// Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
// (X/Z^2, Y/Z^3).
static int ec_GFp_nistp256_point_get_affine_coordinates(
const EC_GROUP *group, const EC_RAW_POINT *point, EC_FELEM *x_out,
EC_FELEM *y_out) {
if (ec_GFp_simple_is_at_infinity(group, point)) {
OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
return 0;
}
fe z1, z2;
fe_from_generic(z1, &point->Z);
fe_inv(z2, z1);
fe_sqr(z1, z2);
// Instead of using |fe_from_montgomery| to convert the |x| coordinate and
// then calling |fe_from_montgomery| again to convert the |y| coordinate
// below, convert the common factor |z1| once now, saving one reduction.
fe_from_montgomery(z1);
if (x_out != NULL) {
fe x;
fe_from_generic(x, &point->X);
fe_mul(x, x, z1);
fe_to_generic(x_out, x);
}
if (y_out != NULL) {
fe y;
fe_from_generic(y, &point->Y);
fe_mul(z1, z1, z2);
fe_mul(y, y, z1);
fe_to_generic(y_out, y);
}
return 1;
}
extern void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
{
fe_mul(r->X, p->X, p->T);
fe_mul(r->Y, p->Y, p->Z);
fe_mul(r->Z, p->Z, p->T);
fe_mul(r->T, p->X, p->Y);
}
void ge_montx_to_p2(ge_p2* p, const fe u, const unsigned char ed_sign_bit)
{
fe x, y, A, v, v2, iv, nx;
fe_frombytes(A, A_bytes);
/* given u, recover edwards y */
/* given u, recover v */
/* given u and v, recover edwards x */
fe_montx_to_edy(y, u); /* y = (u - 1) / (u + 1) */
fe_mont_rhs(v2, u); /* v^2 = u(u^2 + Au + 1) */
fe_sqrt(v, v2); /* v = sqrt(v^2) */
fe_mul(x, u, A); /* x = u * sqrt(-(A+2)) */
fe_invert(iv, v); /* 1/v */
fe_mul(x, x, iv); /* x = (u/v) * sqrt(-(A+2)) */
fe_neg(nx, x); /* negate x to match sign bit */
fe_cmov(x, nx, fe_isnegative(x) ^ ed_sign_bit);
fe_copy(p->X, x);
fe_copy(p->Y, y);
fe_1(p->Z);
/* POSTCONDITION: check that p->X and p->Y satisfy the Ed curve equation */
/* -x^2 + y^2 = 1 + dx^2y^2 */
#ifndef NDEBUG
{
fe one, d, x2, y2, x2y2, dx2y2;
unsigned char dbytes[32] = {
0xa3, 0x78, 0x59, 0x13, 0xca, 0x4d, 0xeb, 0x75,
0xab, 0xd8, 0x41, 0x41, 0x4d, 0x0a, 0x70, 0x00,
0x98, 0xe8, 0x79, 0x77, 0x79, 0x40, 0xc7, 0x8c,
0x73, 0xfe, 0x6f, 0x2b, 0xee, 0x6c, 0x03, 0x52
};
fe_frombytes(d, dbytes);
fe_1(one);
fe_sq(x2, p->X); /* x^2 */
fe_sq(y2, p->Y); /* y^2 */
fe_mul(dx2y2, x2, y2); /* x^2y^2 */
fe_mul(dx2y2, dx2y2, d); /* dx^2y^2 */
fe_add(dx2y2, dx2y2, one); /* dx^2y^2 + 1 */
fe_neg(x2y2, x2); /* -x^2 */
fe_add(x2y2, x2y2, y2); /* -x^2 + y^2 */
assert(fe_isequal(x2y2, dx2y2));
}
#endif
}
void ge_tobytes(unsigned char *s, const ge_p2 *h)
{
fe recip;
fe x;
fe y;
fe_invert(recip, h->Z);
fe_mul(x, h->X, recip);
fe_mul(y, h->Y, recip);
fe_tobytes(s, y);
s[31] ^= fe_isnegative(x) << 7;
}
void fe_mont_rhs(fe v2, fe u) {
fe A, one;
fe u2, Au, inner;
fe_1(one);
fe_0(A);
A[0] = 486662; /* A = 486662 */
fe_sq(u2, u); /* u^2 */
fe_mul(Au, A, u); /* Au */
fe_add(inner, u2, Au); /* u^2 + Au */
fe_add(inner, inner, one); /* u^2 + Au + 1 */
fe_mul(v2, u, inner); /* u(u^2 + Au + 1) */
}
void curve25519_keygen(unsigned char* curve25519_pubkey_out,
const unsigned char* curve25519_privkey_in)
{
ge_p3 ed; /* Ed25519 pubkey point */
fe ed_y_plus_one, one_minus_ed_y, inv_one_minus_ed_y;
fe mont_x;
/* Perform a fixed-base multiplication of the Edwards base point,
(which is efficient due to precalculated tables), then convert
