bitcoin-bitcoin-core/src/ecmult_impl.h

// Copyright (c) 2013 Pieter Wuille
// Distributed under the MIT/X11 software license, see the accompanying
// file COPYING or http://www.opensource.org/licenses/mit-license.php.

#ifndef _SECP256K1_ECMULT_IMPL_H_
#define _SECP256K1_ECMULT_IMPL_H_

#include <assert.h>
#include "num.h"
#include "group.h"
#include "ecmult.h"

// optimal for 128-bit and 256-bit exponents.
#define WINDOW_A 5

// larger numbers may result in slightly better performance, at the cost of
// exponentially larger precomputed tables. WINDOW_G == 14 results in 640 KiB.
#define WINDOW_G 14

/** Fill a table 'pre' with precomputed odd multiples of a. W determines the size of the table.
 *  pre will contains the values [1*a,3*a,5*a,...,(2^(w-1)-1)*a], so it needs place for
 *  2^(w-2) entries.
 *
 *  There are two versions of this function:
 *  - secp256k1_ecmult_precomp_wnaf_gej, which operates on group elements in jacobian notation,
 *    fast to precompute, but slower to use in later additions.
 *  - secp256k1_ecmult_precomp_wnaf_ge, which operates on group elements in affine notations,
 *    (much) slower to precompute, but a bit faster to use in later additions.
 *  To compute a*P + b*G, we use the jacobian version for P, and the affine version for G, as
 *  G is constant, so it only needs to be done once in advance.
 */
void static secp256k1_ecmult_table_precomp_gej(secp256k1_gej_t *pre, const secp256k1_gej_t *a, int w) {
    pre[0] = *a;
    secp256k1_gej_t d; secp256k1_gej_double(&d, &pre[0]);
    for (int i=1; i<(1 << (w-2)); i++)
        secp256k1_gej_add(&pre[i], &d, &pre[i-1]);
}

void static secp256k1_ecmult_table_precomp_ge(secp256k1_ge_t *pre, const secp256k1_gej_t *a, int w) {
    const int table_size = 1 << (w-2);
    secp256k1_gej_t prej[table_size];
    prej[0] = *a;
    secp256k1_gej_t d; secp256k1_gej_double(&d, a);
    for (int i=1; i<table_size; i++) {
        secp256k1_gej_add(&prej[i], &d, &prej[i-1]);
    }
    secp256k1_ge_set_all_gej(table_size, pre, prej);
}

/** The number of entries a table with precomputed multiples needs to have. */
#define ECMULT_TABLE_SIZE(w) (1 << ((w)-2))

/** The following two macro retrieves a particular odd multiple from a table
 *  of precomputed multiples. */
#define ECMULT_TABLE_GET(r,pre,n,w,neg) do { \
    VERIFY_CHECK(((n) & 1) == 1); \
    VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
    VERIFY_CHECK((n) <=  ((1 << ((w)-1)) - 1)); \
    if ((n) > 0) \
        *(r) = (pre)[((n)-1)/2]; \
    else \
        (neg)((r), &(pre)[(-(n)-1)/2]); \
} while(0)

#define ECMULT_TABLE_GET_GEJ(r,pre,n,w) ECMULT_TABLE_GET((r),(pre),(n),(w),secp256k1_gej_neg)
#define ECMULT_TABLE_GET_GE(r,pre,n,w)  ECMULT_TABLE_GET((r),(pre),(n),(w),secp256k1_ge_neg)

typedef struct {
    // For accelerating the computation of a*P + b*G:
    secp256k1_ge_t pre_g[ECMULT_TABLE_SIZE(WINDOW_G)];    // odd multiples of the generator
    secp256k1_ge_t pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)]; // odd multiples of 2^128*generator

    // For accelerating the computation of a*G:
    // To harden against timing attacks, use the following mechanism:
    // * Break up the multiplicand into groups of 4 bits, called n_0, n_1, n_2, ..., n_63.
    // * Compute sum(n_i * 16^i * G + U_i, i=0..63), where:
    //   * U_i = U * 2^i (for i=0..62)
    //   * U_i = U * (1-2^63) (for i=63)
    //   where U is a point with no known corresponding scalar. Note that sum(U_i, i=0..63) = 0.
    // For each i, and each of the 16 possible values of n_i, (n_i * 16^i * G + U_i) is
    // precomputed (call it prec(i, n_i)). The formula now becomes sum(prec(i, n_i), i=0..63).
    // None of the resulting prec group elements have a known scalar, and neither do any of
    // the intermediate sums while computing a*G.
    // To make memory access uniform, the bytes of prec(i, n_i) are sliced per value of n_i.
    unsigned char prec[64][sizeof(secp256k1_ge_t)][16]; // prec[j][k][i] = k'th byte of (16^j * i * G + U_i)
} secp256k1_ecmult_consts_t;

static const secp256k1_ecmult_consts_t *secp256k1_ecmult_consts = NULL;

static void secp256k1_ecmult_start(void) {
    if (secp256k1_ecmult_consts != NULL)
        return;

    secp256k1_ecmult_consts_t *ret = (secp256k1_ecmult_consts_t*)malloc(sizeof(secp256k1_ecmult_consts_t));
    secp256k1_ecmult_consts = ret;

