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bitcoin/src/secp256k1/src/group_impl.h

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/***********************************************************************
* Copyright (c) 2013, 2014 Pieter Wuille *
* Distributed under the MIT software license, see the accompanying *
* file COPYING or https://www.opensource.org/licenses/mit-license.php.*
***********************************************************************/
#ifndef SECP256K1_GROUP_IMPL_H
#define SECP256K1_GROUP_IMPL_H
#include "field.h"
#include "group.h"
#define SECP256K1_G_ORDER_13 SECP256K1_GE_CONST(\
0xc3459c3d, 0x35326167, 0xcd86cce8, 0x07a2417f,\
0x5b8bd567, 0xde8538ee, 0x0d507b0c, 0xd128f5bb,\
0x8e467fec, 0xcd30000a, 0x6cc1184e, 0x25d382c2,\
0xa2f4494e, 0x2fbe9abc, 0x8b64abac, 0xd005fb24\
)
#define SECP256K1_G_ORDER_199 SECP256K1_GE_CONST(\
0x226e653f, 0xc8df7744, 0x9bacbf12, 0x7d1dcbf9,\
0x87f05b2a, 0xe7edbd28, 0x1f564575, 0xc48dcf18,\
0xa13872c2, 0xe933bb17, 0x5d9ffd5b, 0xb5b6e10c,\
0x57fe3c00, 0xbaaaa15a, 0xe003ec3e, 0x9c269bae\
)
/** Generator for secp256k1, value 'g' defined in
* "Standards for Efficient Cryptography" (SEC2) 2.7.1.
*/
#define SECP256K1_G SECP256K1_GE_CONST(\
0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,\
0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,\
0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,\
0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL\
)
/* These exhaustive group test orders and generators are chosen such that:
* - The field size is equal to that of secp256k1, so field code is the same.
* - The curve equation is of the form y^2=x^3+B for some constant B.
* - The subgroup has a generator 2*P, where P.x=1.
* - The subgroup has size less than 1000 to permit exhaustive testing.
* - The subgroup admits an endomorphism of the form lambda*(x,y) == (beta*x,y).
*
* These parameters are generated using sage/gen_exhaustive_groups.sage.
*/
#if defined(EXHAUSTIVE_TEST_ORDER)
# if EXHAUSTIVE_TEST_ORDER == 13
static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G_ORDER_13;
static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(
0x3d3486b2, 0x159a9ca5, 0xc75638be, 0xb23a69bc,
0x946a45ab, 0x24801247, 0xb4ed2b8e, 0x26b6a417
);
# elif EXHAUSTIVE_TEST_ORDER == 199
static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G_ORDER_199;
static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(
0x2cca28fa, 0xfc614b80, 0x2a3db42b, 0x00ba00b1,
0xbea8d943, 0xdace9ab2, 0x9536daea, 0x0074defb
);