to the Curve25519 montgomery-format public key. In particular,
convert Curve25519's "montgomery" x-coordinate into an Ed25519
"edwards" y-coordinate:
mont_x = (ed_y + 1) / (1 - ed_y)
with projective coordinates:
mont_x = (ed_y + ed_z) / (ed_z - ed_y)
NOTE: ed_y=1 is converted to mont_x=0 since fe_invert is mod-exp
*/
ge_scalarmult_base(&ed, curve25519_privkey_in);
fe_add(ed_y_plus_one, ed.Y, ed.Z);
fe_sub(one_minus_ed_y, ed.Z, ed.Y);
fe_invert(inv_one_minus_ed_y, one_minus_ed_y);
fe_mul(mont_x, ed_y_plus_one, inv_one_minus_ed_y);
fe_tobytes(curve25519_pubkey_out, mont_x);
}
extern void ge_p3_to_cached(ge_cached *r,const ge_p3 *p)
{
fe_add(r->YplusX,p->Y,p->X);
fe_sub(r->YminusX,p->Y,p->X);
fe_copy(r->Z,p->Z);
fe_mul(r->T2d,p->T,d2);
}
static void
edwards_to_montgomery(fe montgomeryX, const fe edwardsY, const fe edwardsZ)
{
fe tempX;
fe tempZ;
fe_add(tempX, edwardsZ, edwardsY);
fe_sub(tempZ, edwardsZ, edwardsY);
fe_invert(tempZ, tempZ);
fe_mul(montgomeryX, tempX, tempZ);
}
// point_double calculates 2*(x_in, y_in, z_in)
//
// The method is taken from:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
//
// Coq transcription and correctness proof:
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L93>
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L201>
//
// Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
// while x_out == y_in is not (maybe this works, but it's not tested).
static void point_double(fe x_out, fe y_out, fe z_out,
const fe x_in, const fe y_in, const fe z_in) {
fe delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta;
// delta = z^2
fe_sqr(delta, z_in);
// gamma = y^2
fe_sqr(gamma, y_in);
// beta = x*gamma
fe_mul(beta, x_in, gamma);
// alpha = 3*(x-delta)*(x+delta)
fe_sub(ftmp, x_in, delta);
fe_add(ftmp2, x_in, delta);
fe_add(tmptmp, ftmp2, ftmp2);
fe_add(ftmp2, ftmp2, tmptmp);
fe_mul(alpha, ftmp, ftmp2);
// x' = alpha^2 - 8*beta
fe_sqr(x_out, alpha);
fe_add(fourbeta, beta, beta);
fe_add(fourbeta, fourbeta, fourbeta);
fe_add(tmptmp, fourbeta, fourbeta);
fe_sub(x_out, x_out, tmptmp);
// z' = (y + z)^2 - gamma - delta
fe_add(delta, gamma, delta);
fe_add(ftmp, y_in, z_in);
fe_sqr(z_out, ftmp);
fe_sub(z_out, z_out, delta);
// y' = alpha*(4*beta - x') - 8*gamma^2
fe_sub(y_out, fourbeta, x_out);
fe_add(gamma, gamma, gamma);
fe_sqr(gamma, gamma);
fe_mul(y_out, alpha, y_out);
fe_add(gamma, gamma, gamma);
fe_sub(y_out, y_out, gamma);
}
int curve25519_verify(const unsigned char* signature,
const unsigned char* curve25519_pubkey,
const unsigned char* msg, const unsigned long msg_len)
{
fe mont_x, mont_x_minus_one, mont_x_plus_one, inv_mont_x_plus_one;
fe one;
fe ed_y;
unsigned char ed_pubkey[32];
unsigned long long some_retval;
unsigned char verifybuf[MAX_MSG_LEN + 64]; /* working buffer */
unsigned char verifybuf2[MAX_MSG_LEN + 64]; /* working buffer #2 */
if (msg_len > MAX_MSG_LEN) {
return -1;
}
/* Convert the Curve25519 public key into an Ed25519 public key. In
particular, convert Curve25519's "montgomery" x-coordinate into an
Ed25519 "edwards" y-coordinate:
ed_y = (mont_x - 1) / (mont_x + 1)
NOTE: mont_x=-1 is converted to ed_y=0 since fe_invert is mod-exp
Then move the sign bit into the pubkey from the signature.
