
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for (i = 0; i < 4; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u5); /* 101 */
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for (i = 0; i < 5; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u11); /* 1011 */
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for (i = 0; i < 4; i++) {
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u11); /* 1011 */
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for (i = 0; i < 4; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x3); /* 111 */
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for (i = 0; i < 5; i++) { /* 00 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x3); /* 111 */
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for (i = 0; i < 6; i++) { /* 00 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u13); /* 1101 */
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for (i = 0; i < 4; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u5); /* 101 */
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for (i = 0; i < 3; i++) {
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x3); /* 111 */
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for (i = 0; i < 5; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u9); /* 1001 */
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for (i = 0; i < 6; i++) { /* 000 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u5); /* 101 */
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for (i = 0; i < 10; i++) { /* 0000000 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x3); /* 111 */
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for (i = 0; i < 4; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x3); /* 111 */
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for (i = 0; i < 9; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x8); /* 11111111 */
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for (i = 0; i < 5; i++) { /* 0 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u9); /* 1001 */
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for (i = 0; i < 6; i++) { /* 00 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u11); /* 1011 */
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for (i = 0; i < 4; i++) {
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u13); /* 1101 */
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for (i = 0; i < 5; i++) {
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &x2); /* 11 */
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for (i = 0; i < 6; i++) { /* 00 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u13); /* 1101 */
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for (i = 0; i < 10; i++) { /* 000000 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u13); /* 1101 */
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for (i = 0; i < 4; i++) {
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, &u9); /* 1001 */
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for (i = 0; i < 6; i++) { /* 00000 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(t, t, x); /* 1 */
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for (i = 0; i < 8; i++) { /* 00 */
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secp256k1_scalar_sqr(t, t);
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}
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secp256k1_scalar_mul(r, t, &x6); /* 111111 */
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}
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SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) {
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return !(a->d[0] & 1);
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}
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#endif
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static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) {
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#if defined(USE_SCALAR_INV_BUILTIN)
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secp256k1_scalar_inverse(r, x);
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#elif defined(USE_SCALAR_INV_NUM)
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unsigned char b[32];
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secp256k1_num n, m;
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secp256k1_scalar t = *x;
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secp256k1_scalar_get_b32(b, &t);
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secp256k1_num_set_bin(&n, b, 32);
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secp256k1_scalar_order_get_num(&m);
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secp256k1_num_mod_inverse(&n, &n, &m);
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secp256k1_num_get_bin(b, 32, &n);
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secp256k1_scalar_set_b32(r, b, NULL);
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/* Verify that the inverse was computed correctly, without GMP code. */
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secp256k1_scalar_mul(&t, &t, r);
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CHECK(secp256k1_scalar_is_one(&t));
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#else
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#error "Please select scalar inverse implementation"
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#endif
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}
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#ifdef USE_ENDOMORPHISM
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#if defined(EXHAUSTIVE_TEST_ORDER)
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/**
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* Find k1 and k2 given k, such that k1 + k2 * lambda == k mod n; unlike in the
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* full case we don't bother making k1 and k2 be small, we just want them to be
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* nontrivial to get full test coverage for the exhaustive tests. We therefore
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* (arbitrarily) set k2 = k + 5 and k1 = k - k2 * lambda.
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*/
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static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) {
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*r2 = (*a + 5) % EXHAUSTIVE_TEST_ORDER;
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*r1 = (*a + (EXHAUSTIVE_TEST_ORDER - *r2) * EXHAUSTIVE_TEST_LAMBDA) % EXHAUSTIVE_TEST_ORDER;
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}
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#else
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/**
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* The Secp256k1 curve has an endomorphism, where lambda * (x, y) = (beta * x, y), where
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* lambda is {0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
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* 0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72}
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*
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* "Guide to Elliptic Curve Cryptography" (Hankerson, Menezes, Vanstone) gives an algorithm
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* (algorithm 3.74) to find k1 and k2 given k, such that k1 + k2 * lambda == k mod n, and k1
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* and k2 have a small size.
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* It relies on constants a1, b1, a2, b2. These constants for the value of lambda above are:
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*
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* - a1 = {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
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* - b1 = -{0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3}
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* - a2 = {0x01,0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8}
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* - b2 = {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
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*
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* The algorithm then computes c1 = round(b1 * k / n) and c2 = round(b2 * k / n), and gives
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* k1 = k - (c1*a1 + c2*a2) and k2 = -(c1*b1 + c2*b2). Instead, we use modular arithmetic, and
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* compute k1 as k - k2 * lambda, avoiding the need for constants a1 and a2.
