src/group_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_GROUP_IMPL_H_
#define _SECP256K1_GROUP_IMPL_H_

#include <string.h>

#include "num.h"
#include "field.h"
#include "group.h"

static void secp256k1_ge_set_infinity(secp256k1_ge_t *r) {
    r->infinity = 1;
}

static void secp256k1_ge_set_xy(secp256k1_ge_t *r, const secp256k1_fe_t *x, const secp256k1_fe_t *y) {
    r->infinity = 0;
    r->x = *x;
    r->y = *y;
}

static int secp256k1_ge_is_infinity(const secp256k1_ge_t *a) {
    return a->infinity;
}

static void secp256k1_ge_neg(secp256k1_ge_t *r, const secp256k1_ge_t *a) {
    *r = *a;
    secp256k1_fe_normalize_weak(&r->y);
    secp256k1_fe_negate(&r->y, &r->y, 1);
}

static void secp256k1_ge_get_hex(char *r, int *rlen, const secp256k1_ge_t *a) {
    char cx[65]; int lx=65;
    char cy[65]; int ly=65;
    secp256k1_fe_get_hex(cx, &lx, &a->x);
    secp256k1_fe_get_hex(cy, &ly, &a->y);
    lx = strlen(cx);
    ly = strlen(cy);
    int len = lx + ly + 3 + 1;
    if (*rlen < len) {
        *rlen = len;
        return;
    }
    *rlen = len;
    r[0] = '(';
    memcpy(r+1, cx, lx);
    r[1+lx] = ',';
    memcpy(r+2+lx, cy, ly);
    r[2+lx+ly] = ')';
    r[3+lx+ly] = 0;
}

static void secp256k1_ge_set_gej(secp256k1_ge_t *r, secp256k1_gej_t *a) {
    r->infinity = a->infinity;
    secp256k1_fe_inv(&a->z, &a->z);
    secp256k1_fe_t z2; secp256k1_fe_sqr(&z2, &a->z);
    secp256k1_fe_t z3; secp256k1_fe_mul(&z3, &a->z, &z2);
    secp256k1_fe_mul(&a->x, &a->x, &z2);
    secp256k1_fe_mul(&a->y, &a->y, &z3);
    secp256k1_fe_set_int(&a->z, 1);
    r->x = a->x;
    r->y = a->y;
}

static void secp256k1_ge_set_gej_var(secp256k1_ge_t *r, secp256k1_gej_t *a) {
    r->infinity = a->infinity;
    if (a->infinity) {
        return;
    }
    secp256k1_fe_inv_var(&a->z, &a->z);
    secp256k1_fe_t z2; secp256k1_fe_sqr(&z2, &a->z);
    secp256k1_fe_t z3; secp256k1_fe_mul(&z3, &a->z, &z2);
    secp256k1_fe_mul(&a->x, &a->x, &z2);
    secp256k1_fe_mul(&a->y, &a->y, &z3);
    secp256k1_fe_set_int(&a->z, 1);
    r->x = a->x;
    r->y = a->y;
}

static void secp256k1_ge_set_all_gej_var(size_t len, secp256k1_ge_t r[len], const secp256k1_gej_t a[len]) {
    size_t count = 0;
    secp256k1_fe_t *az = checked_malloc(sizeof(secp256k1_fe_t) * len);
    for (size_t i=0; i<len; i++) {
        if (!a[i].infinity) {
            az[count++] = a[i].z;
        }
    }

    secp256k1_fe_t *azi = checked_malloc(sizeof(secp256k1_fe_t) * count);
    secp256k1_fe_inv_all_var(count, azi, az);
    free(az);

    count = 0;
    for (size_t i=0; i<len; i++) {
        r[i].infinity = a[i].infinity;
        if (!a[i].infinity) {
            secp256k1_fe_t *zi = &azi[count++];
            secp256k1_fe_t zi2; secp256k1_fe_sqr(&zi2, zi);
            secp256k1_fe_t zi3; secp256k1_fe_mul(&zi3, &zi2, zi);
            secp256k1_fe_mul(&r[i].x, &a[i].x, &zi2);
            secp256k1_fe_mul(&r[i].y, &a[i].y, &zi3);
        }
    }
    free(azi);
}

static void secp256k1_gej_set_infinity(secp256k1_gej_t *r) {
    r->infinity = 1;
    secp256k1_fe_set_int(&r->x, 0);
    secp256k1_fe_set_int(&r->y, 0);
    secp256k1_fe_set_int(&r->z, 0);
}

static void secp256k1_gej_set_xy(secp256k1_gej_t *r, const secp256k1_fe_t *x, const secp256k1_fe_t *y) {
    r->infinity = 0;
    r->x = *x;
    r->y = *y;
    secp256k1_fe_set_int(&r->z, 1);
}

static void secp256k1_gej_clear(secp256k1_gej_t *r) {
    r->infinity = 0;
    secp256k1_fe_clear(&r->x);
    secp256k1_fe_clear(&r->y);
    secp256k1_fe_clear(&r->z);
}

static void secp256k1_ge_clear(secp256k1_ge_t *r) {
    r->infinity = 0;
    secp256k1_fe_clear(&r->x);
    secp256k1_fe_clear(&r->y);
}

static int secp256k1_ge_set_xo_var(secp256k1_ge_t *r, const secp256k1_fe_t *x, int odd) {
    r->x = *x;
    secp256k1_fe_t x2; secp256k1_fe_sqr(&x2, x);
    secp256k1_fe_t x3; secp256k1_fe_mul(&x3, x, &x2);
    r->infinity = 0;
    secp256k1_fe_t c; secp256k1_fe_set_int(&c, 7);
    secp256k1_fe_add(&c, &x3);
    if (!secp256k1_fe_sqrt_var(&r->y, &c))
        return 0;
    secp256k1_fe_normalize_var(&r->y);
    if (secp256k1_fe_is_odd(&r->y) != odd)
        secp256k1_fe_negate(&r->y, &r->y, 1);
    return 1;
}

static void secp256k1_gej_set_ge(secp256k1_gej_t *r, const secp256k1_ge_t *a) {
   r->infinity = a->infinity;
   r->x = a->x;
   r->y = a->y;
   secp256k1_fe_set_int(&r->z, 1);
}

