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 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 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
}
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);
}