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*
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* g1, g2 are precomputed constants used to replace division with a rounded multiplication
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* when decomposing the scalar for an endomorphism-based point multiplication.
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*
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* The possibility of using precomputed estimates is mentioned in "Guide to Elliptic Curve
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* Cryptography" (Hankerson, Menezes, Vanstone) in section 3.5.
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*
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* The derivation is described in the paper "Efficient Software Implementation of Public-Key
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* Cryptography on Sensor Networks Using the MSP430X Microcontroller" (Gouvea, Oliveira, Lopez),
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* Section 4.3 (here we use a somewhat higher-precision estimate):
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* d = a1*b2 - b1*a2
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* g1 = round((2^272)*b2/d)
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* g2 = round((2^272)*b1/d)
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*
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* (Note that 'd' is also equal to the curve order here because [a1,b1] and [a2,b2] are found
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* as outputs of the Extended Euclidean Algorithm on inputs 'order' and 'lambda').
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*
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* The function below splits a in r1 and r2, such that r1 + lambda * r2 == a (mod order).
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*/
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static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) {
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secp256k1_scalar c1, c2;
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static const secp256k1_scalar minus_lambda = SECP256K1_SCALAR_CONST(
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0xAC9C52B3UL, 0x3FA3CF1FUL, 0x5AD9E3FDUL, 0x77ED9BA4UL,
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0xA880B9FCUL, 0x8EC739C2UL, 0xE0CFC810UL, 0xB51283CFUL
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);
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static const secp256k1_scalar minus_b1 = SECP256K1_SCALAR_CONST(
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0x00000000UL, 0x00000000UL, 0x00000000UL, 0x00000000UL,
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0xE4437ED6UL, 0x010E8828UL, 0x6F547FA9UL, 0x0ABFE4C3UL
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);
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static const secp256k1_scalar minus_b2 = SECP256K1_SCALAR_CONST(
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0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL,
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0x8A280AC5UL, 0x0774346DUL, 0xD765CDA8UL, 0x3DB1562CUL
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);
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static const secp256k1_scalar g1 = SECP256K1_SCALAR_CONST(
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0x00000000UL, 0x00000000UL, 0x00000000UL, 0x00003086UL,
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0xD221A7D4UL, 0x6BCDE86CUL, 0x90E49284UL, 0xEB153DABUL
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);
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static const secp256k1_scalar g2 = SECP256K1_SCALAR_CONST(
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0x00000000UL, 0x00000000UL, 0x00000000UL, 0x0000E443UL,
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0x7ED6010EUL, 0x88286F54UL, 0x7FA90ABFUL, 0xE4C42212UL
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);
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VERIFY_CHECK(r1 != a);
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VERIFY_CHECK(r2 != a);
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/* these _var calls are constant time since the shift amount is constant */
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secp256k1_scalar_mul_shift_var(&c1, a, &g1, 272);
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secp256k1_scalar_mul_shift_var(&c2, a, &g2, 272);
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secp256k1_scalar_mul(&c1, &c1, &minus_b1);
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secp256k1_scalar_mul(&c2, &c2, &minus_b2);
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secp256k1_scalar_add(r2, &c1, &c2);
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secp256k1_scalar_mul(r1, r2, &minus_lambda);