static int secp256k1_gej_eq_x_var(const secp256k1_fe_t *x, const secp256k1_gej_t *a) {
    VERIFY_CHECK(!a->infinity);
    secp256k1_fe_t r; secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
    secp256k1_fe_t r2 = a->x; secp256k1_fe_normalize_weak(&r2);
    return secp256k1_fe_equal_var(&r, &r2);
}

static void secp256k1_gej_neg(secp256k1_gej_t *r, const secp256k1_gej_t *a) {
    r->infinity = a->infinity;
    r->x = a->x;
    r->y = a->y;
    r->z = a->z;
    secp256k1_fe_normalize_weak(&r->y);
    secp256k1_fe_negate(&r->y, &r->y, 1);
}

static int secp256k1_gej_is_infinity(const secp256k1_gej_t *a) {
    return a->infinity;
}

static int secp256k1_gej_is_valid_var(const secp256k1_gej_t *a) {
    if (a->infinity)
        return 0;
    /** y^2 = x^3 + 7
     *  (Y/Z^3)^2 = (X/Z^2)^3 + 7
     *  Y^2 / Z^6 = X^3 / Z^6 + 7
     *  Y^2 = X^3 + 7*Z^6
     */
    secp256k1_fe_t y2; secp256k1_fe_sqr(&y2, &a->y);
    secp256k1_fe_t x3; secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
    secp256k1_fe_t z2; secp256k1_fe_sqr(&z2, &a->z);
    secp256k1_fe_t z6; secp256k1_fe_sqr(&z6, &z2); secp256k1_fe_mul(&z6, &z6, &z2);
    secp256k1_fe_mul_int(&z6, 7);
    secp256k1_fe_add(&x3, &z6);
    secp256k1_fe_normalize_weak(&x3);
    return secp256k1_fe_equal_var(&y2, &x3);
}

static int secp256k1_ge_is_valid_var(const secp256k1_ge_t *a) {
    if (a->infinity)
        return 0;
    /* y^2 = x^3 + 7 */
    secp256k1_fe_t y2; secp256k1_fe_sqr(&y2, &a->y);
    secp256k1_fe_t x3; secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
    secp256k1_fe_t c; secp256k1_fe_set_int(&c, 7);
    secp256k1_fe_add(&x3, &c);
    secp256k1_fe_normalize_weak(&x3);
    return secp256k1_fe_equal_var(&y2, &x3);
}

static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *a) {
    // For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity,
    // Q must equal -Q, or that Q.y == -(Q.y), or Q.y is 0. For a point on y^2 = x^3 + 7 to have
    // y=0, x^3 must be -7 mod p. However, -7 has no cube root mod p.
    r->infinity = a->infinity;
    if (r->infinity) {
        return;
    }

    secp256k1_fe_t t1,t2,t3,t4;
    secp256k1_fe_mul(&r->z, &a->z, &a->y);
    secp256k1_fe_mul_int(&r->z, 2);       /* Z' = 2*Y*Z (2) */
    secp256k1_fe_sqr(&t1, &a->x);
    secp256k1_fe_mul_int(&t1, 3);         /* T1 = 3*X^2 (3) */
    secp256k1_fe_sqr(&t2, &t1);           /* T2 = 9*X^4 (1) */
    secp256k1_fe_sqr(&t3, &a->y);
    secp256k1_fe_mul_int(&t3, 2);         /* T3 = 2*Y^2 (2) */
    secp256k1_fe_sqr(&t4, &t3);
    secp256k1_fe_mul_int(&t4, 2);         /* T4 = 8*Y^4 (2) */
    secp256k1_fe_mul(&t3, &t3, &a->x);    /* T3 = 2*X*Y^2 (1) */
    r->x = t3;
    secp256k1_fe_mul_int(&r->x, 4);       /* X' = 8*X*Y^2 (4) */
    secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8*X*Y^2 (5) */
    secp256k1_fe_add(&r->x, &t2);         /* X' = 9*X^4 - 8*X*Y^2 (6) */
    secp256k1_fe_negate(&t2, &t2, 1);     /* T2 = -9*X^4 (2) */
    secp256k1_fe_mul_int(&t3, 6);         /* T3 = 12*X*Y^2 (6) */
    secp256k1_fe_add(&t3, &t2);           /* T3 = 12*X*Y^2 - 9*X^4 (8) */
    secp256k1_fe_mul(&r->y, &t1, &t3);    /* Y' = 36*X^3*Y^2 - 27*X^6 (1) */
    secp256k1_fe_negate(&t2, &t4, 2);     /* T2 = -8*Y^4 (3) */
    secp256k1_fe_add(&r->y, &t2);         /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
}

static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_gej_t *b) {
    if (a->infinity) {
        *r = *b;
        return;
    }
    if (b->infinity) {
        *r = *a;
        return;
    }
    r->infinity = 0;
    secp256k1_fe_t z22; secp256k1_fe_sqr(&z22, &b->z);
    secp256k1_fe_t z12; secp256k1_fe_sqr(&z12, &a->z);
    secp256k1_fe_t u1; secp256k1_fe_mul(&u1, &a->x, &z22);
    secp256k1_fe_t u2; secp256k1_fe_mul(&u2, &b->x, &z12);
    secp256k1_fe_t s1; secp256k1_fe_mul(&s1, &a->y, &z22); secp256k1_fe_mul(&s1, &s1, &b->z);
    secp256k1_fe_t s2; secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
    secp256k1_fe_t h; secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
    secp256k1_fe_t i; secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
    if (secp256k1_fe_normalizes_to_zero(&h)) {
        if (secp256k1_fe_normalizes_to_zero(&i)) {
            secp256k1_gej_double_var(r, a);
        } else {
            r->infinity = 1;
        }
        return;
    }
    secp256k1_fe_t i2; secp256k1_fe_sqr(&i2, &i);
    secp256k1_fe_t h2; secp256k1_fe_sqr(&h2, &h);
    secp256k1_fe_t h3; secp256k1_fe_mul(&h3, &h, &h2);
    secp256k1_fe_mul(&r->z, &a->z, &b->z); secp256k1_fe_mul(&r->z, &r->z, &h);
    secp256k1_fe_t t; secp256k1_fe_mul(&t, &u1, &h2);
    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
    secp256k1_fe_add(&r->y, &h3);
}