# else
# error No known generator for the specified exhaustive test group order.
# endif
#else
static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G;
static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 7);
#endif
static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi) {
secp256k1_fe zi2;
secp256k1_fe zi3;
VERIFY_CHECK(!a->infinity);
secp256k1_fe_sqr(&zi2, zi);
secp256k1_fe_mul(&zi3, &zi2, zi);
secp256k1_fe_mul(&r->x, &a->x, &zi2);
secp256k1_fe_mul(&r->y, &a->y, &zi3);
r->infinity = a->infinity;
}
static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y) {
r->infinity = 0;
r->x = *x;
r->y = *y;
}
static int secp256k1_ge_is_infinity(const secp256k1_ge *a) {
return a->infinity;
}
static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a) {
*r = *a;
secp256k1_fe_normalize_weak(&r->y);
secp256k1_fe_negate(&r->y, &r->y, 1);
}
static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a) {
secp256k1_fe z2, z3;
r->infinity = a->infinity;
secp256k1_fe_inv(&a->z, &a->z);
secp256k1_fe_sqr(&z2, &a->z);
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 *r, secp256k1_gej *a) {
secp256k1_fe z2, z3;
if (a->infinity) {
secp256k1_ge_set_infinity(r);
return;
}
secp256k1_fe_inv_var(&a->z, &a->z);
secp256k1_fe_sqr(&z2, &a->z);
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);
secp256k1_ge_set_xy(r, &a->x, &a->y);
}
static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len) {
secp256k1_fe u;
size_t i;
size_t last_i = SIZE_MAX;
for (i = 0; i < len; i++) {
if (a[i].infinity) {
secp256k1_ge_set_infinity(&r[i]);
} else {
/* Use destination's x coordinates as scratch space */
if (last_i == SIZE_MAX) {
r[i].x = a[i].z;
} else {
secp256k1_fe_mul(&r[i].x, &r[last_i].x, &a[i].z);
}
last_i = i;
}
}
if (last_i == SIZE_MAX) {
return;
}
secp256k1_fe_inv_var(&u, &r[last_i].x);
i = last_i;
while (i > 0) {
i--;
if (!a[i].infinity) {
secp256k1_fe_mul(&r[last_i].x, &r[i].x, &u);
secp256k1_fe_mul(&u, &u, &a[last_i].z);
last_i = i;
}
}
VERIFY_CHECK(!a[last_i].infinity);
r[last_i].x = u;
for (i = 0; i < len; i++) {
if (!a[i].infinity) {
secp256k1_ge_set_gej_zinv(&r[i], &a[i], &r[i].x);
}
}
}
static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const secp256k1_fe *zr) {
size_t i = len - 1;
secp256k1_fe zs;
if (len > 0) {
/* Ensure all y values are in weak normal form for fast negation of points */
secp256k1_fe_normalize_weak(&a[i].y);
zs = zr[i];
/* Work our way backwards, using the z-ratios to scale the x/y values. */
while (i > 0) {
secp256k1_gej tmpa;
if (i != len - 1) {
secp256k1_fe_mul(&zs, &zs, &zr[i]);
}
i--;
tmpa.x = a[i].x;
tmpa.y = a[i].y;
tmpa.infinity = 0;
secp256k1_ge_set_gej_zinv(&a[i], &tmpa, &zs);
}
}
}
static void secp256k1_gej_set_infinity(secp256k1_gej *r) {
r->infinity = 1;
secp256k1_fe_clear(&r->x);
secp256k1_fe_clear(&r->y);
secp256k1_fe_clear(&r->z);
}
static void secp256k1_ge_set_infinity(secp256k1_ge *r) {
r->infinity = 1;
secp256k1_fe_clear(&r->x);
secp256k1_fe_clear(&r->y);
}
static void secp256k1_gej_clear(secp256k1_gej *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 *r) {