*/
fe_frombytes(mont_x, curve25519_pubkey);
fe_1(one);
fe_sub(mont_x_minus_one, mont_x, one);
fe_add(mont_x_plus_one, mont_x, one);
fe_invert(inv_mont_x_plus_one, mont_x_plus_one);
fe_mul(ed_y, mont_x_minus_one, inv_mont_x_plus_one);
fe_tobytes(ed_pubkey, ed_y);
/* Copy the sign bit, and remove it from signature */
ed_pubkey[31] &= 0x7F; /* bit should be zero already, but just in case */
ed_pubkey[31] |= (signature[63] & 0x80);
memmove(verifybuf, signature, 64);
verifybuf[63] &= 0x7F;
memmove(verifybuf+64, msg, msg_len);
/* Then perform a normal Ed25519 verification, return 0 on success */
/* The below call has a strange API: */
/* verifybuf = R || S || message */
/* verifybuf2 = internal to next call gets a copy of verifybuf, S gets
replaced with pubkey for hashing, then the whole thing gets zeroized
(if bad sig), or contains a copy of msg (good sig) */
return crypto_sign_open(verifybuf2, &some_retval, verifybuf, 64 + msg_len, ed_pubkey);
}
void fe_sqrt(fe out, const fe a)
{
fe exp, b, b2, bi, i;
#ifndef NDEBUG
fe legendre, zero, one;
#endif
fe_frombytes(i, i_bytes);
fe_pow22523(exp, a); /* b = a^(q-5)/8 */
/* PRECONDITION: legendre symbol == 1 (square) or 0 (a == zero) */
#ifndef NDEBUG
fe_sq(legendre, exp); /* in^((q-5)/4) */
fe_sq(legendre, legendre); /* in^((q-5)/2) */
fe_mul(legendre, legendre, a); /* in^((q-3)/2) */
fe_mul(legendre, legendre, a); /* in^((q-1)/2) */
fe_0(zero);
fe_1(one);
assert(fe_isequal(legendre, zero) || fe_isequal(legendre, one));
#endif
fe_mul(b, a, exp); /* b = a * a^(q-5)/8 */
fe_sq(b2, b); /* b^2 = a * a^(q-1)/4 */
/* note b^4 == a^2, so b^2 == a or -a
* if b^2 != a, multiply it by sqrt(-1) */
fe_mul(bi, b, i);
fe_cmov(b, bi, 1 ^ fe_isequal(b2, a));
fe_copy(out, b);
/* PRECONDITION: out^2 == a */
#ifndef NDEBUG
fe_sq(b2, out);
assert(fe_isequal(a, b2));
#endif
}
void fe_montx_to_edy(fe y, const fe u)
{
/*
y = (u - 1) / (u + 1)
NOTE: u=-1 is converted to y=0 since fe_invert is mod-exp
*/
fe one, um1, up1;
fe_1(one);
fe_sub(um1, u, one);
fe_add(up1, u, one);
fe_invert(up1, up1);
fe_mul(y, um1, up1);
}
static int curve25519(unsigned char* q, unsigned char* n, unsigned char* p)
{
unsigned char e[32];
unsigned int i;
fe x1;
fe x2;
fe z2;
fe x3;
fe z3;
fe tmp0;
fe tmp1;
int pos;
unsigned int swap;
unsigned int b;
for (i = 0;i < 32;++i) e[i] = n[i];
e[0] &= 248;
e[31] &= 127;
e[31] |= 64;
fe_frombytes(x1,p);
fe_1(x2);
fe_0(z2);
fe_copy(x3,x1);
fe_1(z3);
swap = 0;
for (pos = 254;pos >= 0;--pos) {
b = e[pos / 8] >> (pos & 7);
b &= 1;
swap ^= b;
fe_cswap(x2,x3,swap);
fe_cswap(z2,z3,swap);
swap = b;
#include <cyassl/ctaocrypt/ecc25519_montgomery.h>
}
fe_cswap(x2,x3,swap);
fe_cswap(z2,z3,swap);
fe_invert(z2,z2);
fe_mul(x2,x2,z2);
fe_tobytes(q,x2);
return 0;
}
int crypto_sign_ed25519_pk_to_curve25519(unsigned char *curve25519_pk,