static void secp256k1_gej_add_ge_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b) {
    if (a->infinity) {
        r->infinity = b->infinity;
        r->x = b->x;
        r->y = b->y;
        secp256k1_fe_set_int(&r->z, 1);
        return;
    }
    if (b->infinity) {
        *r = *a;
        return;
    }
    r->infinity = 0;
    secp256k1_fe_t z12; secp256k1_fe_sqr(&z12, &a->z);
    secp256k1_fe_t u1 = a->x; secp256k1_fe_normalize_weak(&u1);
    secp256k1_fe_t u2; secp256k1_fe_mul(&u2, &b->x, &z12);
    secp256k1_fe_t s1 = a->y; secp256k1_fe_normalize_weak(&s1);
    secp256k1_fe_t s2; secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
    secp256k1_fe_t h; secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
    secp256k1_fe_t i; secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
    if (secp256k1_fe_normalizes_to_zero(&h)) {
        if (secp256k1_fe_normalizes_to_zero(&i)) {
            secp256k1_gej_double_var(r, a);
        } else {
            r->infinity = 1;
        }
        return;
    }
    secp256k1_fe_t i2; secp256k1_fe_sqr(&i2, &i);
    secp256k1_fe_t h2; secp256k1_fe_sqr(&h2, &h);
    secp256k1_fe_t h3; secp256k1_fe_mul(&h3, &h, &h2);
    r->z = a->z; secp256k1_fe_mul(&r->z, &r->z, &h);
    secp256k1_fe_t t; secp256k1_fe_mul(&t, &u1, &h2);
    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
    secp256k1_fe_add(&r->y, &h3);
}

static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b) {
    VERIFY_CHECK(!b->infinity);
    VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);

    /** In:
     *    Eric Brier and Marc Joye, Weierstrass Elliptic Curves and Side-Channel Attacks.
     *    In D. Naccache and P. Paillier, Eds., Public Key Cryptography, vol. 2274 of Lecture Notes in Computer Science, pages 335-345. Springer-Verlag, 2002.
     *  we find as solution for a unified addition/doubling formula:
     *    lambda = ((x1 + x2)^2 - x1 * x2 + a) / (y1 + y2), with a = 0 for secp256k1's curve equation.
     *    x3 = lambda^2 - (x1 + x2)
     *    2*y3 = lambda * (x1 + x2 - 2 * x3) - (y1 + y2).
     *
     *  Substituting x_i = Xi / Zi^2 and yi = Yi / Zi^3, for i=1,2,3, gives:
     *    U1 = X1*Z2^2, U2 = X2*Z1^2
     *    S1 = Y1*Z2^3, S2 = Y2*Z1^3
     *    Z = Z1*Z2
     *    T = U1+U2
     *    M = S1+S2
     *    Q = T*M^2
     *    R = T^2-U1*U2
     *    X3 = 4*(R^2-Q)
     *    Y3 = 4*(R*(3*Q-2*R^2)-M^4)
     *    Z3 = 2*M*Z
     *  (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)
     */

    secp256k1_fe_t zz; secp256k1_fe_sqr(&zz, &a->z);                /* z = Z1^2 */
    secp256k1_fe_t u1 = a->x; secp256k1_fe_normalize_weak(&u1);     /* u1 = U1 = X1*Z2^2 (1) */
    secp256k1_fe_t u2; secp256k1_fe_mul(&u2, &b->x, &zz);           /* u2 = U2 = X2*Z1^2 (1) */
    secp256k1_fe_t s1 = a->y; secp256k1_fe_normalize_weak(&s1);     /* s1 = S1 = Y1*Z2^3 (1) */
    secp256k1_fe_t s2; secp256k1_fe_mul(&s2, &b->y, &zz);           /* s2 = Y2*Z2^2 (1) */
    secp256k1_fe_mul(&s2, &s2, &a->z);                              /* s2 = S2 = Y2*Z1^3 (1) */
    secp256k1_fe_t z = a->z;                                        /* z = Z = Z1*Z2 (8) */
    secp256k1_fe_t t = u1; secp256k1_fe_add(&t, &u2);               /* t = T = U1+U2 (2) */
    secp256k1_fe_t m = s1; secp256k1_fe_add(&m, &s2);               /* m = M = S1+S2 (2) */
    secp256k1_fe_t n; secp256k1_fe_sqr(&n, &m);                     /* n = M^2 (1) */
    secp256k1_fe_t q; secp256k1_fe_mul(&q, &n, &t);                 /* q = Q = T*M^2 (1) */
    secp256k1_fe_sqr(&n, &n);                                       /* n = M^4 (1) */
    secp256k1_fe_t rr; secp256k1_fe_sqr(&rr, &t);                   /* rr = T^2 (1) */
    secp256k1_fe_mul(&t, &u1, &u2); secp256k1_fe_negate(&t, &t, 1); /* t = -U1*U2 (2) */
    secp256k1_fe_add(&rr, &t);                                      /* rr = R = T^2-U1*U2 (3) */
    secp256k1_fe_sqr(&t, &rr);                                      /* t = R^2 (1) */
    secp256k1_fe_mul(&r->z, &m, &z);                                /* r->z = M*Z (1) */
    int infinity = secp256k1_fe_normalizes_to_zero(&r->z) * (1 - a->infinity);
    secp256k1_fe_mul_int(&r->z, 2 * (1 - a->infinity)); /* r->z = Z3 = 2*M*Z (2) */
    r->x = t;                                           /* r->x = R^2 (1) */
    secp256k1_fe_negate(&q, &q, 1);                     /* q = -Q (2) */
    secp256k1_fe_add(&r->x, &q);                        /* r->x = R^2-Q (3) */
    secp256k1_fe_normalize(&r->x);
    secp256k1_fe_mul_int(&q, 3);                        /* q = -3*Q (6) */
    secp256k1_fe_mul_int(&t, 2);                        /* t = 2*R^2 (2) */
    secp256k1_fe_add(&t, &q);                           /* t = 2*R^2-3*Q (8) */
    secp256k1_fe_mul(&t, &t, &rr);                      /* t = R*(2*R^2-3*Q) (1) */
    secp256k1_fe_add(&t, &n);                           /* t = R*(2*R^2-3*Q)+M^4 (2) */
    secp256k1_fe_negate(&r->y, &t, 2);                  /* r->y = R*(3*Q-2*R^2)-M^4 (3) */
    secp256k1_fe_normalize_weak(&r->y);
    secp256k1_fe_mul_int(&r->x, 4 * (1 - a->infinity)); /* r->x = X3 = 4*(R^2-Q) */
    secp256k1_fe_mul_int(&r->y, 4 * (1 - a->infinity)); /* r->y = Y3 = 4*R*(3*Q-2*R^2)-4*M^4 (4) */