r->infinity = 0;
secp256k1_fe_clear(&r->x);
secp256k1_fe_clear(&r->y);
}
static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) {
secp256k1_fe x2, x3;
r->x = *x;
secp256k1_fe_sqr(&x2, x);
secp256k1_fe_mul(&x3, x, &x2);
r->infinity = 0;
secp256k1_fe_add(&x3, &secp256k1_fe_const_b);
if (!secp256k1_fe_sqrt(&r->y, &x3)) {
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 *r, const secp256k1_ge *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_var(const secp256k1_gej *a, const secp256k1_gej *b) {
secp256k1_gej tmp;
secp256k1_gej_neg(&tmp, a);
secp256k1_gej_add_var(&tmp, &tmp, b, NULL);
return secp256k1_gej_is_infinity(&tmp);
}
static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a) {
secp256k1_fe r, r2;
VERIFY_CHECK(!a->infinity);
secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
r2 = a->x; secp256k1_fe_normalize_weak(&r2);
return secp256k1_fe_equal_var(&r, &r2);
}
static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *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 *a) {
return a->infinity;
}
static int secp256k1_ge_is_valid_var(const secp256k1_ge *a) {
secp256k1_fe y2, x3;
if (a->infinity) {
return 0;
}
/* y^2 = x^3 + 7 */
secp256k1_fe_sqr(&y2, &a->y);
secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
secp256k1_fe_add(&x3, &secp256k1_fe_const_b);
secp256k1_fe_normalize_weak(&x3);
return secp256k1_fe_equal_var(&y2, &x3);
}
static SECP256K1_INLINE void secp256k1_gej_double(secp256k1_gej *r, const secp256k1_gej *a) {
/* Operations: 3 mul, 4 sqr, 8 add/half/mul_int/negate */
secp256k1_fe l, s, t;
r->infinity = a->infinity;
/* Formula used:
* L = (3/2) * X1^2
* S = Y1^2
* T = -X1*S
* X3 = L^2 + 2*T
* Y3 = -(L*(X3 + T) + S^2)
* Z3 = Y1*Z1
*/
secp256k1_fe_mul(&r->z, &a->z, &a->y); /* Z3 = Y1*Z1 (1) */
secp256k1_fe_sqr(&s, &a->y); /* S = Y1^2 (1) */
secp256k1_fe_sqr(&l, &a->x); /* L = X1^2 (1) */
secp256k1_fe_mul_int(&l, 3); /* L = 3*X1^2 (3) */
secp256k1_fe_half(&l); /* L = 3/2*X1^2 (2) */
secp256k1_fe_negate(&t, &s, 1); /* T = -S (2) */
secp256k1_fe_mul(&t, &t, &a->x); /* T = -X1*S (1) */
secp256k1_fe_sqr(&r->x, &l); /* X3 = L^2 (1) */
secp256k1_fe_add(&r->x, &t); /* X3 = L^2 + T (2) */
secp256k1_fe_add(&r->x, &t); /* X3 = L^2 + 2*T (3) */
secp256k1_fe_sqr(&s, &s); /* S' = S^2 (1) */
secp256k1_fe_add(&t, &r->x); /* T' = X3 + T (4) */
secp256k1_fe_mul(&r->y, &t, &l); /* Y3 = L*(X3 + T) (1) */
secp256k1_fe_add(&r->y, &s); /* Y3 = L*(X3 + T) + S^2 (2) */
secp256k1_fe_negate(&r->y, &r->y, 2); /* Y3 = -(L*(X3 + T) + S^2) (3) */
}
static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
/** 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.
*
* Having said this, if this function receives a point on a sextic twist, e.g. by
* a fault attack, it is possible for y to be 0. This happens for y^2 = x^3 + 6,
* since -6 does have a cube root mod p. For this point, this function will not set
* the infinity flag even though the point doubles to infinity, and the result
* point will be gibberish (z = 0 but infinity = 0).
