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scalar_impl.h
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scalar_impl.h
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/***********************************************************************
* Copyright (c) 2014 Pieter Wuille *
* Distributed under the MIT software license, see the accompanying *
* file COPYING or https://www.opensource.org/licenses/mit-license.php.*
***********************************************************************/
#ifndef SECP256K1_SCALAR_IMPL_H
#define SECP256K1_SCALAR_IMPL_H
#ifdef VERIFY
#include <string.h>
#endif
#include "scalar.h"
#include "util.h"
#if defined HAVE_CONFIG_H
#include "libsecp256k1-config.h"
#endif
#if defined(EXHAUSTIVE_TEST_ORDER)
#include "scalar_low_impl.h"
#elif defined(SECP256K1_WIDEMUL_INT128)
#include "scalar_4x64_impl.h"
#elif defined(SECP256K1_WIDEMUL_INT64)
#include "scalar_8x32_impl.h"
#else
#error "Please select wide multiplication implementation"
#endif
static const secp256k1_scalar secp256k1_scalar_one = SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 1);
static const secp256k1_scalar secp256k1_scalar_zero = SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 0);
#ifndef USE_NUM_NONE
static void secp256k1_scalar_get_num(secp256k1_num *r, const secp256k1_scalar *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 *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 int secp256k1_scalar_set_b32_seckey(secp256k1_scalar *r, const unsigned char *bin) {
int overflow;
secp256k1_scalar_set_b32(r, bin, &overflow);
return (!overflow) & (!secp256k1_scalar_is_zero(r));
}
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
}
/* These parameters are generated using sage/gen_exhaustive_groups.sage. */
#if defined(EXHAUSTIVE_TEST_ORDER)
# if EXHAUSTIVE_TEST_ORDER == 13
# define EXHAUSTIVE_TEST_LAMBDA 9
# elif EXHAUSTIVE_TEST_ORDER == 199
# define EXHAUSTIVE_TEST_LAMBDA 92
# else
# error No known lambda for the specified exhaustive test group order.
# endif
/**
* Find r1 and r2 given k, such that r1 + r2 * lambda == k mod n; unlike in the
* full case we don't bother making r1 and r2 be small, we just want them to be
* nontrivial to get full test coverage for the exhaustive tests. We therefore
* (arbitrarily) set r2 = k + 5 (mod n) and r1 = k - r2 * lambda (mod n).
*/
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k) {
*r2 = (*k + 5) % EXHAUSTIVE_TEST_ORDER;
*r1 = (*k + (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: */
static const secp256k1_scalar secp256k1_const_lambda = SECP256K1_SCALAR_CONST(
0x5363AD4CUL, 0xC05C30E0UL, 0xA5261C02UL, 0x8812645AUL,
0x122E22EAUL, 0x20816678UL, 0xDF02967CUL, 0x1B23BD72UL
);
#ifdef VERIFY
static void secp256k1_scalar_split_lambda_verify(const secp256k1_scalar *r1, const secp256k1_scalar *r2, const secp256k1_scalar *k);
#endif
/*
* Both lambda and beta are primitive cube roots of unity. That is lamba^3 == 1 mod n and
* beta^3 == 1 mod p, where n is the curve order and p is the field order.
*
* Futhermore, because (X^3 - 1) = (X - 1)(X^2 + X + 1), the primitive cube roots of unity are
* roots of X^2 + X + 1. Therefore lambda^2 + lamba == -1 mod n and beta^2 + beta == -1 mod p.
* (The other primitive cube roots of unity are lambda^2 and beta^2 respectively.)
*
* Let l = -1/2 + i*sqrt(3)/2, the complex root of X^2 + X + 1. We can define a ring
* homomorphism phi : Z[l] -> Z_n where phi(a + b*l) == a + b*lambda mod n. The kernel of phi
* is a lattice over Z[l] (considering Z[l] as a Z-module). This lattice is generated by a
* reduced basis {a1 + b1*l, a2 + b2*l} where
*
* - 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}
*
* "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 are small in absolute value.
