e0ceeab0d1
- Use defined constants instead of hard-coding their integer value. - Allocate secp256k1 structs on the C stack instead of converting []byte - Remove dead code
701 lines
27 KiB
C
701 lines
27 KiB
C
/**********************************************************************
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* Copyright (c) 2013, 2014 Pieter Wuille *
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* Distributed under the MIT software license, see the accompanying *
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* file COPYING or http://www.opensource.org/licenses/mit-license.php.*
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**********************************************************************/
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#ifndef _SECP256K1_GROUP_IMPL_H_
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#define _SECP256K1_GROUP_IMPL_H_
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#include "num.h"
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#include "field.h"
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#include "group.h"
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/* These points can be generated in sage as follows:
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*
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* 0. Setup a worksheet with the following parameters.
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* b = 4 # whatever CURVE_B will be set to
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* F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
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* C = EllipticCurve ([F (0), F (b)])
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*
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* 1. Determine all the small orders available to you. (If there are
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* no satisfactory ones, go back and change b.)
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* print C.order().factor(limit=1000)
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*
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* 2. Choose an order as one of the prime factors listed in the above step.
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* (You can also multiply some to get a composite order, though the
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* tests will crash trying to invert scalars during signing.) We take a
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* random point and scale it to drop its order to the desired value.
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* There is some probability this won't work; just try again.
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* order = 199
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* P = C.random_point()
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* P = (int(P.order()) / int(order)) * P
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* assert(P.order() == order)
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*
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* 3. Print the values. You'll need to use a vim macro or something to
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* split the hex output into 4-byte chunks.
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* print "%x %x" % P.xy()
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*/
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#if defined(EXHAUSTIVE_TEST_ORDER)
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# if EXHAUSTIVE_TEST_ORDER == 199
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const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
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0xFA7CC9A7, 0x0737F2DB, 0xA749DD39, 0x2B4FB069,
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0x3B017A7D, 0xA808C2F1, 0xFB12940C, 0x9EA66C18,
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0x78AC123A, 0x5ED8AEF3, 0x8732BC91, 0x1F3A2868,
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0x48DF246C, 0x808DAE72, 0xCFE52572, 0x7F0501ED
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);
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const int CURVE_B = 4;
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# elif EXHAUSTIVE_TEST_ORDER == 13
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const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
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0xedc60018, 0xa51a786b, 0x2ea91f4d, 0x4c9416c0,
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0x9de54c3b, 0xa1316554, 0x6cf4345c, 0x7277ef15,
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0x54cb1b6b, 0xdc8c1273, 0x087844ea, 0x43f4603e,
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0x0eaf9a43, 0xf6effe55, 0x939f806d, 0x37adf8ac
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);
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const int CURVE_B = 2;
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# else
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# error No known generator for the specified exhaustive test group order.
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# endif
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#else
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/** Generator for secp256k1, value 'g' defined in
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* "Standards for Efficient Cryptography" (SEC2) 2.7.1.
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*/
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static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
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0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,
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0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,
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0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,
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0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
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);
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const int CURVE_B = 7;
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#endif
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static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi) {
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secp256k1_fe zi2;
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secp256k1_fe zi3;
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secp256k1_fe_sqr(&zi2, zi);
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secp256k1_fe_mul(&zi3, &zi2, zi);
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secp256k1_fe_mul(&r->x, &a->x, &zi2);
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secp256k1_fe_mul(&r->y, &a->y, &zi3);
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r->infinity = a->infinity;
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}
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static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y) {
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r->infinity = 0;
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r->x = *x;
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r->y = *y;
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}
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static int secp256k1_ge_is_infinity(const secp256k1_ge *a) {
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return a->infinity;
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}
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static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a) {
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*r = *a;
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secp256k1_fe_normalize_weak(&r->y);
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secp256k1_fe_negate(&r->y, &r->y, 1);
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}
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static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a) {
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secp256k1_fe z2, z3;
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r->infinity = a->infinity;
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secp256k1_fe_inv(&a->z, &a->z);
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secp256k1_fe_sqr(&z2, &a->z);
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secp256k1_fe_mul(&z3, &a->z, &z2);
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secp256k1_fe_mul(&a->x, &a->x, &z2);
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secp256k1_fe_mul(&a->y, &a->y, &z3);
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secp256k1_fe_set_int(&a->z, 1);
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r->x = a->x;
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r->y = a->y;
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}
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static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a) {
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secp256k1_fe z2, z3;
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r->infinity = a->infinity;
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if (a->infinity) {
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return;
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}
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secp256k1_fe_inv_var(&a->z, &a->z);
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secp256k1_fe_sqr(&z2, &a->z);
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secp256k1_fe_mul(&z3, &a->z, &z2);
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secp256k1_fe_mul(&a->x, &a->x, &z2);
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secp256k1_fe_mul(&a->y, &a->y, &z3);
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secp256k1_fe_set_int(&a->z, 1);
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r->x = a->x;
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r->y = a->y;
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}
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static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len, const secp256k1_callback *cb) {
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secp256k1_fe *az;
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secp256k1_fe *azi;
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size_t i;
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size_t count = 0;
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az = (secp256k1_fe *)checked_malloc(cb, sizeof(secp256k1_fe) * len);
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for (i = 0; i < len; i++) {
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if (!a[i].infinity) {
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az[count++] = a[i].z;
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}
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}
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azi = (secp256k1_fe *)checked_malloc(cb, sizeof(secp256k1_fe) * count);
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secp256k1_fe_inv_all_var(azi, az, count);