const unsigned char *ed25519_pk)
{
ge_p3 A;
fe x;
fe one_minus_y;
ge_frombytes_negate_vartime(&A, ed25519_pk);
fe_1(one_minus_y);
fe_sub(one_minus_y, one_minus_y, A.Y);
fe_invert(one_minus_y, one_minus_y);
fe_1(x);
fe_add(x, x, A.Y);
fe_mul(x, x, one_minus_y);
fe_tobytes(curve25519_pk, x);
return 0;
}
int Sign_publicSigningKeyToCurve25519(uint8_t curve25519keyOut[32], uint8_t publicSigningKey[32])
{
ge_p3 A;
fe x;
fe one_minus_y;
if (ge_frombytes_negate_vartime(&A, publicSigningKey) != 0) {
return -1;
}
fe_1(one_minus_y);
fe_sub(one_minus_y, one_minus_y, A.Y);
fe_invert(one_minus_y, one_minus_y);
fe_1(x);
fe_add(x, x, A.Y);
fe_mul(x, x, one_minus_y);
fe_tobytes(curve25519keyOut, x);
return 0;
}
static int
crypto_scalarmult_curve25519_ref10(unsigned char *q,
const unsigned char *n,
const unsigned char *p)
{
unsigned char e[32];
unsigned int i;
fe x1;
fe x2;
fe z2;
fe x3;
fe z3;
fe tmp0;
fe tmp1;
int pos;
unsigned int swap;
unsigned int b;
for (i = 0;i < 32;++i) e[i] = n[i];
e[0] &= 248;
e[31] &= 127;
e[31] |= 64;
fe_frombytes(x1,p);
fe_1(x2);
fe_0(z2);
fe_copy(x3,x1);
fe_1(z3);
swap = 0;
for (pos = 254;pos >= 0;--pos) {
b = e[pos / 8] >> (pos & 7);
b &= 1;
swap ^= b;
fe_cswap(x2,x3,swap);
fe_cswap(z2,z3,swap);
swap = b;
fe_sub(tmp0,x3,z3);
fe_sub(tmp1,x2,z2);
fe_add(x2,x2,z2);
fe_add(z2,x3,z3);
fe_mul(z3,tmp0,x2);
fe_mul(z2,z2,tmp1);
fe_sq(tmp0,tmp1);
fe_sq(tmp1,x2);
fe_add(x3,z3,z2);
fe_sub(z2,z3,z2);
fe_mul(x2,tmp1,tmp0);
fe_sub(tmp1,tmp1,tmp0);
fe_sq(z2,z2);
fe_mul121666(z3,tmp1);
fe_sq(x3,x3);
fe_add(tmp0,tmp0,z3);
fe_mul(z3,x1,z2);
fe_mul(z2,tmp1,tmp0);
}
fe_cswap(x2,x3,swap);
fe_cswap(z2,z3,swap);
fe_invert(z2,z2);
fe_mul(x2,x2,z2);
fe_tobytes(q,x2);
return 0;
}
void ed25519_key_exchange(unsigned char *shared_secret, const unsigned char *public_key, const unsigned char *private_key) {
unsigned char e[32];
unsigned int i;
fe x1;
fe x2;
fe z2;
fe x3;
fe z3;
fe tmp0;
fe tmp1;
int pos;
unsigned int swap;
unsigned int b;
/* copy the private key and make sure it's valid */
for (i = 0; i < 32; ++i) {
e[i] = private_key[i];
}
e[0] &= 248;
e[31] &= 63;
e[31] |= 64;
/* unpack the public key and convert edwards to montgomery */
/* due to CodesInChaos: montgomeryX = (edwardsY + 1)*inverse(1 - edwardsY) mod p */
fe_frombytes(x1, public_key);
fe_1(tmp1);
fe_add(tmp0, x1, tmp1);
fe_sub(tmp1, tmp1, x1);
fe_invert(tmp1, tmp1);
fe_mul(x1, tmp0, tmp1);
fe_1(x2);
fe_0(z2);
fe_copy(x3, x1);
fe_1(z3);
swap = 0;
for (pos = 254; pos >= 0; --pos) {
b = e[pos / 8] >> (pos & 7);
b &= 1;
swap ^= b;
fe_cswap(x2, x3, swap);
fe_cswap(z2, z3, swap);
swap = b;
/* from montgomery.h */
fe_sub(tmp0, x3, z3);
fe_sub(tmp1, x2, z2);
fe_add(x2, x2, z2);
fe_add(z2, x3, z3);
fe_mul(z3, tmp0, x2);
fe_mul(z2, z2, tmp1);
fe_sq(tmp0, tmp1);
