#include "field.h"
#include "group.h"
-static const secp256k1_ge_t secp256k1_ge_const_g = {
- SECP256K1_FE_CONST(
- 0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,
- 0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL
- ),
- SECP256K1_FE_CONST(
- 0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,
- 0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
- ),
- 0
-};
+/** Generator for secp256k1, value 'g' defined in
+ * "Standards for Efficient Cryptography" (SEC2) 2.7.1.
+ */
+static const secp256k1_ge_t secp256k1_ge_const_g = SECP256K1_GE_CONST(
+ 0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,
+ 0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,
+ 0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,
+ 0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
+);
+
+static void secp256k1_ge_set_gej_zinv(secp256k1_ge_t *r, const secp256k1_gej_t *a, const secp256k1_fe_t *zi) {
+ secp256k1_fe_t zi2;
+ secp256k1_fe_t zi3;
+ 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_infinity(secp256k1_ge_t *r) {
r->infinity = 1;
secp256k1_fe_negate(&r->y, &r->y, 1);
}
-static void secp256k1_ge_get_hex(char *r131, const secp256k1_ge_t *a) {
- r131[0] = '(';
- secp256k1_fe_get_hex(r131 + 1, &a->x);
- r131[65] = ',';
- secp256k1_fe_get_hex(r131 + 66, &a->y);
- r131[130] = ')';
-}
-
static void secp256k1_ge_set_gej(secp256k1_ge_t *r, secp256k1_gej_t *a) {
+ secp256k1_fe_t z2, z3;
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_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);
}
static void secp256k1_ge_set_gej_var(secp256k1_ge_t *r, secp256k1_gej_t *a) {
+ secp256k1_fe_t z2, z3;
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_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);
}
static void secp256k1_ge_set_all_gej_var(size_t len, secp256k1_ge_t *r, const secp256k1_gej_t *a) {
+ secp256k1_fe_t *az;
+ secp256k1_fe_t *azi;
+ size_t i;
size_t count = 0;
- secp256k1_fe_t *az = checked_malloc(sizeof(secp256k1_fe_t) * len);
- for (size_t i=0; i<len; i++) {
+ az = (secp256k1_fe_t *)checked_malloc(sizeof(secp256k1_fe_t) * len);
+ for (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);
+ azi = (secp256k1_fe_t *)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++) {
+ for (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);
+ secp256k1_ge_set_gej_zinv(&r[i], &a[i], &azi[count++]);
}
}
free(azi);
}
+static void secp256k1_ge_set_table_gej_var(size_t len, secp256k1_ge_t *r, const secp256k1_gej_t *a, const secp256k1_fe_t *zr) {
+ size_t i = len - 1;
+ secp256k1_fe_t zi;
+
+ if (len < 1)
+ return;
+
+ /* Compute the inverse of the last z coordinate, and use it to compute the last affine output. */
+ secp256k1_fe_inv(&zi, &a[i].z);
+ secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
+
+ /* Work out way backwards, using the z-ratios to scale the x/y values. */
+ while (i > 0) {
+ secp256k1_fe_mul(&zi, &zi, &zr[i]);
+ i--;
+ secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
+ }
+}
+
+static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge_t *r, secp256k1_fe_t *globalz, const secp256k1_gej_t *a, const secp256k1_fe_t *zr) {
+ size_t i = len - 1;
+ secp256k1_fe_t zs;
+
+ if (len < 1)
+ return;
+
+ /* The z of the final point gives us the "global Z" for the table. */
+ r[i].x = a[i].x;
+ r[i].y = a[i].y;
+ *globalz = a[i].z;
+ r[i].infinity = 0;
+ zs = zr[i];
+
+ /* Work our way backwards, using the z-ratios to scale the x/y values. */
+ while (i > 0) {
+ if (i != len - 1) {
+ secp256k1_fe_mul(&zs, &zs, &zr[i]);
+ }
+ i--;
+ secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zs);
+ }
+}
+