    // get the generator
    const secp256k1_ge_t *g = &secp256k1_ge_consts->g;
    secp256k1_gej_t gj; secp256k1_gej_set_ge(&gj, g);

    // calculate 2^128*generator
    secp256k1_gej_t g_128j = gj;
    for (int i=0; i<128; i++)
        secp256k1_gej_double(&g_128j, &g_128j);

    // precompute the tables with odd multiples
    secp256k1_ecmult_table_precomp_ge(ret->pre_g, &gj, WINDOW_G);
    secp256k1_ecmult_table_precomp_ge(ret->pre_g_128, &g_128j, WINDOW_G);

    // Construct a group element with no known corresponding scalar (nothing up my sleeve).
    secp256k1_gej_t nums_gej;
    {
        static const unsigned char nums_b32[32] = "The scalar for this x is unknown";
        secp256k1_fe_t nums_x;
        secp256k1_fe_set_b32(&nums_x, nums_b32);
        secp256k1_ge_t nums_ge;
        VERIFY_CHECK(secp256k1_ge_set_xo(&nums_ge, &nums_x, 0));
        secp256k1_gej_set_ge(&nums_gej, &nums_ge);
        // Add G to make the bits in x uniformly distributed.
        secp256k1_gej_add_ge(&nums_gej, &nums_gej, g);
    }

    // compute prec.
    secp256k1_ge_t prec[1024];
    {
        secp256k1_gej_t precj[1024]; // Jacobian versions of prec.
        int j = 0;
        secp256k1_gej_t gbase; gbase = gj; // 16^j * G
        secp256k1_gej_t numsbase; numsbase = nums_gej; // 2^j * nums.
        for (int j=0; j<64; j++) {
            // Set precj[j*16 .. j*16+15] to (numsbase, numsbase + gbase, ..., numsbase + 15*gbase).
            precj[j*16] = numsbase;
            for (int i=1; i<16; i++) {
                secp256k1_gej_add(&precj[j*16 + i], &precj[j*16 + i - 1], &gbase);
            }
            // Multiply gbase by 16.
            for (int i=0; i<4; i++) {
                secp256k1_gej_double(&gbase, &gbase);
            }
            // Multiply numbase by 2.
            secp256k1_gej_double(&numsbase, &numsbase);
            if (j == 62) {
                // In the last iteration, numsbase is (1 - 2^j) * nums instead.
                secp256k1_gej_neg(&numsbase, &numsbase);
                secp256k1_gej_add(&numsbase, &numsbase, &nums_gej);
            }
        }
        secp256k1_ge_set_all_gej(1024, prec, precj);
    }
    for (int j=0; j<64; j++) {
        for (int i=0; i<16; i++) {
            const unsigned char* raw = (const unsigned char*)(&prec[j*16 + i]);
            for (int k=0; k<sizeof(secp256k1_ge_t); k++)
                ret->prec[j][k][i] = raw[k];
        }
    }
}

static void secp256k1_ecmult_stop(void) {
    if (secp256k1_ecmult_consts == NULL)
        return;

    secp256k1_ecmult_consts_t *c = (secp256k1_ecmult_consts_t*)secp256k1_ecmult_consts;
    free(c);
    secp256k1_ecmult_consts = NULL;
}

/** Convert a number to WNAF notation. The number becomes represented by sum(2^i * wnaf[i], i=0..bits),
 *  with the following guarantees:
 *  - each wnaf[i] is either 0, or an odd integer between -(1<<(w-1) - 1) and (1<<(w-1) - 1)
 *  - two non-zero entries in wnaf are separated by at least w-1 zeroes.
 *  - the index of the highest non-zero entry in wnaf (=return value-1) is at most bits, where
 *    bits is the number of bits necessary to represent the absolute value of the input.
 */
static int secp256k1_ecmult_wnaf(int *wnaf, const secp256k1_num_t *a, int w) {
    int ret = 0;
    int zeroes = 0;
    secp256k1_num_t x;
    secp256k1_num_init(&x);
    secp256k1_num_copy(&x, a);
    int sign = 1;
    if (secp256k1_num_is_neg(&x)) {
        sign = -1;
        secp256k1_num_negate(&x);
    }
    while (!secp256k1_num_is_zero(&x)) {
        while (!secp256k1_num_is_odd(&x)) {
            zeroes++;
            secp256k1_num_shift(&x, 1);
        }
        int word = secp256k1_num_shift(&x, w);
        while (zeroes) {
            wnaf[ret++] = 0;
            zeroes--;
        }
        if (word & (1 << (w-1))) {
            secp256k1_num_inc(&x);
            wnaf[ret++] = sign * (word - (1 << w));
        } else {
            wnaf[ret++] = sign * word;
        }
        zeroes = w-1;
    }
    secp256k1_num_free(&x);
    return ret;
}