*
* The algorithm computes c1 = round(b2 * k / n) and c2 = round((-b1) * k / n), and gives
* k1 = k - (c1*a1 + c2*a2) and k2 = -(c1*b1 + c2*b2). Instead, we use modular arithmetic, and
* compute r2 = k2 mod n, and r1 = k1 mod n = (k - r2 * lambda) mod n, avoiding the need for
* the 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(2^384 * b2/d)
* g2 = round(2^384 * (-b1)/d)
*
* (Note that d is also equal to the curve order, n, here because [a1,b1] and [a2,b2]
* can be found as outputs of the Extended Euclidean Algorithm on inputs n and lambda).
*
* The function below splits k into r1 and r2, such that
* - r1 + lambda * r2 == k (mod n)
* - either r1 < 2^128 or -r1 mod n < 2^128
* - either r2 < 2^128 or -r2 mod n < 2^128
*
* See proof below.
*/
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k) {
secp256k1_scalar c1, c2;
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(
0x3086D221UL, 0xA7D46BCDUL, 0xE86C90E4UL, 0x9284EB15UL,
0x3DAA8A14UL, 0x71E8CA7FUL, 0xE893209AUL, 0x45DBB031UL
);
static const secp256k1_scalar g2 = SECP256K1_SCALAR_CONST(
0xE4437ED6UL, 0x010E8828UL, 0x6F547FA9UL, 0x0ABFE4C4UL,
0x221208ACUL, 0x9DF506C6UL, 0x1571B4AEUL, 0x8AC47F71UL
);
VERIFY_CHECK(r1 != k);
VERIFY_CHECK(r2 != k);
/* these _var calls are constant time since the shift amount is constant */
secp256k1_scalar_mul_shift_var(&c1, k, &g1, 384);
secp256k1_scalar_mul_shift_var(&c2, k, &g2, 384);
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, &secp256k1_const_lambda);
secp256k1_scalar_negate(r1, r1);
secp256k1_scalar_add(r1, r1, k);
#ifdef VERIFY
secp256k1_scalar_split_lambda_verify(r1, r2, k);
#endif
}
#ifdef VERIFY
/*
* Proof for secp256k1_scalar_split_lambda's bounds.
*
* Let
* - epsilon1 = 2^256 * |g1/2^384 - b2/d|
* - epsilon2 = 2^256 * |g2/2^384 - (-b1)/d|
* - c1 = round(k*g1/2^384)
* - c2 = round(k*g2/2^384)
*
* Lemma 1: |c1 - k*b2/d| < 2^-1 + epsilon1
*
* |c1 - k*b2/d|
* =
* |c1 - k*g1/2^384 + k*g1/2^384 - k*b2/d|
* <= {triangle inequality}
* |c1 - k*g1/2^384| + |k*g1/2^384 - k*b2/d|
* =
* |c1 - k*g1/2^384| + k*|g1/2^384 - b2/d|
* < {rounding in c1 and 0 <= k < 2^256}
* 2^-1 + 2^256 * |g1/2^384 - b2/d|
* = {definition of epsilon1}
* 2^-1 + epsilon1
*
* Lemma 2: |c2 - k*(-b1)/d| < 2^-1 + epsilon2
*
* |c2 - k*(-b1)/d|
* =
* |c2 - k*g2/2^384 + k*g2/2^384 - k*(-b1)/d|
* <= {triangle inequality}
* |c2 - k*g2/2^384| + |k*g2/2^384 - k*(-b1)/d|
* =
* |c2 - k*g2/2^384| + k*|g2/2^384 - (-b1)/d|
* < {rounding in c2 and 0 <= k < 2^256}
* 2^-1 + 2^256 * |g2/2^384 - (-b1)/d|
* = {definition of epsilon2}
* 2^-1 + epsilon2
*
* Let
* - k1 = k - c1*a1 - c2*a2
* - k2 = - c1*b1 - c2*b2
*
* Lemma 3: |k1| < (a1 + a2 + 1)/2 < 2^128
*
* |k1|
* = {definition of k1}
* |k - c1*a1 - c2*a2|
* = {(a1*b2 - b1*a2)/n = 1}