static void secp256k1_gej_set_infinity(secp256k1_gej_t *r) {
r->infinity = 1;
secp256k1_fe_set_int(&r->x, 0);
}
static int secp256k1_ge_set_xo_var(secp256k1_ge_t *r, const secp256k1_fe_t *x, int odd) {
+ secp256k1_fe_t x2, x3, c;
r->x = *x;
- secp256k1_fe_t x2; secp256k1_fe_sqr(&x2, x);
- secp256k1_fe_t x3; secp256k1_fe_mul(&x3, x, &x2);
+ secp256k1_fe_sqr(&x2, x);
+ secp256k1_fe_mul(&x3, x, &x2);
r->infinity = 0;
- secp256k1_fe_t c; secp256k1_fe_set_int(&c, 7);
+ secp256k1_fe_set_int(&c, 7);
secp256k1_fe_add(&c, &x3);
- if (!secp256k1_fe_sqrt_var(&r->y, &c))
+ if (!secp256k1_fe_sqrt_var(&r->y, &c)) {
return 0;
+ }
secp256k1_fe_normalize_var(&r->y);
- if (secp256k1_fe_is_odd(&r->y) != odd)
+ if (secp256k1_fe_is_odd(&r->y) != odd) {
secp256k1_fe_negate(&r->y, &r->y, 1);
+ }
return 1;
}
}
static int secp256k1_gej_eq_x_var(const secp256k1_fe_t *x, const secp256k1_gej_t *a) {
+ secp256k1_fe_t r, r2;
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);
+ 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 int secp256k1_gej_is_valid_var(const secp256k1_gej_t *a) {
- if (a->infinity)
+ secp256k1_fe_t y2, x3, z2, z6;
+ 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_sqr(&y2, &a->y);
+ secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
+ secp256k1_fe_sqr(&z2, &a->z);
+ 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);
}
static int secp256k1_ge_is_valid_var(const secp256k1_ge_t *a) {
- if (a->infinity)
+ secp256k1_fe_t y2, x3, c;
+ 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_sqr(&y2, &a->y);
+ secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
+ 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) {
+static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, secp256k1_fe_t *rzr) {
+ /* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate */
+ secp256k1_fe_t t1,t2,t3,t4;
/** 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) {
+ if (rzr) {
+ secp256k1_fe_set_int(rzr, 1);
+ }
return;
}
- secp256k1_fe_t t1,t2,t3,t4;
+ if (rzr) {
+ *rzr = a->y;
+ secp256k1_fe_normalize_weak(rzr);
+ secp256k1_fe_mul_int(rzr, 2);
+ }
+
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_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) {
+static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_gej_t *b, secp256k1_fe_t *rzr) {
+ /* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */
+ secp256k1_fe_t z22, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
+
if (a->infinity) {
+ VERIFY_CHECK(rzr == NULL);
*r = *b;
return;
}
+
if (b->infinity) {
+ if (rzr) {
+ secp256k1_fe_set_int(rzr, 1);
+ }
*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);
+ 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, &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);
+ secp256k1_gej_double_var(r, a, rzr);
} else {
+ if (rzr) {
+ secp256k1_fe_set_int(rzr, 0);
+ }
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);
+ secp256k1_fe_sqr(&i2, &i);
+ secp256k1_fe_sqr(&h2, &h);
+ secp256k1_fe_mul(&h3, &h, &h2);
+ secp256k1_fe_mul(&h, &h, &b->z);
+ if (rzr) {
+ *rzr = h;
+ }
+ secp256k1_fe_mul(&r->z, &a->z, &h);
+ 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) {
+static void secp256k1_gej_add_ge_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b, secp256k1_fe_t *rzr) {
+ /* 8 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
+ secp256k1_fe_t z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
if (a->infinity) {
- r->infinity = b->infinity;
- r->x = b->x;
- r->y = b->y;
- secp256k1_fe_set_int(&r->z, 1);
+ VERIFY_CHECK(rzr == NULL);
+ secp256k1_gej_set_ge(r, b);
return;
}
+ if (b->infinity) {
+ if (rzr) {
+ secp256k1_fe_set_int(rzr, 1);
+ }
