/********************************************************************** * Copyright (c) 2014 Pieter Wuille * * Distributed under the MIT software license, see the accompanying * * file COPYING or http://www.opensource.org/licenses/mit-license.php.* **********************************************************************/ #ifndef _SECP256K1_SCALAR_IMPL_H_ #define _SECP256K1_SCALAR_IMPL_H_ #include #include "group.h" #include "scalar.h" #if defined HAVE_CONFIG_H #include "libsecp256k1-config.h" #endif #if defined(USE_SCALAR_4X64) #include "scalar_4x64_impl.h" #elif defined(USE_SCALAR_8X32) #include "scalar_8x32_impl.h" #else #error "Please select scalar implementation" #endif typedef struct { #ifndef USE_NUM_NONE secp256k1_num_t order; #endif #ifdef USE_ENDOMORPHISM secp256k1_scalar_t minus_lambda, minus_b1, minus_b2, g1, g2; #endif } secp256k1_scalar_consts_t; static const secp256k1_scalar_consts_t *secp256k1_scalar_consts = NULL; static void secp256k1_scalar_start(void) { if (secp256k1_scalar_consts != NULL) return; /* Allocate. */ secp256k1_scalar_consts_t *ret = (secp256k1_scalar_consts_t*)checked_malloc(sizeof(secp256k1_scalar_consts_t)); #ifndef USE_NUM_NONE static const unsigned char secp256k1_scalar_consts_order[] = { 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 }; secp256k1_num_set_bin(&ret->order, secp256k1_scalar_consts_order, sizeof(secp256k1_scalar_consts_order)); #endif #ifdef USE_ENDOMORPHISM /** * Lambda is a scalar which has the property for secp256k1 that point multiplication by * it is efficiently computable (see secp256k1_gej_mul_lambda). */ static const unsigned char secp256k1_scalar_consts_lambda[32] = { 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'). */ static const unsigned char secp256k1_scalar_consts_minus_b1[32] = { 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00, 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00, 0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28, 0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3 }; static const unsigned char secp256k1_scalar_consts_b2[32] = { 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00, 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00, 0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd, 0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15 }; static const unsigned char secp256k1_scalar_consts_g1[32] = { 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00, 0x00,0x00,0x00,0x00,0x00,0x00,0x30,0x86, 0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c, 0x90,0xe4,0x92,0x84,0xeb,0x15,0x3d,0xab }; static const unsigned char secp256k1_scalar_consts_g2[32] = { 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00, 0x00,0x00,0x00,0x00,0x00,0x00,0xe4,0x43, 0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54, 0x7f,0xa9,0x0a,0xbf,0xe4,0xc4,0x22,0x12 }; secp256k1_scalar_set_b32(&ret->minus_lambda, secp256k1_scalar_consts_lambda, NULL); secp256k1_scalar_negate(&ret->minus_lambda, &ret->minus_lambda); secp256k1_scalar_set_b32(&ret->minus_b1, secp256k1_scalar_consts_minus_b1, NULL); secp256k1_scalar_set_b32(&ret->minus_b2, secp256k1_scalar_consts_b2, NULL); secp256k1_scalar_negate(&ret->minus_b2, &ret->minus_b2); secp256k1_scalar_set_b32(&ret->g1, secp256k1_scalar_consts_g1, NULL); secp256k1_scalar_set_b32(&ret->g2, secp256k1_scalar_consts_g2, NULL); #endif /* Set the global pointer. */ secp256k1_scalar_consts = ret; } static void secp256k1_scalar_stop(void) { if (secp256k1_scalar_consts == NULL) return; secp256k1_scalar_consts_t *c = (secp256k1_scalar_consts_t*)secp256k1_scalar_consts; secp256k1_scalar_consts = NULL; free(c); } #ifndef USE_NUM_NONE static void secp256k1_scalar_get_num(secp256k1_num_t *r, const secp256k1_scalar_t *a) { unsigned char c[32]; secp256k1_scalar_get_b32(c, a); secp256k1_num_set_bin(r, c, 32); } static void secp256k1_scalar_order_get_num(secp256k1_num_t *r) { *r = secp256k1_scalar_consts->order; } #endif static void secp256k1_scalar_inverse(secp256k1_scalar_t *r, const secp256k1_scalar_t *x) { /* First compute x ^ (2^N - 1) for some values of N. */ secp256k1_scalar_t