* |k*(a1*b2 - b1*a2)/n - c1*a1 - c2*a2|
* =
* |a1*(k*b2/n - c1) + a2*(k*(-b1)/n - c2)|
* <= {triangle inequality}
* a1*|k*b2/n - c1| + a2*|k*(-b1)/n - c2|
* < {Lemma 1 and Lemma 2}
* a1*(2^-1 + epslion1) + a2*(2^-1 + epsilon2)
* < {rounding up to an integer}
* (a1 + a2 + 1)/2
* < {rounding up to a power of 2}
* 2^128
*
* Lemma 4: |k2| < (-b1 + b2)/2 + 1 < 2^128
*
* |k2|
* = {definition of k2}
* |- c1*a1 - c2*a2|
* = {(b1*b2 - b1*b2)/n = 0}
* |k*(b1*b2 - b1*b2)/n - c1*b1 - c2*b2|
* =
* |b1*(k*b2/n - c1) + b2*(k*(-b1)/n - c2)|
* <= {triangle inequality}
* (-b1)*|k*b2/n - c1| + b2*|k*(-b1)/n - c2|
* < {Lemma 1 and Lemma 2}
* (-b1)*(2^-1 + epslion1) + b2*(2^-1 + epsilon2)
* < {rounding up to an integer}
* (-b1 + b2)/2 + 1
* < {rounding up to a power of 2}
* 2^128
*
* Let
* - r2 = k2 mod n
* - r1 = k - r2*lambda mod n.
*
* Notice that r1 is defined such that r1 + r2 * lambda == k (mod n).
*
* Lemma 5: r1 == k1 mod n.
*
* r1
* == {definition of r1 and r2}
* k - k2*lambda
* == {definition of k2}
* k - (- c1*b1 - c2*b2)*lambda
* ==
* k + c1*b1*lambda + c2*b2*lambda
* == {a1 + b1*lambda == 0 mod n and a2 + b2*lambda == 0 mod n}
* k - c1*a1 - c2*a2
* == {definition of k1}
* k1
*
* From Lemma 3, Lemma 4, Lemma 5 and the definition of r2, we can conclude that
*
* - either r1 < 2^128 or -r1 mod n < 2^128
* - either r2 < 2^128 or -r2 mod n < 2^128.
*
* Q.E.D.
*/
static void secp256k1_scalar_split_lambda_verify(const secp256k1_scalar *r1, const secp256k1_scalar *r2, const secp256k1_scalar *k) {
secp256k1_scalar s;
unsigned char buf1[32];
unsigned char buf2[32];
/* (a1 + a2 + 1)/2 is 0xa2a8918ca85bafe22016d0b917e4dd77 */
static const unsigned char k1_bound[32] = {
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0xa2, 0xa8, 0x91, 0x8c, 0xa8, 0x5b, 0xaf, 0xe2, 0x20, 0x16, 0xd0, 0xb9, 0x17, 0xe4, 0xdd, 0x77
};
/* (-b1 + b2)/2 + 1 is 0x8a65287bd47179fb2be08846cea267ed */
static const unsigned char k2_bound[32] = {
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x8a, 0x65, 0x28, 0x7b, 0xd4, 0x71, 0x79, 0xfb, 0x2b, 0xe0, 0x88, 0x46, 0xce, 0xa2, 0x67, 0xed
};
secp256k1_scalar_mul(&s, &secp256k1_const_lambda, r2);
secp256k1_scalar_add(&s, &s, r1);
VERIFY_CHECK(secp256k1_scalar_eq(&s, k));
secp256k1_scalar_negate(&s, r1);
secp256k1_scalar_get_b32(buf1, r1);
secp256k1_scalar_get_b32(buf2, &s);
VERIFY_CHECK(secp256k1_memcmp_var(buf1, k1_bound, 32) < 0 || secp256k1_memcmp_var(buf2, k1_bound, 32) < 0);
secp256k1_scalar_negate(&s, r2);
secp256k1_scalar_get_b32(buf1, r2);
secp256k1_scalar_get_b32(buf2, &s);
VERIFY_CHECK(secp256k1_memcmp_var(buf1, k2_bound, 32) < 0 || secp256k1_memcmp_var(buf2, k2_bound, 32) < 0);
}
#endif /* VERIFY */
#endif /* !defined(EXHAUSTIVE_TEST_ORDER) */
#endif /* SECP256K1_SCALAR_IMPL_H */