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/**********************************************************************
* Copyright (c) 2013, 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_REPR_IMPL_H_
#define _SECP256K1_SCALAR_REPR_IMPL_H_
typedef unsigned __int128 uint128_t;
/* Limbs of the secp256k1 order. */
#define SECP256K1_N_0 ((uint64_t)0xBFD25E8CD0364141ULL)
#define SECP256K1_N_1 ((uint64_t)0xBAAEDCE6AF48A03BULL)
#define SECP256K1_N_2 ((uint64_t)0xFFFFFFFFFFFFFFFEULL)
#define SECP256K1_N_3 ((uint64_t)0xFFFFFFFFFFFFFFFFULL)
/* Limbs of 2^256 minus the secp256k1 order. */
#define SECP256K1_N_C_0 (~SECP256K1_N_0 + 1)
#define SECP256K1_N_C_1 (~SECP256K1_N_1)
#define SECP256K1_N_C_2 (1)
/* Limbs of half the secp256k1 order. */
#define SECP256K1_N_H_0 ((uint64_t)0xDFE92F46681B20A0ULL)
#define SECP256K1_N_H_1 ((uint64_t)0x5D576E7357A4501DULL)
#define SECP256K1_N_H_2 ((uint64_t)0xFFFFFFFFFFFFFFFFULL)
#define SECP256K1_N_H_3 ((uint64_t)0x7FFFFFFFFFFFFFFFULL)
SECP256K1_INLINE static void secp256k1_scalar_clear(secp256k1_scalar_t *r) {
r->d[0] = 0;
r->d[1] = 0;
r->d[2] = 0;
r->d[3] = 0;
}
SECP256K1_INLINE static void secp256k1_scalar_set_int(secp256k1_scalar_t *r, unsigned int v) {
r->d[0] = v;
r->d[1] = 0;
r->d[2] = 0;
r->d[3] = 0;
}
SECP256K1_INLINE static unsigned int secp256k1_scalar_get_bits(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count) {
VERIFY_CHECK((offset + count - 1) >> 6 == offset >> 6);
return (a->d[offset >> 6] >> (offset & 0x3F)) & ((((uint64_t)1) << count) - 1);
}
SECP256K1_INLINE static unsigned int secp256k1_scalar_get_bits_var(const secp256k1_scalar_t *a, unsigned int offset, unsigned int count) {
VERIFY_CHECK(count < 32);
VERIFY_CHECK(offset + count <= 256);
if ((offset + count - 1) >> 6 == offset >> 6) {
return secp256k1_scalar_get_bits(a, offset, count);
} else {
VERIFY_CHECK((offset >> 6) + 1 < 4);
return ((a->d[offset >> 6] >> (offset & 0x3F)) | (a->d[(offset >> 6) + 1] << (64 - (offset & 0x3F)))) & ((((uint64_t)1) << count) - 1);
}
}
SECP256K1_INLINE static int secp256k1_scalar_check_overflow(const secp256k1_scalar_t *a) {
int yes = 0;
int no = 0;
no |= (a->d[3] < SECP256K1_N_3); /* No need for a > check. */
no |= (a->d[2] < SECP256K1_N_2);
yes |= (a->d[2] > SECP256K1_N_2) & ~no;
no |= (a->d[1] < SECP256K1_N_1);
yes |= (a->d[1] > SECP256K1_N_1) & ~no;
yes |= (a->d[0] >= SECP256K1_N_0) & ~no;
return yes;
}
SECP256K1_INLINE static int secp256k1_scalar_reduce(secp256k1_scalar_t *r, unsigned int overflow) {
VERIFY_CHECK(overflow <= 1);
uint128_t t = (uint128_t)r->d[0] + overflow * SECP256K1_N_C_0;
r->d[0] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)r->d[1] + overflow * SECP256K1_N_C_1;
r->d[1] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)r->d[2] + overflow * SECP256K1_N_C_2;
r->d[2] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint64_t)r->d[3];
r->d[3] = t & 0xFFFFFFFFFFFFFFFFULL;
return overflow;
}
static void secp256k1_scalar_add(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
uint128_t t = (uint128_t)a->d[0] + b->d[0];
