secp256k1-zkp/src/scalar_impl.h
Andrew Poelstra 4401500060 Add constant-time multiply secp256k1_ecmult_const for ECDH
Designed with clear separation of the wNAF conversion, precomputation
and exponentiation (since the precomp at least we will probably want
to separate in the API for users who reuse points a lot.

Future work:
  - actually separate precomp in the API
  - do multiexp rather than single exponentiation
2015-07-31 12:39:09 -05:00

337 lines
12 KiB
C

/**********************************************************************
* 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 <string.h>
#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
#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);
}
/** secp256k1 curve order, see secp256k1_ecdsa_const_order_as_fe in ecdsa_impl.h */
static void secp256k1_scalar_order_get_num(secp256k1_num_t *r) {
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
};
secp256k1_num_set_bin(r, order, 32);
}
#endif
static void secp256k1_scalar_inverse(secp256k1_scalar_t *r, const secp256k1_scalar_t *x) {
secp256k1_scalar_t *t;
int i;
/* 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 (i = 0; i < 6; i++) {
secp256k1_scalar_sqr(&x15, &x15);
}
secp256k1_scalar_mul(&x15, &x15, &x7);
secp256k1_scalar_sqr(&x30, &x15);
for (i = 0; i < 14; i++) {
secp256k1_scalar_sqr(&x30, &x30);
}
secp256k1_scalar_mul(&x30, &x30, &x15);
secp256k1_scalar_sqr(&x60, &x30);
for (i = 0; i < 29; i++) {
secp256k1_scalar_sqr(&x60, &x60);
}
secp256k1_scalar_mul(&x60, &x60, &x30);
secp256k1_scalar_sqr(&x120, &x60);
for (i = 0; i < 59; i++) {
secp256k1_scalar_sqr(&x120, &x120);
}
secp256k1_scalar_mul(&x120, &x120, &x60);
secp256k1_scalar_sqr(&x127, &x120);
for (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). */
t = &x127;
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 4; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x3); /* 111 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 4; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x3); /* 111 */
for (i = 0; i < 3; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x2); /* 11 */
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 < 4; i++) { /* 00 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x2); /* 11 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 5; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x4); /* 1111 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 3; i++) { /* 00 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 4; i++) { /* 000 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
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 < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 3; i++) { /* 00 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 3; i++) { /* 00 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 5; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x4); /* 1111 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 5; i++) { /* 000 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x2); /* 11 */
for (i = 0; i < 4; i++) { /* 00 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x2); /* 11 */
for (i = 0; i < 2; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
for (i = 0; i < 8; i++) { /* 000000 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x2); /* 11 */
for (i = 0; i < 3; i++) { /* 0 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, &x2); /* 11 */
for (i = 0; i < 3; i++) { /* 00 */
secp256k1_scalar_sqr(t, t);
}
secp256k1_scalar_mul(t, t, x); /* 1 */
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_t *a) {
/* d[0] is present and is the lowest word for all representations */
return !(a->d[0] & 1);
}
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_num_t n, m;
secp256k1_scalar_t 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
/**
* 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_var(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) {
secp256k1_scalar_t c1, c2;
static const secp256k1_scalar_t minus_lambda = SECP256K1_SCALAR_CONST(
0xAC9C52B3UL, 0x3FA3CF1FUL, 0x5AD9E3FDUL, 0x77ED9BA4UL,
0xA880B9FCUL, 0x8EC739C2UL, 0xE0CFC810UL, 0xB51283CFUL
);
static const secp256k1_scalar_t minus_b1 = SECP256K1_SCALAR_CONST(
0x00000000UL, 0x00000000UL, 0x00000000UL, 0x00000000UL,
0xE4437ED6UL, 0x010E8828UL, 0x6F547FA9UL, 0x0ABFE4C3UL
);
static const secp256k1_scalar_t minus_b2 = SECP256K1_SCALAR_CONST(
0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL,
0x8A280AC5UL, 0x0774346DUL, 0xD765CDA8UL, 0x3DB1562CUL
);
static const secp256k1_scalar_t g1 = SECP256K1_SCALAR_CONST(
0x00000000UL, 0x00000000UL, 0x00000000UL, 0x00003086UL,
0xD221A7D4UL, 0x6BCDE86CUL, 0x90E49284UL, 0xEB153DABUL
);
static const secp256k1_scalar_t g2 = SECP256K1_SCALAR_CONST(
0x00000000UL, 0x00000000UL, 0x00000000UL, 0x0000E443UL,
0x7ED6010EUL, 0x88286F54UL, 0x7FA90ABFUL, 0xE4C42212UL
);
VERIFY_CHECK(r1 != a);
VERIFY_CHECK(r2 != a);
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