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_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
}