r->d[0] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)a->d[1] + b->d[1];
r->d[1] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)a->d[2] + b->d[2];
r->d[2] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)a->d[3] + b->d[3];
r->d[3] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
secp256k1_scalar_reduce(r, t + secp256k1_scalar_check_overflow(r));
}
static void secp256k1_scalar_add_bit(secp256k1_scalar_t *r, unsigned int bit) {
VERIFY_CHECK(bit < 256);
uint128_t t = (uint128_t)r->d[0] + (((uint64_t)((bit >> 6) == 0)) << bit);
r->d[0] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)r->d[1] + (((uint64_t)((bit >> 6) == 1)) << (bit & 0x3F));
r->d[1] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)r->d[2] + (((uint64_t)((bit >> 6) == 2)) << (bit & 0x3F));
r->d[2] = t & 0xFFFFFFFFFFFFFFFFULL; t >>= 64;
t += (uint128_t)r->d[3] + (((uint64_t)((bit >> 6) == 3)) << (bit & 0x3F));
r->d[3] = t & 0xFFFFFFFFFFFFFFFFULL;
#ifdef VERIFY
VERIFY_CHECK((t >> 64) == 0);
VERIFY_CHECK(secp256k1_scalar_check_overflow(r) == 0);
#endif
}
static void secp256k1_scalar_set_b32(secp256k1_scalar_t *r, const unsigned char *b32, int *overflow) {
r->d[0] = (uint64_t)b32[31] | (uint64_t)b32[30] << 8 | (uint64_t)b32[29] << 16 | (uint64_t)b32[28] << 24 | (uint64_t)b32[27] << 32 | (uint64_t)b32[26] << 40 | (uint64_t)b32[25] << 48 | (uint64_t)b32[24] << 56;
r->d[1] = (uint64_t)b32[23] | (uint64_t)b32[22] << 8 | (uint64_t)b32[21] << 16 | (uint64_t)b32[20] << 24 | (uint64_t)b32[19] << 32 | (uint64_t)b32[18] << 40 | (uint64_t)b32[17] << 48 | (uint64_t)b32[16] << 56;
r->d[2] = (uint64_t)b32[15] | (uint64_t)b32[14] << 8 | (uint64_t)b32[13] << 16 | (uint64_t)b32[12] << 24 | (uint64_t)b32[11] << 32 | (uint64_t)b32[10] << 40 | (uint64_t)b32[9] << 48 | (uint64_t)b32[8] << 56;
r->d[3] = (uint64_t)b32[7] | (uint64_t)b32[6] << 8 | (uint64_t)b32[5] << 16 | (uint64_t)b32[4] << 24 | (uint64_t)b32[3] << 32 | (uint64_t)b32[2] << 40 | (uint64_t)b32[1] << 48 | (uint64_t)b32[0] << 56;
int over = secp256k1_scalar_reduce(r, secp256k1_scalar_check_overflow(r));
if (overflow) {
*overflow = over;
}
}
static void secp256k1_scalar_get_b32(unsigned char *bin, const secp256k1_scalar_t* a) {
bin[0] = a->d[3] >> 56; bin[1] = a->d[3] >> 48; bin[2] = a->d[3] >> 40; bin[3] = a->d[3] >> 32; bin[4] = a->d[3] >> 24; bin[5] = a->d[3] >> 16; bin[6] = a->d[3] >> 8; bin[7] = a->d[3];
bin[8] = a->d[2] >> 56; bin[9] = a->d[2] >> 48; bin[10] = a->d[2] >> 40; bin[11] = a->d[2] >> 32; bin[12] = a->d[2] >> 24; bin[13] = a->d[2] >> 16; bin[14] = a->d[2] >> 8; bin[15] = a->d[2];
bin[16] = a->d[1] >> 56; bin[17] = a->d[1] >> 48; bin[18] = a->d[1] >> 40; bin[19] = a->d[1] >> 32; bin[20] = a->d[1] >> 24; bin[21] = a->d[1] >> 16; bin[22] = a->d[1] >> 8; bin[23] = a->d[1];
bin[24] = a->d[0] >> 56; bin[25] = a->d[0] >> 48; bin[26] = a->d[0] >> 40; bin[27] = a->d[0] >> 32; bin[28] = a->d[0] >> 24; bin[29] = a->d[0] >> 16; bin[30] = a->d[0] >> 8; bin[31] = a->d[0];
}
SECP256K1_INLINE static int secp256k1_scalar_is_zero(const secp256k1_scalar_t *a) {
return (a->d[0] | a->d[1] | a->d[2] | a->d[3]) == 0;
}
