path: root/src/scalar_4x64_impl.h

/**********************************************************************
 * 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 int 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;
    int overflow = t + secp256k1_scalar_check_overflow(r);
    VERIFY_CHECK(overflow == 0 || overflow == 1);
    secp256k1_scalar_reduce(r, overflow);
    return overflow;
}

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 & 0x3F));
    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