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Diffstat (limited to 'crypto/secp256k1/libsecp256k1/src/group_impl.h')
-rw-r--r-- | crypto/secp256k1/libsecp256k1/src/group_impl.h | 700 |
1 files changed, 0 insertions, 700 deletions
diff --git a/crypto/secp256k1/libsecp256k1/src/group_impl.h b/crypto/secp256k1/libsecp256k1/src/group_impl.h deleted file mode 100644 index 7d723532f..000000000 --- a/crypto/secp256k1/libsecp256k1/src/group_impl.h +++ /dev/null @@ -1,700 +0,0 @@ -/********************************************************************** - * 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 "num.h" -#include "field.h" -#include "group.h" - -/* These points can be generated in sage as follows: - * - * 0. Setup a worksheet with the following parameters. - * b = 4 # whatever CURVE_B will be set to - * F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F) - * C = EllipticCurve ([F (0), F (b)]) - * - * 1. Determine all the small orders available to you. (If there are - * no satisfactory ones, go back and change b.) - * print C.order().factor(limit=1000) - * - * 2. Choose an order as one of the prime factors listed in the above step. - * (You can also multiply some to get a composite order, though the - * tests will crash trying to invert scalars during signing.) We take a - * random point and scale it to drop its order to the desired value. - * There is some probability this won't work; just try again. - * order = 199 - * P = C.random_point() - * P = (int(P.order()) / int(order)) * P - * assert(P.order() == order) - * - * 3. Print the values. You'll need to use a vim macro or something to - * split the hex output into 4-byte chunks. - * print "%x %x" % P.xy() - */ -#if defined(EXHAUSTIVE_TEST_ORDER) -# if EXHAUSTIVE_TEST_ORDER == 199 -const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST( - 0xFA7CC9A7, 0x0737F2DB, 0xA749DD39, 0x2B4FB069, - 0x3B017A7D, 0xA808C2F1, 0xFB12940C, 0x9EA66C18, - 0x78AC123A, 0x5ED8AEF3, 0x8732BC91, 0x1F3A2868, - 0x48DF246C, 0x808DAE72, 0xCFE52572, 0x7F0501ED -); - -const int CURVE_B = 4; -# elif EXHAUSTIVE_TEST_ORDER == 13 -const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST( - 0xedc60018, 0xa51a786b, 0x2ea91f4d, 0x4c9416c0, - 0x9de54c3b, 0xa1316554, 0x6cf4345c, 0x7277ef15, - 0x54cb1b6b, 0xdc8c1273, 0x087844ea, 0x43f4603e, - 0x0eaf9a43, 0xf6effe55, 0x939f806d, 0x37adf8ac -); -const int CURVE_B = 2; -# else -# error No known generator for the specified exhaustive test group order. -# endif -#else -/** Generator for secp256k1, value 'g' defined in - * "Standards for Efficient Cryptography" (SEC2) 2.7.1. - */ -static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST( - 0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL, - 0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL, - 0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL, - 0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL -); - -const int CURVE_B = 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; - 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; - r->infinity = a->infinity; - if (a->infinity) { - 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); - r->x = a->x; - r->y = a->y; -} - -static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len, const secp256k1_callback *cb) { - secp256k1_fe *az; - secp256k1_fe *azi; - size_t i; - size_t count = 0; - az = (secp256k1_fe *)checked_malloc(cb, sizeof(secp256k1_fe) * len); - for (i = 0; i < len; i++) { - if (!a[i].infinity) { - az[count++] = a[i].z; - } - } - - azi = (secp256k1_fe *)checked_malloc(cb, sizeof(secp256k1_fe) * count); - secp256k1_fe_inv_all_var(azi, az, count); - free(az); - - count = 0; - for (i = 0; i < len; i++) { - r[i].infinity = a[i].infinity; - if (!a[i].infinity) { - secp256k1_ge_set_gej_zinv(&r[i], &a[i], &azi[count++]); - } - } - free(azi); -} - -static void secp256k1_ge_set_table_gej_var(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zr, size_t len) { - size_t i = len - 1; - secp256k1_fe zi; - - if (len > 0) { - /* 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 *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr) { - size_t i = len - 1; - secp256k1_fe zs; - - if (len > 0) { - /* 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 *r) { - r->infinity = 1; - secp256k1_fe_clear(&r->x); - secp256k1_fe_clear(&r->y); - secp256k1_fe_clear(&r->z); -} - -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_xquad(secp256k1_ge *r, const secp256k1_fe *x) { - secp256k1_fe x2, x3, c; - r->x = *x; - secp256k1_fe_sqr(&x2, x); - secp256k1_fe_mul(&x3, x, &x2); - r->infinity = 