/**
* Method based on FFT to find the determinant and the first coefficient of w(x) this suffices to
* find w(x) completely
*
* @param q the polynomial
* @param n the degree
* @return The determinant and the first coefficient of w(x).
*/
static BigInteger[] gzModZ2(Polynomial q, int n) {
int i;
int N = 1 << n;
Polynomial V = new Polynomial(q.coeffs); // V = q
Polynomial U = new Polynomial(1); // U = 1
U.setCoeff(0, new BigInteger("1"));
Polynomial F = new Polynomial(N); // F(x) = x^N +1
F.setCoeff(0, new BigInteger("1"));
F.setCoeff(N, new BigInteger("1"));
Polynomial V2 = new Polynomial(N);
while (N > 1) {
V2 = new Polynomial(V.coeffs);
for (i = 1; i <= V2.degree; i += 2) { // set V2(x) := V(-x)
V2.coeffs[i] = V2.coeffs[i].negate(); // negate odd coefficients
}
V = Polynomial.mod(Polynomial.mult(V, V2), F); // V := V(x) * V(-x) mod f(x)
U = Polynomial.mod(Polynomial.mult(U, V2), F); // U := U(x) * V(-x) mod f(x)
// Sanity-check: verify that the odd coefficients in V are zero
for (i = 1; i <= V.degree; i += 2)
if (!V.coeffs[i].equals(new BigInteger("0"))) {
return null;
}
// "Compress" the non-zero coefficients of V
for (i = 1; i <= V.degree / 2; i++) V.coeffs[i] = V.coeffs[2 * i];
for (; i <= V.degree; i++) V.coeffs[i] = new BigInteger("0");
V.normalize();
// Set U to the "compressed" ( U(x) + U(-x) ) /2
for (i = 0; i <= U.degree / 2; i++) U.coeffs[i] = U.coeffs[2 * i];
for (; i <= U.degree; i++) U.coeffs[i] = new BigInteger("0");
U.normalize();
// Set N := N/2 and update F accordingly
F.coeffs[N] = new BigInteger("0");
N >>= 1;
F.coeffs[N] = new BigInteger("1");
F.normalize();
}
return new BigInteger[] {V.coeffs[0], U.coeffs[0]};
}