static void secp256k1_scalar_negate(secp256k1_scalar_t *r, const secp256k1_scalar_t *a) {
uint64_t nonzero = 0xFFFFFFFFFFFFFFFFULL * (secp256k1_scalar_is_zero(a) == 0);
uint128_t t = (uint128_t)(~a->d[0]) + SECP256K1_N_0 + 1;
r->d[0] = t & nonzero; t >>= 64;
t += (uint128_t)(~a->d[1]) + SECP256K1_N_1;
r->d[1] = t & nonzero; t >>= 64;
t += (uint128_t)(~a->d[2]) + SECP256K1_N_2;
r->d[2] = t & nonzero; t >>= 64;
t += (uint128_t)(~a->d[3]) + SECP256K1_N_3;
r->d[3] = t & nonzero;
}
SECP256K1_INLINE static int secp256k1_scalar_is_one(const secp256k1_scalar_t *a) {
return ((a->d[0] ^ 1) | a->d[1] | a->d[2] | a->d[3]) == 0;
}
static int secp256k1_scalar_is_high(const secp256k1_scalar_t *a) {
int yes = 0;
int no = 0;
no |= (a->d[3] < SECP256K1_N_H_3);
yes |= (a->d[3] > SECP256K1_N_H_3) & ~no;
no |= (a->d[2] < SECP256K1_N_H_2) & ~yes; /* No need for a > check. */
no |= (a->d[1] < SECP256K1_N_H_1) & ~yes;
yes |= (a->d[1] > SECP256K1_N_H_1) & ~no;
yes |= (a->d[0] > SECP256K1_N_H_0) & ~no;
return yes;
}
/* Inspired by the macros in OpenSSL's crypto/bn/asm/x86_64-gcc.c. */
/** Add a*b to the number defined by (c0,c1,c2). c2 must never overflow. */
#define muladd(a,b) { \
uint64_t tl, th; \
{ \
uint128_t t = (uint128_t)a * b; \
th = t >> 64; /* at most 0xFFFFFFFFFFFFFFFE */ \
tl = t; \
} \
c0 += tl; /* overflow is handled on the next line */ \
th += (c0 < tl) ? 1 : 0; /* at most 0xFFFFFFFFFFFFFFFF */ \
c1 += th; /* overflow is handled on the next line */ \
c2 += (c1 < th) ? 1 : 0; /* never overflows by contract (verified in the next line) */ \
VERIFY_CHECK((c1 >= th) || (c2 != 0)); \
}
/** Add a*b to the number defined by (c0,c1). c1 must never overflow. */
#define muladd_fast(a,b) { \
uint64_t tl, th; \
{ \
uint128_t t = (uint128_t)a * b; \
th = t >> 64; /* at most 0xFFFFFFFFFFFFFFFE */ \
tl = t; \
} \
c0 += tl; /* overflow is handled on the next line */ \
th += (c0 < tl) ? 1 : 0; /* at most 0xFFFFFFFFFFFFFFFF */ \
c1 += th; /* never overflows by contract (verified in the next line) */ \
VERIFY_CHECK(c1 >= th); \
}
/** Add 2*a*b to the number defined by (c0,c1,c2). c2 must never overflow. */
#define muladd2(a,b) { \
uint64_t tl, th; \
{ \
uint128_t t = (uint128_t)a * b; \
th = t >> 64; /* at most 0xFFFFFFFFFFFFFFFE */ \
tl = t; \
} \
uint64_t th2 = th + th; /* at most 0xFFFFFFFFFFFFFFFE (in case th was 0x7FFFFFFFFFFFFFFF) */ \
c2 += (th2 < th) ? 1 : 0; /* never overflows by contract (verified the next line) */ \
VERIFY_CHECK((th2 >= th) || (c2 != 0)); \
uint64_t tl2 = tl + tl; /* at most 0xFFFFFFFFFFFFFFFE (in case the lowest 63 bits of tl were 0x7FFFFFFFFFFFFFFF) */ \
th2 += (tl2 < tl) ? 1 : 0; /* at most 0xFFFFFFFFFFFFFFFF */ \
c0 += tl2; /* overflow is handled on the next line */ \
th2 += (c0 < tl2) ? 1 : 0; /* second overflow is handled on the next line */ \
c2 += (c0 < tl2) & (th2 == 0); /* never overflows by contract (verified the next line) */ \
VERIFY_CHECK((c0 >= tl2) || (th2 != 0) || (c2 != 0)); \
c1 += th2; /* overflow is handled on the next line */ \
c2 += (c1 < th2) ? 1 : 0; /* never overflows by contract (verified the next line) */ \
