/********************************************************************** * 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 #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; iinfinity = 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_var(&h)) { if (secp256k1_fe_normalizes_to_zero_var(&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_var(&h)) { if (secp256k1_fe_normalizes_to_zero_var(&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