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secp256k1_scalar_add(r1, r1, a);
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}
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#endif
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#endif
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#endif /* SECP256K1_SCALAR_IMPL_H */
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#else
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/**********************************************************************
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* Copyright (c) 2014 Pieter Wuille *
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* Distributed under the MIT software license, see the accompanying *
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* file COPYING or https://www.gnu.org/licenses/gpl-3.0.en.html*
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**********************************************************************/
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#ifndef SECP256K1_SCALAR_IMPL_H
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#define SECP256K1_SCALAR_IMPL_H
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#include "group.h"
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#include "scalar.h"
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#if defined HAVE_CONFIG_H
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#include "libsecp256k1-config.h"
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#endif
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#if defined(EXHAUSTIVE_TEST_ORDER)
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#include "scalar_low_impl.h"
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#elif defined(USE_SCALAR_4X64)
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#include "scalar_4x64_impl.h"
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#elif defined(USE_SCALAR_8X32)
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#include "scalar_8x32_impl.h"
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#else
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#error "Please select scalar implementation"
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#endif
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#ifndef USE_NUM_NONE
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static void secp256k1_scalar_get_num(secp256k1_num *r, const secp256k1_scalar *a) {
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unsigned char c[32];
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secp256k1_scalar_get_b32(c, a);
|
|
secp256k1_num_set_bin(r, c, 32);
|
|
}
|
|
|
|
/** secp256k1 curve order, see secp256k1_ecdsa_const_order_as_fe in ecdsa_impl.h */
|
|
static void secp256k1_scalar_order_get_num(secp256k1_num *r) {
|
|
#if defined(EXHAUSTIVE_TEST_ORDER)
|
|
static const unsigned char order[32] = {
|
|
0,0,0,0,0,0,0,0,
|
|
0,0,0,0,0,0,0,0,
|
|
0,0,0,0,0,0,0,0,
|
|
0,0,0,0,0,0,0,EXHAUSTIVE_TEST_ORDER
|
|
};
|
|
#else
|
|
static const unsigned char order[32] = {
|
|
0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
|
|
0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,
|
|
0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B,
|
|
0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41
|
|
};
|
|
#endif
|
|
secp256k1_num_set_bin(r, order, 32);
|
|
}
|
|
#endif
|
|
|
|
static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) {
|
|
#if defined(EXHAUSTIVE_TEST_ORDER)
|
|
int i;
|
|
*r = 0;
|
|
for (i = 0; i < EXHAUSTIVE_TEST_ORDER; i++)
|
|
if ((i * *x) % EXHAUSTIVE_TEST_ORDER == 1)
|
|
*r = i;
|
|
/* If this VERIFY_CHECK triggers we were given a noninvertible scalar (and thus
|
|
* have a composite group order; fix it in exhaustive_tests.c). */
|
|
VERIFY_CHECK(*r != 0);
|
|
}
|
|
#else
|
|
secp256k1_scalar *t;
|
|
int i;
|
|
/* First compute xN as x ^ (2^N - 1) for some values of N,
|
|
* and uM as x ^ M for some values of M. */
|
|
secp256k1_scalar x2, x3, x6, x8, x14, x28, x56, x112, x126;
|
|
secp256k1_scalar u2, u5, u9, u11, u13;
|
|
|
|
secp256k1_scalar_sqr(&u2, x);
|
|
secp256k1_scalar_mul(&x2, &u2, x);
|
|
secp256k1_scalar_mul(&u5, &u2, &x2);
|
|
secp256k1_scalar_mul(&x3, &u5, &u2);
|
|
secp256k1_scalar_mul(&u9, &x3, &u2);
|
|
secp256k1_scalar_mul(&u11, &u9, &u2);
|
|
secp256k1_scalar_mul(&u13, &u11, &u2);
|
|
|
|
secp256k1_scalar_sqr(&x6, &u13);
|
|
secp256k1_scalar_sqr(&x6, &x6);
|
|
secp256k1_scalar_mul(&x6, &x6, &u11);
|
|
|
|
secp256k1_scalar_sqr(&x8, &x6);
|
|
secp256k1_scalar_sqr(&x8, &x8);
|
|
secp256k1_scalar_mul(&x8, &x8, &x2);
|
|
|
|
secp256k1_scalar_sqr(&x14, &x8);
|
|
for (i = 0; i < 5; i++) {
|
|
secp256k1_scalar_sqr(&x14, &x14);
|
|
}
|
|
secp256k1_scalar_mul(&x14, &x14, &x6);
|
|
|
|
secp256k1_scalar_sqr(&x28, &x14);
|
|
for (i = 0; i < 13; i++) {
|
|
secp256k1_scalar_sqr(&x28, &x28);
|
|
}
|
|
secp256k1_scalar_mul(&x28, &x28, &x14);