    /** In case a->infinity == 1, the above code results in r->x, r->y, and r->z all equal to 0.
     *  Add b->x to x, b->y to y, and 1 to z in that case.
     */
    t = b->x; secp256k1_fe_mul_int(&t, a->infinity);
    secp256k1_fe_add(&r->x, &t);
    t = b->y; secp256k1_fe_mul_int(&t, a->infinity);
    secp256k1_fe_add(&r->y, &t);
    secp256k1_fe_set_int(&t, a->infinity);
    secp256k1_fe_add(&r->z, &t);
    r->infinity = infinity;
}


static void secp256k1_gej_get_hex(char *r, int *rlen, const secp256k1_gej_t *a) {
    secp256k1_gej_t c = *a;
    secp256k1_ge_t t; secp256k1_ge_set_gej(&t, &c);
    secp256k1_ge_get_hex(r, rlen, &t);
}

#ifdef USE_ENDOMORPHISM
static void secp256k1_gej_mul_lambda(secp256k1_gej_t *r, const secp256k1_gej_t *a) {
    const secp256k1_fe_t *beta = &secp256k1_ge_consts->beta;
    *r = *a;
    secp256k1_fe_mul(&r->x, &r->x, beta);
}
#endif

static void secp256k1_ge_start(void) {
    static const unsigned char secp256k1_ge_consts_g_x[] = {
        0x79,0xBE,0x66,0x7E,0xF9,0xDC,0xBB,0xAC,
        0x55,0xA0,0x62,0x95,0xCE,0x87,0x0B,0x07,
        0x02,0x9B,0xFC,0xDB,0x2D,0xCE,0x28,0xD9,
        0x59,0xF2,0x81,0x5B,0x16,0xF8,0x17,0x98
    };
    static const unsigned char secp256k1_ge_consts_g_y[] = {
        0x48,0x3A,0xDA,0x77,0x26,0xA3,0xC4,0x65,
        0x5D,0xA4,0xFB,0xFC,0x0E,0x11,0x08,0xA8,
        0xFD,0x17,0xB4,0x48,0xA6,0x85,0x54,0x19,
        0x9C,0x47,0xD0,0x8F,0xFB,0x10,0xD4,0xB8
    };
#ifdef USE_ENDOMORPHISM
    /* properties of secp256k1's efficiently computable endomorphism */
    static const unsigned char secp256k1_ge_consts_beta[] = {
        0x7a,0xe9,0x6a,0x2b,0x65,0x7c,0x07,0x10,
        0x6e,0x64,0x47,0x9e,0xac,0x34,0x34,0xe9,
        0x9c,0xf0,0x49,0x75,0x12,0xf5,0x89,0x95,
        0xc1,0x39,0x6c,0x28,0x71,0x95,0x01,0xee
    };
#endif
    if (secp256k1_ge_consts == NULL) {
        secp256k1_ge_consts_t *ret = (secp256k1_ge_consts_t*)checked_malloc(sizeof(secp256k1_ge_consts_t));
#ifdef USE_ENDOMORPHISM
        VERIFY_CHECK(secp256k1_fe_set_b32(&ret->beta, secp256k1_ge_consts_beta));
#endif
        secp256k1_fe_t g_x, g_y;
        VERIFY_CHECK(secp256k1_fe_set_b32(&g_x, secp256k1_ge_consts_g_x));
        VERIFY_CHECK(secp256k1_fe_set_b32(&g_y, secp256k1_ge_consts_g_y));
        secp256k1_ge_set_xy(&ret->g, &g_x, &g_y);
        secp256k1_ge_consts = ret;
    }
}

static void secp256k1_ge_stop(void) {
    if (secp256k1_ge_consts != NULL) {
        secp256k1_ge_consts_t *c = (secp256k1_ge_consts_t*)secp256k1_ge_consts;
        free((void*)c);
        secp256k1_ge_consts = NULL;
    }
}