*/
if (a->infinity) {
secp256k1_gej_set_infinity(r);
if (rzr != NULL) {
secp256k1_fe_set_int(rzr, 1);
}
return;
}
if (rzr != NULL) {
*rzr = a->y;
secp256k1_fe_normalize_weak(rzr);
}
secp256k1_gej_double(r, a);
}
static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr) {
/* 12 mul, 4 sqr, 11 add/negate/normalizes_to_zero (ignoring special cases) */
secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, h2, h3, t;
if (a->infinity) {
VERIFY_CHECK(rzr == NULL);
*r = *b;
return;
}
if (b->infinity) {
if (rzr != NULL) {
secp256k1_fe_set_int(rzr, 1);
}
*r = *a;
return;
}
secp256k1_fe_sqr(&z22, &b->z);
secp256k1_fe_sqr(&z12, &a->z);
secp256k1_fe_mul(&u1, &a->x, &z22);
secp256k1_fe_mul(&u2, &b->x, &z12);
secp256k1_fe_mul(&s1, &a->y, &z22); secp256k1_fe_mul(&s1, &s1, &b->z);
secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
secp256k1_fe_negate(&i, &s2, 1); secp256k1_fe_add(&i, &s1);
if (secp256k1_fe_normalizes_to_zero_var(&h)) {
if (secp256k1_fe_normalizes_to_zero_var(&i)) {
secp256k1_gej_double_var(r, a, rzr);
} else {
if (rzr != NULL) {
secp256k1_fe_set_int(rzr, 0);
}
secp256k1_gej_set_infinity(r);
}
return;
}
r->infinity = 0;
secp256k1_fe_mul(&t, &h, &b->z);
if (rzr != NULL) {
*rzr = t;
}
secp256k1_fe_mul(&r->z, &a->z, &t);
secp256k1_fe_sqr(&h2, &h);
secp256k1_fe_negate(&h2, &h2, 1);
secp256k1_fe_mul(&h3, &h2, &h);
secp256k1_fe_mul(&t, &u1, &h2);
secp256k1_fe_sqr(&r->x, &i);
secp256k1_fe_add(&r->x, &h3);
secp256k1_fe_add(&r->x, &t);
secp256k1_fe_add(&r->x, &t);
secp256k1_fe_add(&t, &r->x);
secp256k1_fe_mul(&r->y, &t, &i);
secp256k1_fe_mul(&h3, &h3, &s1);
secp256k1_fe_add(&r->y, &h3);
}
static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr) {
/* 8 mul, 3 sqr, 13 add/negate/normalize_weak/normalizes_to_zero (ignoring special cases) */
secp256k1_fe z12, u1, u2, s1, s2, h, i, h2, h3, t;
if (a->infinity) {
VERIFY_CHECK(rzr == NULL);
secp256k1_gej_set_ge(r, b);
return;
}
if (b->infinity) {
if (rzr != NULL) {
secp256k1_fe_set_int(rzr, 1);
}
*r = *a;
return;
}
secp256k1_fe_sqr(&z12, &a->z);
u1 = a->x; secp256k1_fe_normalize_weak(&u1);
secp256k1_fe_mul(&u2, &b->x, &z12);
s1 = a->y; secp256k1_fe_normalize_weak(&s1);
secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
secp256k1_fe_negate(&i, &s2, 1); secp256k1_fe_add(&i, &s1);
if (secp256k1_fe_normalizes_to_zero_var(&h)) {
if (secp256k1_fe_normalizes_to_zero_var(&i)) {
secp256k1_gej_double_var(r, a, rzr);
} else {
if (rzr != NULL) {
secp256k1_fe_set_int(rzr, 0);
}
secp256k1_gej_set_infinity(r);
}
return;
}
r->infinity = 0;
if (rzr != NULL) {
*rzr = h;
}
secp256k1_fe_mul(&r->z, &a->z, &h);
secp256k1_fe_sqr(&h2, &h);
secp256k1_fe_negate(&h2, &h2, 1);
secp256k1_fe_mul(&h3, &h2, &h);
secp256k1_fe_mul(&t, &u1, &h2);
secp256k1_fe_sqr(&r->x, &i);
secp256k1_fe_add(&r->x, &h3);
secp256k1_fe_add(&r->x, &t);
secp256k1_fe_add(&r->x, &t);
secp256k1_fe_add(&t, &r->x);
secp256k1_fe_mul(&r->y, &t, &i);
secp256k1_fe_mul(&h3, &h3, &s1);
secp256k1_fe_add(&r->y, &h3);
}
static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv) {
/* 9 mul, 3 sqr, 13 add/negate/normalize_weak/normalizes_to_zero (ignoring special cases) */
secp256k1_fe az, z12, u1, u2, s1, s2, h, i, h2, h3, t;
if (a->infinity) {
secp256k1_fe bzinv2, bzinv3;
r->infinity = b->infinity;
secp256k1_fe_sqr(&bzinv2, bzinv);
secp256k1_fe_mul(&bzinv3, &bzinv2, bzinv);
secp256k1_fe_mul(&r->x, &b->x, &bzinv2);
secp256k1_fe_mul(&r->y, &b->y, &bzinv3);
secp256k1_fe_set_int(&r->z, 1);
return;
}
if (b->infinity) {
*r = *a;
return;
}
/** We need to calculate (rx,ry,rz) = (ax,ay,az) + (bx,by,1/bzinv). Due to
* secp256k1's isomorphism we can multiply the Z coordinates on both sides
* by bzinv, and get: (rx,ry,rz*bzinv) = (ax,ay,az*bzinv) + (bx,by,1).