0; - secp256k1_fe_set_int(&c, CURVE_B); - secp256k1_fe_add(&c, &x3); - return secp256k1_fe_sqrt(&r->y, &c); -} - -static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) { - if (!secp256k1_ge_set_xquad(r, x)) { - 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_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_gej_is_valid_var(const secp256k1_gej *a) { - secp256k1_fe 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_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, CURVE_B); - 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 *a) { - secp256k1_fe y2, x3, c; - 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_set_int(&c, CURVE_B); - 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 *r, const secp256k1_gej *a, secp256k1_fe *rzr) { - /* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate. - * - * Note that there is an implementation described at - * https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l - * which trades a multiply for a square, but in practice this is actually slower, - * mainly because it requires more normalizations. - */ - secp256k1_fe 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. - * - * 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). - */ - r->infinity = a->infinity; - if (r->infinity) { - if (rzr != NULL) { - secp256k1_fe_set_int(rzr, 1); - } - return; - } - - if (rzr != NULL) { - *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_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 SECP256K1_INLINE void secp256k1_gej_double_nonzero(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) { - VERIFY_CHECK(!secp256k1_gej_is_infinity(a)); - secp256k1_gej_double_var(r, a, rzr); -} - -static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr) { - /* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */ - secp256k1_fe 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 != NULL) { - secp256k1_fe_set_int(rzr, 1); - } - *r = *a; - return; - } - - r->infinity = 0; - 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, rzr); - } else { - if (rzr != NULL) { - 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); - secp256k1_fe_mul(&h, &h, &b->z); - if (rzr != NULL) { - *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 *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr) { - /* 8 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */ - secp256k1_fe z12, u1, u2, s1, s2, h, i, i2, 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; - } - 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 != NULL) { - 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 != NULL) { - *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 *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv) { - /* 9 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */ - secp256k1_fe az, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t; - - if (b->infinity) { - *r = *a; - return; - } - 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; - } - r->infinity = 0; - - /** 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, NULL); - } else { - r->infinity = 1; - } - return; - } - 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_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 *r, const secp256k1_gej *a, const secp256k1_ge *b) { - /* Operations: 7 mul, 5 sqr, 4 normalize, 21 mul_int/add/negate/cmov */ - static const secp256k1_fe fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1); - 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 = 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.) - * - * 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_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(&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); /* 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, 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_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_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); -} - -#ifdef USE_ENDOMORPHISM -static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a) { - static const secp256k1_fe beta = SECP256K1_FE_CONST( - 0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul, - 0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul - ); - *r = *a; - secp256k1_fe_mul(&r->x, &r->x, &beta); -} -#endif - -static int secp256k1_gej_has_quad_y_var(const secp256k1_gej *a) { - secp256k1_fe yz; - - if (a->infinity) { - return 0; - } - - /* We rely on the fact that the Jacobi symbol of 1 / a->z^3 is the same as - * that of a->z. Thus a->y / a->z^3 is a quadratic residue iff a->y * a->z - is */ - secp256k1_fe_mul(&yz, &a->y, &a->z); - return secp256k1_fe_is_quad_var(&yz); -} - -#endif |