|
|
|
|
secp256k1_scalar_sqr(&x56, &x28);
|
|
for (i = 0; i < 27; i++) {
|
|
secp256k1_scalar_sqr(&x56, &x56);
|
|
}
|
|
secp256k1_scalar_mul(&x56, &x56, &x28);
|
|
|
|
secp256k1_scalar_sqr(&x112, &x56);
|
|
for (i = 0; i < 55; i++) {
|
|
secp256k1_scalar_sqr(&x112, &x112);
|
|
}
|
|
secp256k1_scalar_mul(&x112, &x112, &x56);
|
|
|
|
secp256k1_scalar_sqr(&x126, &x112);
|
|
for (i = 0; i < 13; i++) {
|
|
secp256k1_scalar_sqr(&x126, &x126);
|
|
}
|
|
secp256k1_scalar_mul(&x126, &x126, &x14);
|
|
|
|
/* Then accumulate the final result (t starts at x126). */
|
|
t = &x126;
|
|
for (i = 0; i < 3; i++) {
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u5); /* 101 */
|
|
for (i = 0; i < 4; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x3); /* 111 */
|
|
for (i = 0; i < 4; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u5); /* 101 */
|
|
for (i = 0; i < 5; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u11); /* 1011 */
|
|
for (i = 0; i < 4; i++) {
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u11); /* 1011 */
|
|
for (i = 0; i < 4; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x3); /* 111 */
|
|
for (i = 0; i < 5; i++) { /* 00 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x3); /* 111 */
|
|
for (i = 0; i < 6; i++) { /* 00 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u13); /* 1101 */
|
|
for (i = 0; i < 4; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u5); /* 101 */
|
|
for (i = 0; i < 3; i++) {
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x3); /* 111 */
|
|
for (i = 0; i < 5; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u9); /* 1001 */
|
|
for (i = 0; i < 6; i++) { /* 000 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u5); /* 101 */
|
|
for (i = 0; i < 10; i++) { /* 0000000 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x3); /* 111 */
|
|
for (i = 0; i < 4; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x3); /* 111 */
|
|
for (i = 0; i < 9; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x8); /* 11111111 */
|
|
for (i = 0; i < 5; i++) { /* 0 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u9); /* 1001 */
|
|
for (i = 0; i < 6; i++) { /* 00 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u11); /* 1011 */
|
|
for (i = 0; i < 4; i++) {
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u13); /* 1101 */
|
|
for (i = 0; i < 5; i++) {
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &x2); /* 11 */
|
|
for (i = 0; i < 6; i++) { /* 00 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u13); /* 1101 */
|
|
for (i = 0; i < 10; i++) { /* 000000 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u13); /* 1101 */
|
|
for (i = 0; i < 4; i++) {
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, &u9); /* 1001 */
|
|
for (i = 0; i < 6; i++) { /* 00000 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(t, t, x); /* 1 */
|
|
for (i = 0; i < 8; i++) { /* 00 */
|
|
secp256k1_scalar_sqr(t, t);
|
|
}
|
|
secp256k1_scalar_mul(r, t, &x6); /* 111111 */
|
|
}
|
|
|
|
SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) {
|
|
return !(a->d[0] & 1);
|
|
}
|
|
#endif
|
|
|
|
static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) {
|
|
#if defined(USE_SCALAR_INV_BUILTIN)
|
|
secp256k1_scalar_inverse(r, x);
|
|
#elif defined(USE_SCALAR_INV_NUM)
|
|
unsigned char b[32];
|
|
secp256k1_num n, m;
|
|
secp256k1_scalar t = *x;
|
|
secp256k1_scalar_get_b32(b, &t);
|
|
secp256k1_num_set_bin(&n, b, 32);
|
|
secp256k1_scalar_order_get_num(&m);
|
|
secp256k1_num_mod_inverse(&n, &n, &m);
|
|
secp256k1_num_get_bin(b, 32, &n);
|
|
secp256k1_scalar_set_b32(r, b, NULL);
|
|
/* Verify that the inverse was computed correctly, without GMP code. */
|
|
secp256k1_scalar_mul(&t, &t, r);
|
|
CHECK(secp256k1_scalar_is_one(&t));
|
|
#else
|
|
#error "Please select scalar inverse implementation"
|
|
#endif
|
|
}
|
|
|
|
#ifdef USE_ENDOMORPHISM
|
|
#if defined(EXHAUSTIVE_TEST_ORDER)
|
|
/**
|
|
* Find k1 and k2 given k, such that k1 + k2 * lambda == k mod n; unlike in the
|
|
* full case we don't bother making k1 and k2 be small, we just want them to be
|
|
* nontrivial to get full test coverage for the exhaustive tests. We therefore
|
|
* (arbitrarily) set k2 = k + 5 and k1 = k - k2 * lambda.
|
|
*/
|
|
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) {
|
|
*r2 = (*a + 5) % EXHAUSTIVE_TEST_ORDER;
|
|
*r1 = (*a + (EXHAUSTIVE_TEST_ORDER - *r2) * EXHAUSTIVE_TEST_LAMBDA) % EXHAUSTIVE_TEST_ORDER;
|
|
}
|
|
#else
|
|
/**
|
|
* The Secp256k1 curve has an endomorphism, where lambda * (x, y) = (beta * x, y), where
|
|
* lambda is {0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
|
|
* 0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72}
|
|
*
|
|
* "Guide to Elliptic Curve Cryptography" (Hankerson, Menezes, Vanstone) gives an algorithm
|
|
* (algorithm 3.74) to find k1 and k2 given k, such that k1 + k2 * lambda == k mod n, and k1
|
|
* and k2 have a small size.