fe_sq(tmp1, x2);
fe_add(x3, z3, z2);
fe_sub(z2, z3, z2);
fe_mul(x2, tmp1, tmp0);
fe_sub(tmp1, tmp1, tmp0);
fe_sq(z2, z2);
fe_mul121666(z3, tmp1);
fe_sq(x3, x3);
fe_add(tmp0, tmp0, z3);
fe_mul(z3, x1, z2);
fe_mul(z2, tmp1, tmp0);
}
fe_cswap(x2, x3, swap);
fe_cswap(z2, z3, swap);
fe_invert(z2, z2);
fe_mul(x2, x2, z2);
fe_tobytes(shared_secret, x2);
}
// fe_inv calculates |out| = |in|^{-1}
//
// Based on Fermat's Little Theorem:
// a^p = a (mod p)
// a^{p-1} = 1 (mod p)
// a^{p-2} = a^{-1} (mod p)
static void fe_inv(fe out, const fe in) {
fe ftmp, ftmp2;
// each e_I will hold |in|^{2^I - 1}
fe e2, e4, e8, e16, e32, e64;
fe_sqr(ftmp, in); // 2^1
fe_mul(ftmp, in, ftmp); // 2^2 - 2^0
fe_copy(e2, ftmp);
fe_sqr(ftmp, ftmp); // 2^3 - 2^1
fe_sqr(ftmp, ftmp); // 2^4 - 2^2
fe_mul(ftmp, ftmp, e2); // 2^4 - 2^0
fe_copy(e4, ftmp);
fe_sqr(ftmp, ftmp); // 2^5 - 2^1
fe_sqr(ftmp, ftmp); // 2^6 - 2^2
fe_sqr(ftmp, ftmp); // 2^7 - 2^3
fe_sqr(ftmp, ftmp); // 2^8 - 2^4
fe_mul(ftmp, ftmp, e4); // 2^8 - 2^0
fe_copy(e8, ftmp);
for (size_t i = 0; i < 8; i++) {
fe_sqr(ftmp, ftmp);
} // 2^16 - 2^8
fe_mul(ftmp, ftmp, e8); // 2^16 - 2^0
fe_copy(e16, ftmp);
for (size_t i = 0; i < 16; i++) {
fe_sqr(ftmp, ftmp);
} // 2^32 - 2^16
fe_mul(ftmp, ftmp, e16); // 2^32 - 2^0
fe_copy(e32, ftmp);
for (size_t i = 0; i < 32; i++) {
fe_sqr(ftmp, ftmp);
} // 2^64 - 2^32
fe_copy(e64, ftmp);
fe_mul(ftmp, ftmp, in); // 2^64 - 2^32 + 2^0
for (size_t i = 0; i < 192; i++) {
fe_sqr(ftmp, ftmp);
} // 2^256 - 2^224 + 2^192
fe_mul(ftmp2, e64, e32); // 2^64 - 2^0
for (size_t i = 0; i < 16; i++) {
fe_sqr(ftmp2, ftmp2);
} // 2^80 - 2^16
fe_mul(ftmp2, ftmp2, e16); // 2^80 - 2^0
for (size_t i = 0; i < 8; i++) {
fe_sqr(ftmp2, ftmp2);
} // 2^88 - 2^8
fe_mul(ftmp2, ftmp2, e8); // 2^88 - 2^0
for (size_t i = 0; i < 4; i++) {
fe_sqr(ftmp2, ftmp2);
} // 2^92 - 2^4
fe_mul(ftmp2, ftmp2, e4); // 2^92 - 2^0
fe_sqr(ftmp2, ftmp2); // 2^93 - 2^1
fe_sqr(ftmp2, ftmp2); // 2^94 - 2^2
fe_mul(ftmp2, ftmp2, e2); // 2^94 - 2^0
fe_sqr(ftmp2, ftmp2); // 2^95 - 2^1
fe_sqr(ftmp2, ftmp2); // 2^96 - 2^2
fe_mul(ftmp2, ftmp2, in); // 2^96 - 3
fe_mul(out, ftmp2, ftmp); // 2^256 - 2^224 + 2^192 + 2^96 - 3
}
// point_add calcuates (x1, y1, z1) + (x2, y2, z2)
//
// The method is taken from:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
// adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
//
// Coq transcription and correctness proof:
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L135>
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L205>
//
// This function includes a branch for checking whether the two input points
// are equal, (while not equal to the point at infinity). This case never
// happens during single point multiplication, so there is no timing leak for
// ECDH or ECDSA signing.