#endif
Commit	Line	Data
	1	/**********************************************************************
	2	* Copyright (c) 2013, 2014 Pieter Wuille *
	3	* Distributed under the MIT software license, see the accompanying *
	4	* file COPYING or http://www.opensource.org/licenses/mit-license.php.*
	5	**********************************************************************/
	6
	7	#ifndef _SECP256K1_GROUP_IMPL_H_
	8	#define _SECP256K1_GROUP_IMPL_H_
	9
	10	#include <string.h>
	11
	12	#include "num.h"
	13	#include "field.h"
	14	#include "group.h"
	15
	16	static void secp256k1_ge_set_infinity(secp256k1_ge_t *r) {
	17	r->infinity = 1;
	18	}
	19
	20	static void secp256k1_ge_set_xy(secp256k1_ge_t r, const secp256k1_fe_t x, const secp256k1_fe_t *y) {
	21	r->infinity = 0;
	22	r->x = *x;
	23	r->y = *y;
	24	}
	25
	26	static int secp256k1_ge_is_infinity(const secp256k1_ge_t *a) {
	27	return a->infinity;
	28	}
	29
	30	static void secp256k1_ge_neg(secp256k1_ge_t r, const secp256k1_ge_t a) {
	31	r = a;
	32	secp256k1_fe_normalize_weak(&r->y);
	33	secp256k1_fe_negate(&r->y, &r->y, 1);
	34	}
	35
	36	static void secp256k1_ge_get_hex(char r, int rlen, const secp256k1_ge_t *a) {
	37	char cx[65]; int lx=65;
	38	char cy[65]; int ly=65;
	39	secp256k1_fe_get_hex(cx, &lx, &a->x);
	40	secp256k1_fe_get_hex(cy, &ly, &a->y);
	41	lx = strlen(cx);
	42	ly = strlen(cy);
	43	int len = lx + ly + 3 + 1;
	44	if (*rlen < len) {
	45	*rlen = len;
	46	return;
	47	}
	48	*rlen = len;
	49	r[0] = '(';
	50	memcpy(r+1, cx, lx);
	51	r[1+lx] = ',';
	52	memcpy(r+2+lx, cy, ly);
	53	r[2+lx+ly] = ')';
	54	r[3+lx+ly] = 0;
	55	}
	56
	57	static void secp256k1_ge_set_gej(secp256k1_ge_t r, secp256k1_gej_t a) {
	58	r->infinity = a->infinity;
	59	secp256k1_fe_inv(&a->z, &a->z);
	60	secp256k1_fe_t z2; secp256k1_fe_sqr(&z2, &a->z);
	61	secp256k1_fe_t z3; secp256k1_fe_mul(&z3, &a->z, &z2);
	62	secp256k1_fe_mul(&a->x, &a->x, &z2);
	63	secp256k1_fe_mul(&a->y, &a->y, &z3);
	64	secp256k1_fe_set_int(&a->z, 1);
	65	r->x = a->x;
	66	r->y = a->y;
	67	}
	68
	69	static void secp256k1_ge_set_gej_var(secp256k1_ge_t r, secp256k1_gej_t a) {
	70	r->infinity = a->infinity;
	71	if (a->infinity) {
	72	return;
	73	}
	74	secp256k1_fe_inv_var(&a->z, &a->z);
	75	secp256k1_fe_t z2; secp256k1_fe_sqr(&z2, &a->z);
	76	secp256k1_fe_t z3; secp256k1_fe_mul(&z3, &a->z, &z2);
	77	secp256k1_fe_mul(&a->x, &a->x, &z2);
	78	secp256k1_fe_mul(&a->y, &a->y, &z3);
	79	secp256k1_fe_set_int(&a->z, 1);
	80	r->x = a->x;
	81	r->y = a->y;
	82	}
	83
	84	static void secp256k1_ge_set_all_gej_var(size_t len, secp256k1_ge_t r[len], const secp256k1_gej_t a[len]) {
	85	size_t count = 0;
	86	secp256k1_fe_t az = checked_malloc(sizeof(secp256k1_fe_t) len);
	87	for (size_t i=0; i<len; i++) {
	88	if (!a[i].infinity) {
	89	az[count++] = a[i].z;
	90	}
	91	}
	92
	93	secp256k1_fe_t azi = checked_malloc(sizeof(secp256k1_fe_t) count);
	94	secp256k1_fe_inv_all_var(count, azi, az);
	95	free(az);
	96
	97	count = 0;
	98	for (size_t i=0; i<len; i++) {
	99	r[i].infinity = a[i].infinity;
	100	if (!a[i].infinity) {
	101	secp256k1_fe_t *zi = &azi[count++];
	102	secp256k1_fe_t zi2; secp256k1_fe_sqr(&zi2, zi);
	103	secp256k1_fe_t zi3; secp256k1_fe_mul(&zi3, &zi2, zi);
	104	secp256k1_fe_mul(&r[i].x, &a[i].x, &zi2);
	105	secp256k1_fe_mul(&r[i].y, &a[i].y, &zi3);
	106	}
	107	}
	108	free(azi);
	109	}
	110
	111	static void secp256k1_gej_set_infinity(secp256k1_gej_t *r) {
	112	r->infinity = 1;
	113	secp256k1_fe_set_int(&r->x, 0);
	114	secp256k1_fe_set_int(&r->y, 0);
	115	secp256k1_fe_set_int(&r->z, 0);
	116	}
	117
	118	static void secp256k1_gej_set_xy(secp256k1_gej_t r, const secp256k1_fe_t x, const secp256k1_fe_t *y) {
	119	r->infinity = 0;
	120	r->x = *x;
	121	r->y = *y;
	122	secp256k1_fe_set_int(&r->z, 1);
	123	}
	124
	125	static void secp256k1_gej_clear(secp256k1_gej_t *r) {
	126	r->infinity = 0;
	127	secp256k1_fe_clear(&r->x);
	128	secp256k1_fe_clear(&r->y);
	129	secp256k1_fe_clear(&r->z);
	130	}
	131
	132	static void secp256k1_ge_clear(secp256k1_ge_t *r) {
	133	r->infinity = 0;
	134	secp256k1_fe_clear(&r->x);
	135	secp256k1_fe_clear(&r->y);
	136	}
	137
	138	static int secp256k1_ge_set_xo_var(secp256k1_ge_t r, const secp256k1_fe_t x, int odd) {
	139	r->x = *x;
	140	secp256k1_fe_t x2; secp256k1_fe_sqr(&x2, x);