x2, x3, x4, x6, x7, x8, x15, x30, x60, x120, x127; secp256k1_scalar_sqr(&x2, x); secp256k1_scalar_mul(&x2, &x2, x); secp256k1_scalar_sqr(&x3, &x2); secp256k1_scalar_mul(&x3, &x3, x); secp256k1_scalar_sqr(&x4, &x3); secp256k1_scalar_mul(&x4, &x4, x); secp256k1_scalar_sqr(&x6, &x4); secp256k1_scalar_sqr(&x6, &x6); secp256k1_scalar_mul(&x6, &x6, &x2); secp256k1_scalar_sqr(&x7, &x6); secp256k1_scalar_mul(&x7, &x7, x); secp256k1_scalar_sqr(&x8, &x7); secp256k1_scalar_mul(&x8, &x8, x); secp256k1_scalar_sqr(&x15, &x8); for (int i=0; i<6; i++) secp256k1_scalar_sqr(&x15, &x15); secp256k1_scalar_mul(&x15, &x15, &x7); secp256k1_scalar_sqr(&x30, &x15); for (int i=0; i<14; i++) secp256k1_scalar_sqr(&x30, &x30); secp256k1_scalar_mul(&x30, &x30, &x15); secp256k1_scalar_sqr(&x60, &x30); for (int i=0; i<29; i++) secp256k1_scalar_sqr(&x60, &x60); secp256k1_scalar_mul(&x60, &x60, &x30); secp256k1_scalar_sqr(&x120, &x60); for (int i=0; i<59; i++) secp256k1_scalar_sqr(&x120, &x120); secp256k1_scalar_mul(&x120, &x120, &x60); secp256k1_scalar_sqr(&x127, &x120); for (int i=0; i<6; i++) secp256k1_scalar_sqr(&x127, &x127); secp256k1_scalar_mul(&x127, &x127, &x7); /* Then accumulate the final result (t starts at x127). */ secp256k1_scalar_t *t = &x127; for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<4; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x3); /* 111 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<4; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x3); /* 111 */ for (int i=0; i<3; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x2); /* 11 */ for (int i=0; i<4; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x3); /* 111 */ for (int i=0; i<5; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x3); /* 111 */ for (int i=0; i<4; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x2); /* 11 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<5; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x4); /* 1111 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<3; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<4; i++) /* 000 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<10; i++) /* 0000000 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x3); /* 111 */ for (int i=0; i<4; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x3); /* 111 */ for (int i=0; i<9; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x8); /* 11111111 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<3; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<3; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<5; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x4); /* 1111 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<5; i++) /* 000 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x2); /* 11 */ for (int i=0; i<4; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x2); /* 11 */ for (int i=0; i<2; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<8; i++) /* 000000 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x2); /* 11 */ for (int i=0; i<3; i++) /* 0 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, &x2); /* 11 */ for (int i=0; i<3; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<6; i++) /* 00000 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(t, t, x); /* 1 */ for (int i=0; i<8; i++) /* 00 */ secp256k1_scalar_sqr(t, t); secp256k1_scalar_mul(r, t, &x6); /* 111111 */ } static void secp256k1_scalar_inverse_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *x) { #if defined(USE_SCALAR_INV_BUILTIN) secp256k1_scalar_inverse(r, x); #elif defined(USE_SCALAR_INV_NUM) unsigned char b[32]; secp256k1_scalar_get_b32(b, x); secp256k1_num_t n; secp256k1_num_set_bin(&n, b, 32); secp256k1_num_mod_inverse(&n, &n, &secp256k1_scalar_consts->order); secp256k1_num_get_bin(b, 32, &n); secp256k1_scalar_set_b32(r, b, NULL); #else #error "Please select scalar inverse implementation" #endif } #ifdef USE_ENDOMORPHISM static void secp256k1_scalar_split_lambda_var(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) { VERIFY_CHECK(r1 != a); VERIFY_CHECK(r2 != a); secp256k1_scalar_t c1, c2; secp256k1_scalar_mul_shift_var(&c1, a, &secp256k1_scalar_consts->g1, 272); secp256k1_scalar_mul_shift_var(&c2, a, &secp256k1_scalar_consts->g2, 272); secp256k1_scalar_mul(&c1, &c1, &secp256k1_scalar_consts->minus_b1); secp256k1_scalar_mul(&c2, &c2, &secp256k1_scalar_consts->minus_b2); secp256k1_scalar_add(r2, &c1, &c2); secp256k1_scalar_mul(r1, r2, &secp256k1_scalar_consts->minus_lambda); secp256k1_scalar_add(r1, r1, a); } #endif #endif