* This means that (rx,ry,rz) can be calculated as
* (ax,ay,az*bzinv) + (bx,by,1), when not applying the bzinv factor to rz.
* The variable az below holds the modified Z coordinate for a, which is used
* for the computation of rx and ry, but not for rz.
*/
secp256k1_fe_mul(&az, &a->z, bzinv);
secp256k1_fe_sqr(&z12, &az);
u1 = a->x; secp256k1_fe_normalize_weak(&u1);
secp256k1_fe_mul(&u2, &b->x, &z12);
s1 = a->y; secp256k1_fe_normalize_weak(&s1);
secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &az);
secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
secp256k1_fe_negate(&i, &s2, 1); secp256k1_fe_add(&i, &s1);
if (secp256k1_fe_normalizes_to_zero_var(&h)) {
if (secp256k1_fe_normalizes_to_zero_var(&i)) {
secp256k1_gej_double_var(r, a, NULL);
} else {
secp256k1_gej_set_infinity(r);
}
return;
}
r->infinity = 0;
secp256k1_fe_mul(&r->z, &a->z, &h);
secp256k1_fe_sqr(&h2, &h);
secp256k1_fe_negate(&h2, &h2, 1);
secp256k1_fe_mul(&h3, &h2, &h);
secp256k1_fe_mul(&t, &u1, &h2);
secp256k1_fe_sqr(&r->x, &i);
secp256k1_fe_add(&r->x, &h3);
secp256k1_fe_add(&r->x, &t);
secp256k1_fe_add(&r->x, &t);
secp256k1_fe_add(&t, &r->x);
secp256k1_fe_mul(&r->y, &t, &i);
secp256k1_fe_mul(&h3, &h3, &s1);
secp256k1_fe_add(&r->y, &h3);
}
static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b) {
/* Operations: 7 mul, 5 sqr, 24 add/cmov/half/mul_int/negate/normalize_weak/normalizes_to_zero */
secp256k1_fe zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
secp256k1_fe m_alt, rr_alt;
int infinity, degenerate;
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 = R^2+Q
* Y3 = -(R*(2*X3+Q)+M^4)/2
* Z3 = M*Z
* (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)
*
* This formula has the benefit of being the same for both addition
* of distinct points and doubling. However, it breaks down in the
* case that either point is infinity, or that y1 = -y2. We handle
* these cases in the following ways:
*
* - If b is infinity we simply bail by means of a VERIFY_CHECK.
*
* - If a is infinity, we detect this, and at the end of the
* computation replace the result (which will be meaningless,
* but we compute to be constant-time) with b.x : b.y : 1.
*
* - If a = -b, we have y1 = -y2, which is a degenerate case.
* But here the answer is infinity, so we simply set the
* infinity flag of the result, overriding the computed values
* without even needing to cmov.
*
* - If y1 = -y2 but x1 != x2, which does occur thanks to certain
* properties of our curve (specifically, 1 has nontrivial cube
* roots in our field, and the curve equation has no x coefficient)
* then the answer is not infinity but also not given by the above
* equation. In this case, we cmov in place an alternate expression
* for lambda. Specifically (y1 - y2)/(x1 - x2). Where both these
* expressions for lambda are defined, they are equal, and can be
* obtained from each other by multiplication by (y1 + y2)/(y1 + y2)
* then substitution of x^3 + 7 for y^2 (using the curve equation).