	141	secp256k1_fe_t x3; secp256k1_fe_mul(&x3, x, &x2);
	142	r->infinity = 0;
	143	secp256k1_fe_t c; secp256k1_fe_set_int(&c, 7);
	144	secp256k1_fe_add(&c, &x3);
	145	if (!secp256k1_fe_sqrt_var(&r->y, &c))
	146	return 0;
	147	secp256k1_fe_normalize_var(&r->y);
	148	if (secp256k1_fe_is_odd(&r->y) != odd)
	149	secp256k1_fe_negate(&r->y, &r->y, 1);
	150	return 1;
	151	}
	152
	153	static void secp256k1_gej_set_ge(secp256k1_gej_t r, const secp256k1_ge_t a) {
	154	r->infinity = a->infinity;
	155	r->x = a->x;
	156	r->y = a->y;
	157	secp256k1_fe_set_int(&r->z, 1);
	158	}
	159
	160	static int secp256k1_gej_eq_x_var(const secp256k1_fe_t x, const secp256k1_gej_t a) {
	161	VERIFY_CHECK(!a->infinity);
	162	secp256k1_fe_t r; secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
	163	secp256k1_fe_t r2 = a->x; secp256k1_fe_normalize_weak(&r2);
	164	return secp256k1_fe_equal_var(&r, &r2);
	165	}
	166
	167	static void secp256k1_gej_neg(secp256k1_gej_t r, const secp256k1_gej_t a) {
	168	r->infinity = a->infinity;
	169	r->x = a->x;
	170	r->y = a->y;
	171	r->z = a->z;
	172	secp256k1_fe_normalize_weak(&r->y);
	173	secp256k1_fe_negate(&r->y, &r->y, 1);
	174	}
	175
	176	static int secp256k1_gej_is_infinity(const secp256k1_gej_t *a) {
	177	return a->infinity;
	178	}
	179
	180	static int secp256k1_gej_is_valid_var(const secp256k1_gej_t *a) {
	181	if (a->infinity)
	182	return 0;
	183	/** y^2 = x^3 + 7
	184	* (Y/Z^3)^2 = (X/Z^2)^3 + 7
	185	* Y^2 / Z^6 = X^3 / Z^6 + 7
	186	* Y^2 = X^3 + 7*Z^6
	187	*/
	188	secp256k1_fe_t y2; secp256k1_fe_sqr(&y2, &a->y);
	189	secp256k1_fe_t x3; secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
	190	secp256k1_fe_t z2; secp256k1_fe_sqr(&z2, &a->z);
	191	secp256k1_fe_t z6; secp256k1_fe_sqr(&z6, &z2); secp256k1_fe_mul(&z6, &z6, &z2);
	192	secp256k1_fe_mul_int(&z6, 7);
	193	secp256k1_fe_add(&x3, &z6);
	194	secp256k1_fe_normalize_weak(&x3);
	195	return secp256k1_fe_equal_var(&y2, &x3);
	196	}
	197
	198	static int secp256k1_ge_is_valid_var(const secp256k1_ge_t *a) {
	199	if (a->infinity)
	200	return 0;
	201	/* y^2 = x^3 + 7 */
	202	secp256k1_fe_t y2; secp256k1_fe_sqr(&y2, &a->y);
	203	secp256k1_fe_t x3; secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
	204	secp256k1_fe_t c; secp256k1_fe_set_int(&c, 7);
	205	secp256k1_fe_add(&x3, &c);
	206	secp256k1_fe_normalize_weak(&x3);
	207	return secp256k1_fe_equal_var(&y2, &x3);
	208	}
	209
	210	static void secp256k1_gej_double_var(secp256k1_gej_t r, const secp256k1_gej_t a) {
	211	// For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity,
	212	// Q must equal -Q, or that Q.y == -(Q.y), or Q.y is 0. For a point on y^2 = x^3 + 7 to have
	213	// y=0, x^3 must be -7 mod p. However, -7 has no cube root mod p.
	214	r->infinity = a->infinity;
	215	if (r->infinity) {
	216	return;
	217	}
	218
	219	secp256k1_fe_t t1,t2,t3,t4;
	220	secp256k1_fe_mul(&r->z, &a->z, &a->y);
	221	secp256k1_fe_mul_int(&r->z, 2); /* Z' = 2YZ (2) */
	222	secp256k1_fe_sqr(&t1, &a->x);
	223	secp256k1_fe_mul_int(&t1, 3); /* T1 = 3X^2 (3) /
	224	secp256k1_fe_sqr(&t2, &t1); /* T2 = 9X^4 (1) /
	225	secp256k1_fe_sqr(&t3, &a->y);
	226	secp256k1_fe_mul_int(&t3, 2); /* T3 = 2Y^2 (2) /
	227	secp256k1_fe_sqr(&t4, &t3);
	228	secp256k1_fe_mul_int(&t4, 2); /* T4 = 8Y^4 (2) /
	229	secp256k1_fe_mul(&t3, &t3, &a->x); /* T3 = 2XY^2 (1) */
	230	r->x = t3;
	231	secp256k1_fe_mul_int(&r->x, 4); /* X' = 8XY^2 (4) */
	232	secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8XY^2 (5) */
	233	secp256k1_fe_add(&r->x, &t2); /* X' = 9X^4 - 8XY^2 (6) /
	234	secp256k1_fe_negate(&t2, &t2, 1); /* T2 = -9X^4 (2) /
	235	secp256k1_fe_mul_int(&t3, 6); /* T3 = 12XY^2 (6) */
	236	secp256k1_fe_add(&t3, &t2); /* T3 = 12XY^2 - 9X^4 (8) /
	237	secp256k1_fe_mul(&r->y, &t1, &t3); /* Y' = 36X^3Y^2 - 27X^6 (1) /
	238	secp256k1_fe_negate(&t2, &t4, 2); /* T2 = -8Y^4 (3) /
	239	secp256k1_fe_add(&r->y, &t2); /* Y' = 36X^3Y^2 - 27X^6 - 8Y^4 (4) */
	240	}
	241
	242	static void secp256k1_gej_add_var(secp256k1_gej_t r, const secp256k1_gej_t a, const secp256k1_gej_t *b) {
	243	if (a->infinity) {
	244	r = b;
	245	return;
	246	}
	247	if (b->infinity) {
	248	r = a;
	249	return;
	250	}
	251	r->infinity = 0;
	252	secp256k1_fe_t z22; secp256k1_fe_sqr(&z22, &b->z);
	253	secp256k1_fe_t z12; secp256k1_fe_sqr(&z12, &a->z);