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free(az);
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count = 0;
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for (i = 0; i < len; i++) {
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r[i].infinity = a[i].infinity;
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if (!a[i].infinity) {
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secp256k1_ge_set_gej_zinv(&r[i], &a[i], &azi[count++]);
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}
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}
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free(azi);
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}
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static void secp256k1_ge_set_table_gej_var(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zr, size_t len) {
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size_t i = len - 1;
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secp256k1_fe zi;
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if (len > 0) {
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/* Compute the inverse of the last z coordinate, and use it to compute the last affine output. */
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secp256k1_fe_inv(&zi, &a[i].z);
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secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
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/* Work out way backwards, using the z-ratios to scale the x/y values. */
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while (i > 0) {
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secp256k1_fe_mul(&zi, &zi, &zr[i]);
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i--;
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secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
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}
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}
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}
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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) {
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size_t i = len - 1;
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secp256k1_fe zs;
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if (len > 0) {
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/* The z of the final point gives us the "global Z" for the table. */
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r[i].x = a[i].x;
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r[i].y = a[i].y;
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*globalz = a[i].z;
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r[i].infinity = 0;
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zs = zr[i];
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/* Work our way backwards, using the z-ratios to scale the x/y values. */
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while (i > 0) {
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if (i != len - 1) {
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secp256k1_fe_mul(&zs, &zs, &zr[i]);
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}
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i--;
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secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zs);
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}
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}
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}
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static void secp256k1_gej_set_infinity(secp256k1_gej *r) {
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r->infinity = 1;
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secp256k1_fe_clear(&r->x);
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secp256k1_fe_clear(&r->y);
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secp256k1_fe_clear(&r->z);
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}
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static void secp256k1_gej_clear(secp256k1_gej *r) {
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r->infinity = 0;
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secp256k1_fe_clear(&r->x);
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secp256k1_fe_clear(&r->y);
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secp256k1_fe_clear(&r->z);
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}
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static void secp256k1_ge_clear(secp256k1_ge *r) {
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r->infinity = 0;
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secp256k1_fe_clear(&r->x);
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secp256k1_fe_clear(&r->y);
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}
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static int secp256k1_ge_set_xquad(secp256k1_ge *r, const secp256k1_fe *x) {
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secp256k1_fe x2, x3, c;
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r->x = *x;
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secp256k1_fe_sqr(&x2, x);
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secp256k1_fe_mul(&x3, x, &x2);
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r->infinity = 0;
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secp256k1_fe_set_int(&c, CURVE_B);
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secp256k1_fe_add(&c, &x3);
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return secp256k1_fe_sqrt(&r->y, &c);
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}
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static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) {
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if (!secp256k1_ge_set_xquad(r, x)) {
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return 0;
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}
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secp256k1_fe_normalize_var(&r->y);
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if (secp256k1_fe_is_odd(&r->y) != odd) {
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secp256k1_fe_negate(&r->y, &r->y, 1);
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}
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return 1;
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}
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static void secp256k1_gej_set_ge(secp256k1_gej *r, const secp256k1_ge *a) {
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r->infinity = a->infinity;
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r->x = a->x;
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r->y = a->y;
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secp256k1_fe_set_int(&r->z, 1);
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}
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static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a) {
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secp256k1_fe r, r2;
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VERIFY_CHECK(!a->infinity);
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secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
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r2 = a->x; secp256k1_fe_normalize_weak(&r2);
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return secp256k1_fe_equal_var(&r, &r2);
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}
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static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *a) {
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r->infinity = a->infinity;
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r->x = a->x;
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r->y = a->y;
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r->z = a->z;
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secp256k1_fe_normalize_weak(&r->y);
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secp256k1_fe_negate(&r->y, &r->y, 1);
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}
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static int secp256k1_gej_is_infinity(const secp256k1_gej *a) {
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return a->infinity;
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}
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static int secp256k1_gej_is_valid_var(const secp256k1_gej *a) {
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secp256k1_fe y2, x3, z2, z6;
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if (a->infinity) {
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return 0;
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}
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/** y^2 = x^3 + 7
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* (Y/Z^3)^2 = (X/Z^2)^3 + 7
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* Y^2 / Z^6 = X^3 / Z^6 + 7
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* Y^2 = X^3 + 7*Z^6
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*/
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secp256k1_fe_sqr(&y2, &a->y);
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secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
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secp256k1_fe_sqr(&z2, &a->z);
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secp256k1_fe_sqr(&z6, &z2); secp256k1_fe_mul(&z6, &z6, &z2);
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secp256k1_fe_mul_int(&z6, CURVE_B);
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secp256k1_fe_add(&x3, &z6);
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secp256k1_fe_normalize_weak(&x3);
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return secp256k1_fe_equal_var(&y2, &x3);
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}
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static int secp256k1_ge_is_valid_var(const secp256k1_ge *a) {
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secp256k1_fe y2, x3, c;
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if (a->infinity) {
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return 0;
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}
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/* y^2 = x^3 + 7 */
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secp256k1_fe_sqr(&y2, &a->y);
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secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
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secp256k1_fe_set_int(&c, CURVE_B);
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secp256k1_fe_add(&x3, &c);
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secp256k1_fe_normalize_weak(&x3);
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return secp256k1_fe_equal_var(&y2, &x3);
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}
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static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
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/* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate.