static void point_add(fe x3, fe y3, fe z3, const fe x1,
const fe y1, const fe z1, const int mixed,
const fe x2, const fe y2, const fe z2) {
fe x_out, y_out, z_out;
limb_t z1nz = fe_nz(z1);
limb_t z2nz = fe_nz(z2);
// z1z1 = z1z1 = z1**2
fe z1z1; fe_sqr(z1z1, z1);
fe u1, s1, two_z1z2;
if (!mixed) {
// z2z2 = z2**2
fe z2z2; fe_sqr(z2z2, z2);
// u1 = x1*z2z2
fe_mul(u1, x1, z2z2);
// two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2
fe_add(two_z1z2, z1, z2);
fe_sqr(two_z1z2, two_z1z2);
fe_sub(two_z1z2, two_z1z2, z1z1);
fe_sub(two_z1z2, two_z1z2, z2z2);
// s1 = y1 * z2**3
fe_mul(s1, z2, z2z2);
fe_mul(s1, s1, y1);
} else {
// We'll assume z2 = 1 (special case z2 = 0 is handled later).
// u1 = x1*z2z2
fe_copy(u1, x1);
// two_z1z2 = 2z1z2
fe_add(two_z1z2, z1, z1);
// s1 = y1 * z2**3
fe_copy(s1, y1);
}
// u2 = x2*z1z1
fe u2; fe_mul(u2, x2, z1z1);
// h = u2 - u1
fe h; fe_sub(h, u2, u1);
limb_t xneq = fe_nz(h);
// z_out = two_z1z2 * h
fe_mul(z_out, h, two_z1z2);
// z1z1z1 = z1 * z1z1
fe z1z1z1; fe_mul(z1z1z1, z1, z1z1);
// s2 = y2 * z1**3
fe s2; fe_mul(s2, y2, z1z1z1);
// r = (s2 - s1)*2
fe r;
fe_sub(r, s2, s1);
fe_add(r, r, r);
limb_t yneq = fe_nz(r);
if (!xneq && !yneq && z1nz && z2nz) {
point_double(x3, y3, z3, x1, y1, z1);
return;
}
// I = (2h)**2
fe i;
fe_add(i, h, h);
fe_sqr(i, i);
// J = h * I
fe j; fe_mul(j, h, i);
// V = U1 * I
fe v; fe_mul(v, u1, i);
// x_out = r**2 - J - 2V
fe_sqr(x_out, r);
fe_sub(x_out, x_out, j);
fe_sub(x_out, x_out, v);
fe_sub(x_out, x_out, v);
// y_out = r(V-x_out) - 2 * s1 * J
fe_sub(y_out, v, x_out);
fe_mul(y_out, y_out, r);
fe s1j;
fe_mul(s1j, s1, j);
fe_sub(y_out, y_out, s1j);
fe_sub(y_out, y_out, s1j);
fe_cmovznz(x_out, z1nz, x2, x_out);
fe_cmovznz(x3, z2nz, x1, x_out);
fe_cmovznz(y_out, z1nz, y2, y_out);
fe_cmovznz(y3, z2nz, y1, y_out);
fe_cmovznz(z_out, z1nz, z2, z_out);
fe_cmovznz(z3, z2nz, z1, z_out);
}
