SQUARE_ROOTS = {3: 189, 5: 12, 15: 225, 17: 53, 51: 51, 85: 15, 255: 3}

def precompute_x(n):

	"""Return a geometric sequence [1, w, w**2, ..., w**(n-1)], where w**n==1.

	This can be done only for certain values of n."""

	w = SQUARE_ROOTS[n]  # primitive N-th square root of 1

	return list(itertools.accumulate([1]+[w]*(n-1), gfmul))

def complex_dft(p):

	"""Quadratic formula from the definition. The basic case in complex numbers."""

	for i in range(N2):

		if N1*i % N2 == 1:

			return i

	raise ValueError("Failed to find an inverse to {0} mod {1}.".format(N1, N2))

	"""https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm"""

	if len(divisors) == 1:

	N = len(p)

	N1 = divisors[0]

	N2 = N//N1

	N1_inv = compute_inverse(N1, N2)

	N2_inv = compute_inverse(N2, N1)

	ys = []

	for n1 in range(N1):  # compute rows

		p_ = [p[(n2*N1+n1*N2) % N] for n2 in range(N2)]

	for k2 in range(N2):  # compute cols

		p_ = [row[k2] for row in ys]

		for (yi, row) in zip(y_, ys):  # update col

			row[k2] = yi

	# remap and output

	res = [0]*N

	for k1 in range(N1):

	def test_complex_prime_fft(self):

		random.seed(1918)

		for divisors in [[3], [2, 3], [3, 5], [3, 5, 17], [2, 3, 5, 7, 11]]:

			n = functools.reduce(operator.mul, divisors)

			coefficients = [random.randint(-128, 127) for i in range(n)]

			b = complex_dft(coefficients)

			all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))

		random.seed(1918)

		x = {i: precompute_x(i) for i in [3, 5, 15, 17]}  # all sets of xs

		for n in [3, 5, 15, 17]:

			coefficients = [random.randint(0, 255) for i in range(n)]

			self.assertEqual(

				dft(coefficients),

				batch_evaluate(coefficients[::-1], x[n])

			)