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*
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* Note that there is an implementation described at
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* https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
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* which trades a multiply for a square, but in practice this is actually slower,
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* mainly because it requires more normalizations.
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*/
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secp256k1_fe t1,t2,t3,t4;
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/** For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity,
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* 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
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* y=0, x^3 must be -7 mod p. However, -7 has no cube root mod p.
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*
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* Having said this, if this function receives a point on a sextic twist, e.g. by
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* a fault attack, it is possible for y to be 0. This happens for y^2 = x^3 + 6,
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* since -6 does have a cube root mod p. For this point, this function will not set
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* the infinity flag even though the point doubles to infinity, and the result
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* point will be gibberish (z = 0 but infinity = 0).
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*/
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r->infinity = a->infinity;
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if (r->infinity) {
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if (rzr != NULL) {
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secp256k1_fe_set_int(rzr, 1);
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}
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return;
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}
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if (rzr != NULL) {
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*rzr = a->y;
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secp256k1_fe_normalize_weak(rzr);
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secp256k1_fe_mul_int(rzr, 2);
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}
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secp256k1_fe_mul(&r->z, &a->z, &a->y);
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secp256k1_fe_mul_int(&r->z, 2); /* Z' = 2*Y*Z (2) */
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secp256k1_fe_sqr(&t1, &a->x);
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secp256k1_fe_mul_int(&t1, 3); /* T1 = 3*X^2 (3) */
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secp256k1_fe_sqr(&t2, &t1); /* T2 = 9*X^4 (1) */
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secp256k1_fe_sqr(&t3, &a->y);
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secp256k1_fe_mul_int(&t3, 2); /* T3 = 2*Y^2 (2) */
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secp256k1_fe_sqr(&t4, &t3);
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secp256k1_fe_mul_int(&t4, 2); /* T4 = 8*Y^4 (2) */
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secp256k1_fe_mul(&t3, &t3, &a->x); /* T3 = 2*X*Y^2 (1) */
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r->x = t3;
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secp256k1_fe_mul_int(&r->x, 4); /* X' = 8*X*Y^2 (4) */
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secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8*X*Y^2 (5) */
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secp256k1_fe_add(&r->x, &t2); /* X' = 9*X^4 - 8*X*Y^2 (6) */
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secp256k1_fe_negate(&t2, &t2, 1); /* T2 = -9*X^4 (2) */
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secp256k1_fe_mul_int(&t3, 6); /* T3 = 12*X*Y^2 (6) */
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secp256k1_fe_add(&t3, &t2); /* T3 = 12*X*Y^2 - 9*X^4 (8) */
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secp256k1_fe_mul(&r->y, &t1, &t3); /* Y' = 36*X^3*Y^2 - 27*X^6 (1) */
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secp256k1_fe_negate(&t2, &t4, 2); /* T2 = -8*Y^4 (3) */
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secp256k1_fe_add(&r->y, &t2); /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
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}
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static SECP256K1_INLINE void secp256k1_gej_double_nonzero(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
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VERIFY_CHECK(!secp256k1_gej_is_infinity(a));
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secp256k1_gej_double_var(r, a, rzr);
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}
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static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr) {
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/* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */
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secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
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if (a->infinity) {
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VERIFY_CHECK(rzr == NULL);
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*r = *b;
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return;
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}
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if (b->infinity) {
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|
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
|