+ *r = *a;
+ return;
+ }
+ r->infinity = 0;
+
+ 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, &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, rzr);
+ } else {
+ if (rzr) {
+ secp256k1_fe_set_int(rzr, 0);
+ }
+ r->infinity = 1;
+ }
+ return;
+ }
+ secp256k1_fe_sqr(&i2, &i);
+ secp256k1_fe_sqr(&h2, &h);
+ secp256k1_fe_mul(&h3, &h, &h2);
+ if (rzr) {
+ *rzr = h;
+ }
+ secp256k1_fe_mul(&r->z, &a->z, &h);
+ 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_zinv_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b, const secp256k1_fe_t *bzinv) {
+ /* 9 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
+ secp256k1_fe_t az, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
+
if (b->infinity) {
*r = *a;
return;
}
+ if (a->infinity) {
+ secp256k1_fe_t 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;
+ }
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);
+
+ /** 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, &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);
+ secp256k1_gej_double_var(r, a, NULL);
} 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_sqr(&i2, &i);
+ secp256k1_fe_sqr(&h2, &h);
+ 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);
+ 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) {
+ /* Operations: 7 mul, 5 sqr, 4 normalize, 21 mul_int/add/negate/cmov */
+ static const secp256k1_fe_t fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
+ secp256k1_fe_t zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
+ secp256k1_fe_t m_alt, rr_alt;
+ int infinity, degenerate;
VERIFY_CHECK(!b->infinity);
VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);
* 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.)
+ *
+ * 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_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_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*Z2^2 (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_mul(&q, &n, &t); /* 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 = Malt*Z (1) */
+ infinity = secp256k1_fe_normalizes_to_zero(&r->z) * (1 - a->infinity);
+ secp256k1_fe_mul_int(&r->z, 2); /* r->z = Z3 = 2*Malt*Z (2) */
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_add(&t, &q); /* t = Ralt^2-Q (3) */
+ secp256k1_fe_normalize_weak(&t);
+ r->x = t; /* r->x = Ralt^2-Q (1) */
+ secp256k1_fe_mul_int(&t, 2); /* t = 2*x3 (2) */
+ secp256k1_fe_add(&t, &q); /* t = 2*x3 - Q: (4) */
+ 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*(Q - 2x3) - M^3*Malt (4) */
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) */
+ secp256k1_fe_mul_int(&r->x, 4); /* r->x = X3 = 4*(Ralt^2-Q) */
+ secp256k1_fe_mul_int(&r->y, 4); /* r->y = Y3 = 4*Ralt*(Q - 2x3) - 4*M^3*Malt (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);
+ /** 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, &fe_1, a->infinity);
r->infinity = infinity;
}
-static void secp256k1_gej_get_hex(char *r131, const secp256k1_gej_t *a) {
- secp256k1_gej_t c = *a;
- secp256k1_ge_t t; secp256k1_ge_set_gej(&t, &c);
- secp256k1_ge_get_hex(r131, &t);
+static void secp256k1_gej_rescale(secp256k1_gej_t *r, const secp256k1_fe_t *s) {
+ /* Operations: 4 mul, 1 sqr */
+ secp256k1_fe_t 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_t *r, const secp256k1_ge_t *a) {
}
#ifdef USE_ENDOMORPHISM
-static void secp256k1_gej_mul_lambda(secp256k1_gej_t *r, const secp256k1_gej_t *a) {
+static void secp256k1_ge_mul_lambda(secp256k1_ge_t *r, const secp256k1_ge_t *a) {
static const secp256k1_fe_t beta = SECP256K1_FE_CONST(
0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul
projects / secp256k1.git / blobdiff