VERIFY_CHECK((c1 >= th2) || (c2 != 0)); \
}
/** Add a to the number defined by (c0,c1,c2). c2 must never overflow. */
#define sumadd(a) { \
c0 += (a); /* overflow is handled on the next line */ \
unsigned int over = (c0 < (a)) ? 1 : 0; \
c1 += over; /* overflow is handled on the next line */ \
c2 += (c1 < over) ? 1 : 0; /* never overflows by contract */ \
}
/** Add a to the number defined by (c0,c1). c1 must never overflow, c2 must be zero. */
#define sumadd_fast(a) { \
c0 += (a); /* overflow is handled on the next line */ \
c1 += (c0 < (a)) ? 1 : 0; /* never overflows by contract (verified the next line) */ \
VERIFY_CHECK((c1 != 0) | (c0 >= (a))); \
VERIFY_CHECK(c2 == 0); \
}
/** Extract the lowest 64 bits of (c0,c1,c2) into n, and left shift the number 64 bits. */
#define extract(n) { \
(n) = c0; \
c0 = c1; \
c1 = c2; \
c2 = 0; \
}
/** Extract the lowest 64 bits of (c0,c1,c2) into n, and left shift the number 64 bits. c2 is required to be zero. */
#define extract_fast(n) { \
(n) = c0; \
c0 = c1; \
c1 = 0; \
VERIFY_CHECK(c2 == 0); \
}
static void secp256k1_scalar_reduce_512(secp256k1_scalar_t *r, const uint64_t *l) {
uint64_t n0 = l[4], n1 = l[5], n2 = l[6], n3 = l[7];
/* 160 bit accumulator. */
uint64_t c0, c1;
uint32_t c2;
/* Reduce 512 bits into 385. */
/* m[0..6] = l[0..3] + n[0..3] * SECP256K1_N_C. */
c0 = l[0]; c1 = 0; c2 = 0;
muladd_fast(n0, SECP256K1_N_C_0);
uint64_t m0; extract_fast(m0);
sumadd_fast(l[1]);
muladd(n1, SECP256K1_N_C_0);
muladd(n0, SECP256K1_N_C_1);
uint64_t m1; extract(m1);
sumadd(l[2]);
muladd(n2, SECP256K1_N_C_0);
muladd(n1, SECP256K1_N_C_1);
sumadd(n0);
uint64_t m2; extract(m2);
sumadd(l[3]);
muladd(n3, SECP256K1_N_C_0);
muladd(n2, SECP256K1_N_C_1);
sumadd(n1);
uint64_t m3; extract(m3);
muladd(n3, SECP256K1_N_C_1);
sumadd(n2);
uint64_t m4; extract(m4);
sumadd_fast(n3);
uint64_t m5; extract_fast(m5);
VERIFY_CHECK(c0 <= 1);
uint32_t m6 = c0;
/* Reduce 385 bits into 258. */
/* p[0..4] = m[0..3] + m[4..6] * SECP256K1_N_C. */
c0 = m0; c1 = 0; c2 = 0;
muladd_fast(m4, SECP256K1_N_C_0);
uint64_t p0; extract_fast(p0);
sumadd_fast(m1);
muladd(m5, SECP256K1_N_C_0);
muladd(m4, SECP256K1_N_C_1);
uint64_t p1; extract(p1);
sumadd(m2);
muladd(m6, SECP256K1_N_C_0);
muladd(m5, SECP256K1_N_C_1);
sumadd(m4);
uint64_t p2; extract(p2);
sumadd_fast(m3);
muladd_fast(m6, SECP256K1_N_C_1);
sumadd_fast(m5);
uint64_t p3; extract_fast(p3);
uint32_t p4 = c0 + m6;
VERIFY_CHECK(p4 <= 2);
/* Reduce 258 bits into 256. */
/* r[0..3] = p[0..3] + p[4] * SECP256K1_N_C. */
uint128_t c = p0 + (uint128_t)SECP256K1_N_C_0 * p4;
r->d[0] = c & 0xFFFFFFFFFFFFFFFFULL; c >>= 64;
c += p1 + (uint128_t)SECP256K1_N_C_1 * p4;
r->d[1] = c & 0xFFFFFFFFFFFFFFFFULL; c >>= 64;
c += p2 + (uint128_t)p4;
r->d[2] = c & 0xFFFFFFFFFFFFFFFFULL; c >>= 64;
c += p3;
r->d[3] = c & 0xFFFFFFFFFFFFFFFFULL; c >>= 64;
/* Final reduction of r. */
secp256k1_scalar_reduce(r, c + secp256k1_scalar_check_overflow(r));
}
static void secp256k1_scalar_mul_512(uint64_t l[8], const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
/* 160 bit accumulator. */