void static secp256k1_ecmult_gen(secp256k1_gej_t *r, const secp256k1_num_t *gn) {
    secp256k1_num_t n;
    secp256k1_num_init(&n);
    secp256k1_num_copy(&n, gn);
    const secp256k1_ecmult_consts_t *c = secp256k1_ecmult_consts;
    secp256k1_gej_set_infinity(r);
    secp256k1_ge_t add;
    int bits;
    for (int j=0; j<64; j++) {
        bits = secp256k1_num_shift(&n, 4);
        for (int k=0; k<sizeof(secp256k1_ge_t); k++)
            ((unsigned char*)(&add))[k] = c->prec[j][k][bits];
        secp256k1_gej_add_ge(r, r, &add);
    }
    bits = 0;
    secp256k1_ge_clear(&add);
    secp256k1_num_clear(&n);
    secp256k1_num_free(&n);
}

void static secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_num_t *na, const secp256k1_num_t *ng) {
    const secp256k1_ecmult_consts_t *c = secp256k1_ecmult_consts;

#ifdef USE_ENDOMORPHISM
    secp256k1_num_t na_1, na_lam;
    secp256k1_num_init(&na_1);
    secp256k1_num_init(&na_lam);
    // split na into na_1 and na_lam (where na = na_1 + na_lam*lambda, and na_1 and na_lam are ~128 bit)
    secp256k1_gej_split_exp(&na_1, &na_lam, na);

    // build wnaf representation for na_1 and na_lam.
    int wnaf_na_1[129];   int bits_na_1   = secp256k1_ecmult_wnaf(wnaf_na_1,   &na_1,   WINDOW_A);
    int wnaf_na_lam[129]; int bits_na_lam = secp256k1_ecmult_wnaf(wnaf_na_lam, &na_lam, WINDOW_A);
    int bits = bits_na_1;
    if (bits_na_lam > bits) bits = bits_na_lam;
#else
    // build wnaf representation for na.
    int wnaf_na[257];     int bits_na     = secp256k1_ecmult_wnaf(wnaf_na,     na,      WINDOW_A);
    int bits = bits_na;
#endif

    // calculate odd multiples of a
    secp256k1_gej_t pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
    secp256k1_ecmult_table_precomp_gej(pre_a, a, WINDOW_A);

#ifdef USE_ENDOMORPHISM
    secp256k1_gej_t pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
    for (int i=0; i<ECMULT_TABLE_SIZE(WINDOW_A); i++)
        secp256k1_gej_mul_lambda(&pre_a_lam[i], &pre_a[i]);
#endif

    // Splitted G factors.
    secp256k1_num_t ng_1, ng_128;
    secp256k1_num_init(&ng_1);
    secp256k1_num_init(&ng_128);

    // split ng into ng_1 and ng_128 (where gn = gn_1 + gn_128*2^128, and gn_1 and gn_128 are ~128 bit)
    secp256k1_num_split(&ng_1, &ng_128, ng, 128);

    // Build wnaf representation for ng_1 and ng_128
    int wnaf_ng_1[129];   int bits_ng_1   = secp256k1_ecmult_wnaf(wnaf_ng_1,   &ng_1,   WINDOW_G);
    int wnaf_ng_128[129]; int bits_ng_128 = secp256k1_ecmult_wnaf(wnaf_ng_128, &ng_128, WINDOW_G);
    if (bits_ng_1 > bits) bits = bits_ng_1;
    if (bits_ng_128 > bits) bits = bits_ng_128;

    secp256k1_gej_set_infinity(r);
    secp256k1_gej_t tmpj;
    secp256k1_ge_t tmpa;

    for (int i=bits-1; i>=0; i--) {
        secp256k1_gej_double(r, r);
        int n;
#ifdef USE_ENDOMORPHISM
        if (i < bits_na_1 && (n = wnaf_na_1[i])) {
            ECMULT_TABLE_GET_GEJ(&tmpj, pre_a, n, WINDOW_A);
            secp256k1_gej_add(r, r, &tmpj);
        }
        if (i < bits_na_lam && (n = wnaf_na_lam[i])) {
            ECMULT_TABLE_GET_GEJ(&tmpj, pre_a_lam, n, WINDOW_A);
            secp256k1_gej_add(r, r, &tmpj);
        }
#else
        if (i < bits_na && (n = wnaf_na[i])) {
            ECMULT_TABLE_GET_GEJ(&tmpj, pre_a, n, WINDOW_A);
            secp256k1_gej_add(r, r, &tmpj);
        }
#endif
        if (i < bits_ng_1 && (n = wnaf_ng_1[i])) {
            ECMULT_TABLE_GET_GE(&tmpa, c->pre_g, n, WINDOW_G);
            secp256k1_gej_add_ge(r, r, &tmpa);
        }
        if (i < bits_ng_128 && (n = wnaf_ng_128[i])) {
            ECMULT_TABLE_GET_GE(&tmpa, c->pre_g_128, n, WINDOW_G);
            secp256k1_gej_add_ge(r, r, &tmpa);
        }
    }

#ifdef USE_ENDOMORPHISM
    secp256k1_num_free(&na_1);
    secp256k1_num_free(&na_lam);
#endif
    secp256k1_num_free(&ng_1);
    secp256k1_num_free(&ng_128);
}

#endif