* For all pairs of nonzero points (a, b) at least one is defined,
* so this covers everything.
*/
secp256k1_fe_sqr(&zz, &a->z); /* z = Z1^2 */
u1 = a->x; secp256k1_fe_normalize_weak(&u1); /* u1 = U1 = X1*Z2^2 (1) */
secp256k1_fe_mul(&u2, &b->x, &zz); /* u2 = U2 = X2*Z1^2 (1) */
s1 = a->y; secp256k1_fe_normalize_weak(&s1); /* s1 = S1 = Y1*Z2^3 (1) */
secp256k1_fe_mul(&s2, &b->y, &zz); /* s2 = Y2*Z1^2 (1) */
secp256k1_fe_mul(&s2, &s2, &a->z); /* s2 = S2 = Y2*Z1^3 (1) */
t = u1; secp256k1_fe_add(&t, &u2); /* t = T = U1+U2 (2) */
m = s1; secp256k1_fe_add(&m, &s2); /* m = M = S1+S2 (2) */
secp256k1_fe_sqr(&rr, &t); /* rr = T^2 (1) */
secp256k1_fe_negate(&m_alt, &u2, 1); /* Malt = -X2*Z1^2 */
secp256k1_fe_mul(&tt, &u1, &m_alt); /* tt = -U1*U2 (2) */
secp256k1_fe_add(&rr, &tt); /* rr = R = T^2-U1*U2 (3) */
/** If lambda = R/M = 0/0 we have a problem (except in the "trivial"
* case that Z = z1z2 = 0, and this is special-cased later on). */
degenerate = secp256k1_fe_normalizes_to_zero(&m) &
secp256k1_fe_normalizes_to_zero(&rr);
/* This only occurs when y1 == -y2 and x1^3 == x2^3, but x1 != x2.
* This means either x1 == beta*x2 or beta*x1 == x2, where beta is
* a nontrivial cube root of one. In either case, an alternate
* non-indeterminate expression for lambda is (y1 - y2)/(x1 - x2),
* so we set R/M equal to this. */
rr_alt = s1;
secp256k1_fe_mul_int(&rr_alt, 2); /* rr = Y1*Z2^3 - Y2*Z1^3 (2) */
secp256k1_fe_add(&m_alt, &u1); /* Malt = X1*Z2^2 - X2*Z1^2 */
secp256k1_fe_cmov(&rr_alt, &rr, !degenerate);
secp256k1_fe_cmov(&m_alt, &m, !degenerate);
/* Now Ralt / Malt = lambda and is guaranteed not to be 0/0.
* From here on out Ralt and Malt represent the numerator
* and denominator of lambda; R and M represent the explicit
* expressions x1^2 + x2^2 + x1x2 and y1 + y2. */
secp256k1_fe_sqr(&n, &m_alt); /* n = Malt^2 (1) */
secp256k1_fe_negate(&q, &t, 2); /* q = -T (3) */
secp256k1_fe_mul(&q, &q, &n); /* q = Q = -T*Malt^2 (1) */
/* These two lines use the observation that either M == Malt or M == 0,
* so M^3 * Malt is either Malt^4 (which is computed by squaring), or
* zero (which is "computed" by cmov). So the cost is one squaring
* versus two multiplications. */
secp256k1_fe_sqr(&n, &n);
secp256k1_fe_cmov(&n, &m, degenerate); /* n = M^3 * Malt (2) */
secp256k1_fe_sqr(&t, &rr_alt); /* t = Ralt^2 (1) */
secp256k1_fe_mul(&r->z, &a->z, &m_alt); /* r->z = Z3 = Malt*Z (1) */
infinity = secp256k1_fe_normalizes_to_zero(&r->z) & ~a->infinity;
secp256k1_fe_add(&t, &q); /* t = Ralt^2 + Q (2) */
r->x = t; /* r->x = X3 = Ralt^2 + Q (2) */
secp256k1_fe_mul_int(&t, 2); /* t = 2*X3 (4) */
secp256k1_fe_add(&t, &q); /* t = 2*X3 + Q (5) */