uint64_t c0 = 0, c1 = 0;
uint32_t c2 = 0;
/* l[0..7] = a[0..3] * b[0..3]. */
muladd_fast(a->d[0], b->d[0]);
extract_fast(l[0]);
muladd(a->d[0], b->d[1]);
muladd(a->d[1], b->d[0]);
extract(l[1]);
muladd(a->d[0], b->d[2]);
muladd(a->d[1], b->d[1]);
muladd(a->d[2], b->d[0]);
extract(l[2]);
muladd(a->d[0], b->d[3]);
muladd(a->d[1], b->d[2]);
muladd(a->d[2], b->d[1]);
muladd(a->d[3], b->d[0]);
extract(l[3]);
muladd(a->d[1], b->d[3]);
muladd(a->d[2], b->d[2]);
muladd(a->d[3], b->d[1]);
extract(l[4]);
muladd(a->d[2], b->d[3]);
muladd(a->d[3], b->d[2]);
extract(l[5]);
muladd_fast(a->d[3], b->d[3]);
extract_fast(l[6]);
VERIFY_CHECK(c1 <= 0);
l[7] = c0;
}
static void secp256k1_scalar_sqr_512(uint64_t l[8], const secp256k1_scalar_t *a) {
/* 160 bit accumulator. */
uint64_t c0 = 0, c1 = 0;
uint32_t c2 = 0;
/* l[0..7] = a[0..3] * b[0..3]. */
muladd_fast(a->d[0], a->d[0]);
extract_fast(l[0]);
muladd2(a->d[0], a->d[1]);
extract(l[1]);
muladd2(a->d[0], a->d[2]);
muladd(a->d[1], a->d[1]);
extract(l[2]);
muladd2(a->d[0], a->d[3]);
muladd2(a->d[1], a->d[2]);
extract(l[3]);
muladd2(a->d[1], a->d[3]);
muladd(a->d[2], a->d[2]);
extract(l[4]);
muladd2(a->d[2], a->d[3]);
extract(l[5]);
muladd_fast(a->d[3], a->d[3]);
extract_fast(l[6]);
VERIFY_CHECK(c1 == 0);
l[7] = c0;
}
#undef sumadd
#undef sumadd_fast
#undef muladd
#undef muladd_fast
#undef muladd2
#undef extract
#undef extract_fast
static void secp256k1_scalar_mul(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
uint64_t l[8];
secp256k1_scalar_mul_512(l, a, b);
secp256k1_scalar_reduce_512(r, l);
}
static void secp256k1_scalar_sqr(secp256k1_scalar_t *r, const secp256k1_scalar_t *a) {
uint64_t l[8];
secp256k1_scalar_sqr_512(l, a);
secp256k1_scalar_reduce_512(r, l);
}
static void secp256k1_scalar_split_128(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) {
r1->d[0] = a->d[0];
r1->d[1] = a->d[1];
r1->d[2] = 0;
r1->d[3] = 0;
r2->d[0] = a->d[2];
r2->d[1] = a->d[3];
r2->d[2] = 0;
r2->d[3] = 0;
}
SECP256K1_INLINE static int secp256k1_scalar_eq(const secp256k1_scalar_t *a, const secp256k1_scalar_t *b) {
return ((a->d[0] ^ b->d[0]) | (a->d[1] ^ b->d[1]) | (a->d[2] ^ b->d[2]) | (a->d[3] ^ b->d[3])) == 0;
}
SECP256K1_INLINE static void secp256k1_scalar_mul_shift_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *a, const secp256k1_scalar_t *b, unsigned int shift) {
VERIFY_CHECK(shift >= 256);
uint64_t l[8];
secp256k1_scalar_mul_512(l, a, b);
unsigned int shiftlimbs = shift >> 6;
unsigned int shiftlow = shift & 0x3F;
unsigned int shifthigh = 64 - shiftlow;
r->d[0] = shift < 512 ? (l[0 + shiftlimbs] >> shiftlow | (shift < 448 && shiftlow ? (l[1 + shiftlimbs] << shifthigh) : 0)) : 0;
r->d[1] = shift < 448 ? (l[1 + shiftlimbs] >> shiftlow | (shift < 384 && shiftlow ? (l[2 + shiftlimbs] << shifthigh) : 0)) : 0;
r->d[2] = shift < 384 ? (l[2 + shiftlimbs] >> shiftlow | (shift < 320 && shiftlow ? (l[3 + shiftlimbs] << shifthigh) : 0)) : 0;
r->d[3] = shift < 320 ? (l[3 + shiftlimbs] >> shiftlow) : 0;
if ((l[(shift - 1) >> 6] >> ((shift - 1) & 0x3f)) & 1) {
secp256k1_scalar_add_bit(r, 0);
}
}
#endif