bitcoin-bitcoin-core/src/ecmult.cpp

#include <sstream>
#include <algorithm>

#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 == 13 results in 640 KiB.
#define WINDOW_G 14

extern "C" {

/** 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_ge_t *a, int w) {
    pre[0] = *a;
    secp256k1_gej_t x; secp256k1_gej_set_ge(&x, a);
    secp256k1_gej_t d; secp256k1_gej_double(&d, &x);
    for (int i=1; i<(1 << (w-2)); i++) {
        secp256k1_gej_add_ge(&x, &d, &pre[i-1]);
        secp256k1_ge_set_gej(&pre[i], &x);
    }
}

/** 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 { \
    assert(((n) & 1) == 1); \
    assert((n) >= -((1 << ((w)-1)) - 1)); \
    assert((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 {
    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
    secp256k1_ge_t prec[64][16]; // prec[j][i] = 16^j * (i+1) * G
    secp256k1_ge_t fin; // -(sum(prec[j][0], j=0..63))
} secp256k1_ecmult_consts_t;

static 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;

    // calculate 2^128*generator
    secp256k1_gej_t g_128j; secp256k1_gej_set_ge(&g_128j, g);
    for (int i=0; i<128; i++)
        secp256k1_gej_double(&g_128j, &g_128j);
    secp256k1_ge_t g_128; secp256k1_ge_set_gej(&g_128, &g_128j);

    // precompute the tables with odd multiples
    secp256k1_ecmult_table_precomp_ge(ret->pre_g, g, WINDOW_G);
    secp256k1_ecmult_table_precomp_ge(ret->pre_g_128, &g_128, WINDOW_G);

    // compute prec and fin
    secp256k1_gej_t gg; secp256k1_gej_set_ge(&gg, g);
    secp256k1_ge_t ad = *g;
    secp256k1_gej_t fn; secp256k1_gej_set_infinity(&fn);
    for (int j=0; j<64; j++) {
        secp256k1_ge_set_gej(&ret->prec[j][0], &gg);
        secp256k1_gej_add(&fn, &fn, &gg);
        for (int i=1; i<16; i++) {
            secp256k1_gej_add_ge(&gg, &gg, &ad);
            secp256k1_ge_set_gej(&ret->prec[j][i], &gg);
        }
        ad = ret->prec[j][15];
    }
    secp256k1_ge_set_gej(&ret->fin, &fn);
    secp256k1_ge_neg(&ret->fin, &ret->fin);
}

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

    free(secp256k1_ecmult_consts);
    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_ge(r, &c->prec[0][secp256k1_num_shift(&n, 4)]);
    for (int j=1; j<64; j++)
        secp256k1_gej_add_ge(r, r, &c->prec[j][secp256k1_num_shift(&n, 4)]);
    secp256k1_num_free(&n);
    secp256k1_gej_add_ge(r, r, &c->fin);
}

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;

    secp256k1_num_t na_1, na_lam;
    secp256k1_num_t ng_1, ng_128;
    secp256k1_num_init(&na_1);
    secp256k1_num_init(&na_lam);
    secp256k1_num_init(&ng_1);
    secp256k1_num_init(&ng_128);

    // 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);
    // 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 na_1, na_lam, ng_1, ng_128
    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 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);

    // calculate a_lam = a*lambda
    secp256k1_gej_t a_lam; secp256k1_gej_mul_lambda(&a_lam, a);

    // calculate odd multiples of a and a_lam
    secp256k1_gej_t pre_a_1[ECMULT_TABLE_SIZE(WINDOW_A)], pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
    secp256k1_ecmult_table_precomp_gej(pre_a_1,   a,      WINDOW_A);
    secp256k1_ecmult_table_precomp_gej(pre_a_lam, &a_lam, WINDOW_A);

    int bits = std::max(std::max(bits_na_1, bits_na_lam), std::max(bits_ng_1, 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;
        if (i < bits_na_1 && (n = wnaf_na_1[i])) {
            ECMULT_TABLE_GET_GEJ(&tmpj, pre_a_1, 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);
        }
        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);
        }
    }

    secp256k1_num_free(&na_1);
    secp256k1_num_free(&na_lam);
    secp256k1_num_free(&ng_1);
    secp256k1_num_free(&ng_128);
}

}