|
|
* It relies on constants a1, b1, a2, b2. These constants for the value of lambda above are:
|
|
*
|
|
* - a1 = {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
|
|
* - b1 = -{0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3}
|
|
* - a2 = {0x01,0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8}
|
|
* - b2 = {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
|
|
*
|
|
* The algorithm then computes c1 = round(b1 * k / n) and c2 = round(b2 * k / n), and gives
|
|
* k1 = k - (c1*a1 + c2*a2) and k2 = -(c1*b1 + c2*b2). Instead, we use modular arithmetic, and
|
|
* compute k1 as k - k2 * lambda, avoiding the need for constants a1 and a2.
|
|
*
|
|
* g1, g2 are precomputed constants used to replace division with a rounded multiplication
|
|
* when decomposing the scalar for an endomorphism-based point multiplication.
|
|
*
|
|
* The possibility of using precomputed estimates is mentioned in "Guide to Elliptic Curve
|
|
* Cryptography" (Hankerson, Menezes, Vanstone) in section 3.5.
|
|
*
|
|
* The derivation is described in the paper "Efficient Software Implementation of Public-Key
|
|
* Cryptography on Sensor Networks Using the MSP430X Microcontroller" (Gouvea, Oliveira, Lopez),
|
|
* Section 4.3 (here we use a somewhat higher-precision estimate):
|
|
* d = a1*b2 - b1*a2
|
|
* g1 = round((2^272)*b2/d)
|
|
* g2 = round((2^272)*b1/d)
|
|
*
|
|
* (Note that 'd' is also equal to the curve order here because [a1,b1] and [a2,b2] are found
|
|
* as outputs of the Extended Euclidean Algorithm on inputs 'order' and 'lambda').
|
|
*
|
|
* The function below splits a in r1 and r2, such that r1 + lambda * r2 == a (mod order).
|
|
*/
|
|
|
|
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) {
|
|
secp256k1_scalar c1, c2;
|
|
static const secp256k1_scalar minus_lambda = SECP256K1_SCALAR_CONST(
|
|
0xAC9C52B3UL, 0x3FA3CF1FUL, 0x5AD9E3FDUL, 0x77ED9BA4UL,
|
|
0xA880B9FCUL, 0x8EC739C2UL, 0xE0CFC810UL, 0xB51283CFUL
|
|
);
|
|
static const secp256k1_scalar minus_b1 = SECP256K1_SCALAR_CONST(
|
|
0x00000000UL, 0x00000000UL, 0x00000000UL, 0x00000000UL,
|
|
0xE4437ED6UL, 0x010E8828UL, 0x6F547FA9UL, 0x0ABFE4C3UL
|
|
);
|
|
static const secp256k1_scalar minus_b2 = SECP256K1_SCALAR_CONST(
|
|
0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL,
|
|
0x8A280AC5UL, 0x0774346DUL, 0xD765CDA8UL, 0x3DB1562CUL
|
|
);
|
|
static const secp256k1_scalar g1 = SECP256K1_SCALAR_CONST(
|
|
0x00000000UL, 0x00000000UL, 0x00000000UL, 0x00003086UL,
|
|
0xD221A7D4UL, 0x6BCDE86CUL, 0x90E49284UL, 0xEB153DABUL
|
|
);
|
|
static const secp256k1_scalar g2 = SECP256K1_SCALAR_CONST(
|
|
0x00000000UL, 0x00000000UL, 0x00000000UL, 0x0000E443UL,
|
|
0x7ED6010EUL, 0x88286F54UL, 0x7FA90ABFUL, 0xE4C42212UL
|
|
);
|
|
VERIFY_CHECK(r1 != a);
|
|
VERIFY_CHECK(r2 != a);
|
|
/* these _var calls are constant time since the shift amount is constant */
|
|
secp256k1_scalar_mul_shift_var(&c1, a, &g1, 272);
|
|
secp256k1_scalar_mul_shift_var(&c2, a, &g2, 272);
|
|
secp256k1_scalar_mul(&c1, &c1, &minus_b1);
|
|
secp256k1_scalar_mul(&c2, &c2, &minus_b2);
|
|
secp256k1_scalar_add(r2, &c1, &c2);
|
|
secp256k1_scalar_mul(r1, r2, &minus_lambda);
|
|
secp256k1_scalar_add(r1, r1, a);
|
|
}
|
|
#endif
|
|
#endif
|
|
|
|
#endif /* SECP256K1_SCALAR_IMPL_H */
|
|
#endif
|
|
|
|
|