secp256k1_fe_mul(&t, &t, &rr_alt); /* t = Ralt*(2*X3 + Q) (1) */
secp256k1_fe_add(&t, &n); /* t = Ralt*(2*X3 + Q) + M^3*Malt (3) */
secp256k1_fe_negate(&r->y, &t, 3); /* r->y = -(Ralt*(2*X3 + Q) + M^3*Malt) (4) */
secp256k1_fe_half(&r->y); /* r->y = Y3 = -(Ralt*(2*X3 + Q) + M^3*Malt)/2 (3) */
/** In case a->infinity == 1, replace r with (b->x, b->y, 1). */
secp256k1_fe_cmov(&r->x, &b->x, a->infinity);
secp256k1_fe_cmov(&r->y, &b->y, a->infinity);
secp256k1_fe_cmov(&r->z, &secp256k1_fe_one, a->infinity);
r->infinity = infinity;
}
static void secp256k1_gej_rescale(secp256k1_gej *r, const secp256k1_fe *s) {
/* Operations: 4 mul, 1 sqr */
secp256k1_fe zz;
VERIFY_CHECK(!secp256k1_fe_is_zero(s));
secp256k1_fe_sqr(&zz, s);
secp256k1_fe_mul(&r->x, &r->x, &zz); /* r->x *= s^2 */
secp256k1_fe_mul(&r->y, &r->y, &zz);
secp256k1_fe_mul(&r->y, &r->y, s); /* r->y *= s^3 */
secp256k1_fe_mul(&r->z, &r->z, s); /* r->z *= s */
}
static void secp256k1_ge_to_storage(secp256k1_ge_storage *r, const secp256k1_ge *a) {
secp256k1_fe x, y;
VERIFY_CHECK(!a->infinity);
x = a->x;
secp256k1_fe_normalize(&x);
y = a->y;
secp256k1_fe_normalize(&y);
secp256k1_fe_to_storage(&r->x, &x);
secp256k1_fe_to_storage(&r->y, &y);
}
static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storage *a) {
secp256k1_fe_from_storage(&r->x, &a->x);
secp256k1_fe_from_storage(&r->y, &a->y);
r->infinity = 0;
}
static SECP256K1_INLINE void secp256k1_gej_cmov(secp256k1_gej *r, const secp256k1_gej *a, int flag) {
secp256k1_fe_cmov(&r->x, &a->x, flag);
secp256k1_fe_cmov(&r->y, &a->y, flag);
secp256k1_fe_cmov(&r->z, &a->z, flag);
r->infinity ^= (r->infinity ^ a->infinity) & flag;
}
static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag) {
secp256k1_fe_storage_cmov(&r->x, &a->x, flag);
secp256k1_fe_storage_cmov(&r->y, &a->y, flag);
}
static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a) {
*r = *a;
secp256k1_fe_mul(&r->x, &r->x, &secp256k1_const_beta);
}
static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge) {
#ifdef EXHAUSTIVE_TEST_ORDER
secp256k1_gej out;
int i;
/* A very simple EC multiplication ladder that avoids a dependency on ecmult. */
secp256k1_gej_set_infinity(&out);
for (i = 0; i < 32; ++i) {
secp256k1_gej_double_var(&out, &out, NULL);
if ((((uint32_t)EXHAUSTIVE_TEST_ORDER) >> (31 - i)) & 1) {
secp256k1_gej_add_ge_var(&out, &out, ge, NULL);
}
}
return secp256k1_gej_is_infinity(&out);
#else
(void)ge;
/* The real secp256k1 group has cofactor 1, so the subgroup is the entire curve. */
return 1;
#endif
}
#endif /* SECP256K1_GROUP_IMPL_H */
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