	254	secp256k1_fe_t u1; secp256k1_fe_mul(&u1, &a->x, &z22);
	255	secp256k1_fe_t u2; secp256k1_fe_mul(&u2, &b->x, &z12);
	256	secp256k1_fe_t s1; secp256k1_fe_mul(&s1, &a->y, &z22); secp256k1_fe_mul(&s1, &s1, &b->z);
	257	secp256k1_fe_t s2; secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
	258	secp256k1_fe_t h; secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
	259	secp256k1_fe_t i; secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
	260	if (secp256k1_fe_normalizes_to_zero(&h)) {
	261	if (secp256k1_fe_normalizes_to_zero(&i)) {
	262	secp256k1_gej_double_var(r, a);
	263	} else {
	264	r->infinity = 1;
	265	}
	266	return;
	267	}
	268	secp256k1_fe_t i2; secp256k1_fe_sqr(&i2, &i);
	269	secp256k1_fe_t h2; secp256k1_fe_sqr(&h2, &h);
	270	secp256k1_fe_t h3; secp256k1_fe_mul(&h3, &h, &h2);
	271	secp256k1_fe_mul(&r->z, &a->z, &b->z); secp256k1_fe_mul(&r->z, &r->z, &h);
	272	secp256k1_fe_t t; secp256k1_fe_mul(&t, &u1, &h2);
	273	r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
	274	secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
	275	secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
	276	secp256k1_fe_add(&r->y, &h3);
	277	}
	278
	279	static void secp256k1_gej_add_ge_var(secp256k1_gej_t r, const secp256k1_gej_t a, const secp256k1_ge_t *b) {
	280	if (a->infinity) {
	281	r->infinity = b->infinity;
	282	r->x = b->x;
	283	r->y = b->y;
	284	secp256k1_fe_set_int(&r->z, 1);
	285	return;
	286	}
	287	if (b->infinity) {
	288	r = a;
	289	return;
	290	}
	291	r->infinity = 0;
	292	secp256k1_fe_t z12; secp256k1_fe_sqr(&z12, &a->z);
	293	secp256k1_fe_t u1 = a->x; secp256k1_fe_normalize_weak(&u1);
	294	secp256k1_fe_t u2; secp256k1_fe_mul(&u2, &b->x, &z12);
	295	secp256k1_fe_t s1 = a->y; secp256k1_fe_normalize_weak(&s1);
	296	secp256k1_fe_t s2; secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
	297	secp256k1_fe_t h; secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
	298	secp256k1_fe_t i; secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
	299	if (secp256k1_fe_normalizes_to_zero(&h)) {
	300	if (secp256k1_fe_normalizes_to_zero(&i)) {
	301	secp256k1_gej_double_var(r, a);
	302	} else {
	303	r->infinity = 1;
	304	}
	305	return;
	306	}
	307	secp256k1_fe_t i2; secp256k1_fe_sqr(&i2, &i);
	308	secp256k1_fe_t h2; secp256k1_fe_sqr(&h2, &h);
	309	secp256k1_fe_t h3; secp256k1_fe_mul(&h3, &h, &h2);
	310	r->z = a->z; secp256k1_fe_mul(&r->z, &r->z, &h);
	311	secp256k1_fe_t t; secp256k1_fe_mul(&t, &u1, &h2);
	312	r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
	313	secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
	314	secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
	315	secp256k1_fe_add(&r->y, &h3);
	316	}
	317
	318	static void secp256k1_gej_add_ge(secp256k1_gej_t r, const secp256k1_gej_t a, const secp256k1_ge_t *b) {
	319	VERIFY_CHECK(!b->infinity);
	320	VERIFY_CHECK(a->infinity == 0 \|\| a->infinity == 1);
	321
	322	/** In:
	323	* Eric Brier and Marc Joye, Weierstrass Elliptic Curves and Side-Channel Attacks.
	324	* In D. Naccache and P. Paillier, Eds., Public Key Cryptography, vol. 2274 of Lecture Notes in Computer Science, pages 335-345. Springer-Verlag, 2002.
	325	* we find as solution for a unified addition/doubling formula:
	326	* lambda = ((x1 + x2)^2 - x1 * x2 + a) / (y1 + y2), with a = 0 for secp256k1's curve equation.
	327	* x3 = lambda^2 - (x1 + x2)
	328	* 2y3 = lambda (x1 + x2 - 2 * x3) - (y1 + y2).
	329	*
	330	* Substituting x_i = Xi / Zi^2 and yi = Yi / Zi^3, for i=1,2,3, gives:
	331	* U1 = X1Z2^2, U2 = X2Z1^2
	332	* S1 = Y1Z2^3, S2 = Y2Z1^3
	333	* Z = Z1*Z2
	334	* T = U1+U2
	335	* M = S1+S2
	336	* Q = T*M^2
	337	* R = T^2-U1*U2
	338	* X3 = 4*(R^2-Q)
	339	* Y3 = 4(R(3Q-2R^2)-M^4)
	340	* Z3 = 2MZ
	341	* (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)
	342	*/
	343
	344	secp256k1_fe_t zz; secp256k1_fe_sqr(&zz, &a->z); /* z = Z1^2 */
	345	secp256k1_fe_t u1 = a->x; secp256k1_fe_normalize_weak(&u1); /* u1 = U1 = X1Z2^2 (1) /
	346	secp256k1_fe_t u2; secp256k1_fe_mul(&u2, &b->x, &zz); /* u2 = U2 = X2Z1^2 (1) /
	347	secp256k1_fe_t s1 = a->y; secp256k1_fe_normalize_weak(&s1); /* s1 = S1 = Y1Z2^3 (1) /
	348	secp256k1_fe_t s2; secp256k1_fe_mul(&s2, &b->y, &zz); /* s2 = Y2Z2^2 (1) /
	349	secp256k1_fe_mul(&s2, &s2, &a->z); /* s2 = S2 = Y2Z1^3 (1) /
