Changeset - 25c5d4c877c6
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Laman - 4 years ago 2020-11-29 21:10:38

dft pluggable into prime_fft
2 files changed with 16 insertions and 7 deletions:
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src/shamira/fft.py
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import math
 
import cmath
 
import itertools
 

	
 
from .gf256 import gfmul, gfpow
 

	
 
# values of n-th square roots
 
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."""
 
	assert n in SQUARE_ROOTS
 
	assert n in SQUARE_ROOTS, 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."""
 
	N = len(p)
 
	w = cmath.exp(-2*math.pi*1j/N)  # primitive N-th square root of 1
 
	y = [0]*N
 
	for k in range(N):
 
		xk = w**k
 
		for n in range(N):
 
			y[k] += p[n] * xk**n
 
	return y
 

	
 

	
 
def dft(p):
 
	"""Quadratic formula from the definition. In GF256."""
 
	N = len(p)
 
	x = precompute_x(N)
 
	y = [0]*N
 
	for k in range(N):
 
		for n in range(N):
 
			y[k] ^= gfmul(p[n], gfpow(x[k], n))
 
	return y
 

	
 

	
 
def compute_inverse(N1, N2):
 
	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))
 

	
 

	
 
def prime_fft(p, divisors):
 
def prime_fft(p, divisors, basic_dft=dft):
 
	"""https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm"""
 
	if len(divisors) == 1:
 
		return complex_dft(p)
 
		return basic_dft(p)
 
	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)]
 
		ys.append(prime_fft(p_, divisors[1:]))
 
		ys.append(prime_fft(p_, divisors[1:], basic_dft))
 

	
 
	for k2 in range(N2):  # compute cols
 
		p_ = [row[k2] for row in ys]
 
		y_ = complex_dft(p_)
 
		y_ = basic_dft(p_)
 
		for (yi, row) in zip(y_, ys):  # update col
 
			row[k2] = yi
 

	
 
	# remap and output
 
	res = [0]*N
 
	for k1 in range(N1):
 
		for k2 in range(N2):
 
			res[(k1*N2*N2_inv+k2*N1*N1_inv) % N] = ys[k1][k2]
 
	return res
src/shamira/tests/test_fft.py
Show inline comments
 
@@ -10,38 +10,47 @@ from ..fft import *
 

	
 

	
 
def batch_evaluate(coefs, xs):
 
	return [evaluate(coefs, x) for x in xs]
 

	
 

	
 
class TestFFT(TestCase):
 
	def test_complex_dft(self):
 
		self.assertEqual(complex_dft([0]), [0+0j])
 
		self.assertEqual(complex_dft([1]), [1+0j])
 
		self.assertEqual(complex_dft([2]), [2+0j])
 
		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1]), [4+0j, 2+0j]))
 
		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1, 4]), [8+0j, 0.5+2.59807621j, 0.5-2.59807621j]))
 
		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1, 4, 1]), [9+0j, -1+0j, 5+0j, -1+0j]))
 
		all(self.assertAlmostEqual(a, b) for (a, b) in zip(
 
			complex_dft([3, 1, 4, 1, 5]),
 
			[14+0j, 0.80901699+2.04087031j, -0.30901699+5.20431056j, -0.30901699-5.20431056j, 0.80901699-2.04087031j]
 
		))
 

	
 
	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)]
 
			a = prime_fft(coefficients, divisors)
 
			a = prime_fft(coefficients, divisors, complex_dft)
 
			b = complex_dft(coefficients)
 
			all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))
 

	
 
	def test_dft(self):
 
	def test_finite_dft(self):
 
		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])
 
			)
 

	
 
	def test_finite_prime_fft(self):
 
		random.seed(1918)
 
		for divisors in [[3], [3, 5], [3, 17], [5, 17], [3, 5, 17]]:
 
			n = functools.reduce(operator.mul, divisors)
 
			coefficients = [random.randint(0, 255) for i in range(n)]
 
			a = prime_fft(coefficients, divisors)
 
			b = dft(coefficients)
 
			all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))
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