	350	secp256k1_fe_t z = a->z; /* z = Z = Z1Z2 (8) /
	351	secp256k1_fe_t t = u1; secp256k1_fe_add(&t, &u2); /* t = T = U1+U2 (2) */
	352	secp256k1_fe_t m = s1; secp256k1_fe_add(&m, &s2); /* m = M = S1+S2 (2) */
	353	secp256k1_fe_t n; secp256k1_fe_sqr(&n, &m); /* n = M^2 (1) */
	354	secp256k1_fe_t q; secp256k1_fe_mul(&q, &n, &t); /* q = Q = TM^2 (1) /
	355	secp256k1_fe_sqr(&n, &n); /* n = M^4 (1) */
	356	secp256k1_fe_t rr; secp256k1_fe_sqr(&rr, &t); /* rr = T^2 (1) */
	357	secp256k1_fe_mul(&t, &u1, &u2); secp256k1_fe_negate(&t, &t, 1); /* t = -U1U2 (2) /
	358	secp256k1_fe_add(&rr, &t); /* rr = R = T^2-U1U2 (3) /
	359	secp256k1_fe_sqr(&t, &rr); /* t = R^2 (1) */
	360	secp256k1_fe_mul(&r->z, &m, &z); /* r->z = MZ (1) /
	361	int infinity = secp256k1_fe_normalizes_to_zero(&r->z) * (1 - a->infinity);
	362	secp256k1_fe_mul_int(&r->z, 2 * (1 - a->infinity)); /* r->z = Z3 = 2MZ (2) */
	363	r->x = t; /* r->x = R^2 (1) */
	364	secp256k1_fe_negate(&q, &q, 1); /* q = -Q (2) */
	365	secp256k1_fe_add(&r->x, &q); /* r->x = R^2-Q (3) */
	366	secp256k1_fe_normalize(&r->x);
	367	secp256k1_fe_mul_int(&q, 3); /* q = -3Q (6) /
	368	secp256k1_fe_mul_int(&t, 2); /* t = 2R^2 (2) /
	369	secp256k1_fe_add(&t, &q); /* t = 2R^2-3Q (8) */
	370	secp256k1_fe_mul(&t, &t, &rr); /* t = R(2R^2-3Q) (1) /
	371	secp256k1_fe_add(&t, &n); /* t = R(2R^2-3Q)+M^4 (2) /
	372	secp256k1_fe_negate(&r->y, &t, 2); /* r->y = R(3Q-2R^2)-M^4 (3) /
	373	secp256k1_fe_normalize_weak(&r->y);
	374	secp256k1_fe_mul_int(&r->x, 4 * (1 - a->infinity)); /* r->x = X3 = 4(R^2-Q) /
	375	secp256k1_fe_mul_int(&r->y, 4 * (1 - a->infinity)); /* r->y = Y3 = 4R(3Q-2R^2)-4M^4 (4) /
	376
	377	/** In case a->infinity == 1, the above code results in r->x, r->y, and r->z all equal to 0.
	378	* Add b->x to x, b->y to y, and 1 to z in that case.
	379	*/
	380	t = b->x; secp256k1_fe_mul_int(&t, a->infinity);
	381	secp256k1_fe_add(&r->x, &t);
	382	t = b->y; secp256k1_fe_mul_int(&t, a->infinity);
	383	secp256k1_fe_add(&r->y, &t);
	384	secp256k1_fe_set_int(&t, a->infinity);
	385	secp256k1_fe_add(&r->z, &t);
	386	r->infinity = infinity;
	387	}
	388
	389
	390
	391	static void secp256k1_gej_get_hex(char r, int rlen, const secp256k1_gej_t *a) {
	392	secp256k1_gej_t c = *a;
	393	secp256k1_ge_t t; secp256k1_ge_set_gej(&t, &c);
	394	secp256k1_ge_get_hex(r, rlen, &t);
	395	}
	396
	397	#ifdef USE_ENDOMORPHISM
	398	static void secp256k1_gej_mul_lambda(secp256k1_gej_t r, const secp256k1_gej_t a) {
	399	const secp256k1_fe_t *beta = &secp256k1_ge_consts->beta;
	400	r = a;
	401	secp256k1_fe_mul(&r->x, &r->x, beta);
	402	}
	403	#endif
	404
	405	static void secp256k1_ge_start(void) {
	406	static const unsigned char secp256k1_ge_consts_g_x[] = {
	407	0x79,0xBE,0x66,0x7E,0xF9,0xDC,0xBB,0xAC,
	408	0x55,0xA0,0x62,0x95,0xCE,0x87,0x0B,0x07,
	409	0x02,0x9B,0xFC,0xDB,0x2D,0xCE,0x28,0xD9,
	410	0x59,0xF2,0x81,0x5B,0x16,0xF8,0x17,0x98
	411	};
	412	static const unsigned char secp256k1_ge_consts_g_y[] = {
	413	0x48,0x3A,0xDA,0x77,0x26,0xA3,0xC4,0x65,
	414	0x5D,0xA4,0xFB,0xFC,0x0E,0x11,0x08,0xA8,
	415	0xFD,0x17,0xB4,0x48,0xA6,0x85,0x54,0x19,
	416	0x9C,0x47,0xD0,0x8F,0xFB,0x10,0xD4,0xB8
	417	};
	418	#ifdef USE_ENDOMORPHISM
	419	/* properties of secp256k1's efficiently computable endomorphism */
	420	static const unsigned char secp256k1_ge_consts_beta[] = {
	421	0x7a,0xe9,0x6a,0x2b,0x65,0x7c,0x07,0x10,
	422	0x6e,0x64,0x47,0x9e,0xac,0x34,0x34,0xe9,
	423	0x9c,0xf0,0x49,0x75,0x12,0xf5,0x89,0x95,
	424	0xc1,0x39,0x6c,0x28,0x71,0x95,0x01,0xee
	425	};
	426	#endif
	427	if (secp256k1_ge_consts == NULL) {
	428	secp256k1_ge_consts_t ret = (secp256k1_ge_consts_t)checked_malloc(sizeof(secp256k1_ge_consts_t));
	429	#ifdef USE_ENDOMORPHISM
	430	VERIFY_CHECK(secp256k1_fe_set_b32(&ret->beta, secp256k1_ge_consts_beta));
	431	#endif
	432	secp256k1_fe_t g_x, g_y;
	433	VERIFY_CHECK(secp256k1_fe_set_b32(&g_x, secp256k1_ge_consts_g_x));
	434	VERIFY_CHECK(secp256k1_fe_set_b32(&g_y, secp256k1_ge_consts_g_y));
	435	secp256k1_ge_set_xy(&ret->g, &g_x, &g_y);
	436	secp256k1_ge_consts = ret;
	437	}
	438	}
	439
	440	static void secp256k1_ge_stop(void) {
	441	if (secp256k1_ge_consts != NULL) {
	442	secp256k1_ge_consts_t c = (secp256k1_ge_consts_t)secp256k1_ge_consts;
	443	free((void*)c);
	444	secp256k1_ge_consts = NULL;
	445	}
	446	}
	447
	448	#endif