diff --git a/src/shamira/fft.py b/src/shamira/fft.py
--- a/src/shamira/fft.py
+++ b/src/shamira/fft.py
@@ -1,17 +1,41 @@
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
+ 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."""
+ """Quadratic formula from the definition. In GF256."""
N = len(p)
- w = cmath.exp(-2*math.pi*1j/N) # primitive N-th square root of 1
- x = [0]*N
+ x = precompute_x(N)
y = [0]*N
for k in range(N):
- x[k] = w**k
for n in range(N):
- y[k] += p[n] * x[k]**n
+ y[k] ^= gfmul(p[n], gfpow(x[k], n))
return y
@@ -25,7 +49,7 @@ def compute_inverse(N1, N2):
def prime_fft(p, divisors):
"""https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm"""
if len(divisors) == 1:
- return dft(p)
+ return complex_dft(p)
N = len(p)
N1 = divisors[0]
N2 = N//N1
@@ -39,7 +63,7 @@ def prime_fft(p, divisors):
for k2 in range(N2): # compute cols
p_ = [row[k2] for row in ys]
- y_ = dft(p_)
+ y_ = complex_dft(p_)
for (yi, row) in zip(y_, ys): # update col
row[k2] = yi
@@ -49,58 +73,3 @@ def prime_fft(p, divisors):
for k2 in range(N2):
res[(k1*N2*N2_inv+k2*N1*N1_inv) % N] = ys[k1][k2]
return res
-
-
-# arr = [9,8,5,3,8,4,9,7,0,9,5,6,6,2,4]
-# print(dft(arr))
-# print()
-# print(prime_fft(arr,[3,5]))
-
-"""
-def fft(x):
- N = len(x)
- if N <= 1: return x
- even = fft(x[0::2])
- odd = fft(x[1::2])
- T = [cmath.exp(-2j*math.pi*k/N)*odd[k] for k in range(N//2)]
- return [even[k] + T[k] for k in range(N//2)] + \
- [even[k] - T[k] for k in range(N//2)]
-
-
-def fft_mixed(x, r1, r2):
- A = dft(x)
- for k0 in range(0, r2):
- pass
-
-
-def bit_reverse(x, width):
- y = 0
- for i in range(width):
- y <<= 1
- y |= x&1
- x >>= 1
- return y
-
-
-def bit_reverse_seq(x):
- n = len(x)
- width = round(math.log2(n))
- return [x[bit_reverse(i, width)] for i in range(n)]
-
-
-def fft2(x):
- y = bit_reverse_seq(x)
- n = len(x)
- omegas = [(-2*math.pi*1j/n)**i for i in range(0, n)]
- b = 1
- while b<n:
- for j in range(0, n, 2*b):
- for k in range(0, b):
- alpha = omegas[n*k//(2*b)]
- u = y[j+k]
- t = alpha*y[j+k+b]
- y[j+k] = u+t
- y[j+k+b] = u-t
- b *= 2
- return y
-"""
diff --git a/src/shamira/gf256.py b/src/shamira/gf256.py
--- a/src/shamira/gf256.py
+++ b/src/shamira/gf256.py
@@ -38,6 +38,19 @@ def gfmul(a, b):
return E[t]
+def gfpow(x, k):
+ """Compute x**k."""
+ i = 1
+ res = 1
+ while i <= k:
+ if k&i:
+ res = gfmul(res, x)
+ x = gfmul(x, x)
+ i <<= 1
+
+ return res
+
+
def evaluate(coefs, x):
"""Evaluate polynomial's value at x.
diff --git a/src/shamira/tests/test_fft.py b/src/shamira/tests/test_fft.py
--- a/src/shamira/tests/test_fft.py
+++ b/src/shamira/tests/test_fft.py
@@ -1,19 +1,47 @@
# GNU GPLv3, see LICENSE
+import random
+import functools
+import operator
from unittest import TestCase
+from ..gf256 import evaluate
from ..fft import *
+def batch_evaluate(coefs, xs):
+ return [evaluate(coefs, x) for x in xs]
+
+
class TestFFT(TestCase):
- def test_dft(self):
- self.assertEqual(dft([0]), [0+0j])
- self.assertEqual(dft([1]), [1+0j])
- self.assertEqual(dft([2]), [2+0j])
- all(self.assertAlmostEqual(a, b) for (a, b) in zip(dft([3, 1]), [4+0j, 2+0j]))
- all(self.assertAlmostEqual(a, b) for (a, b) in zip(dft([3, 1, 4]), [8+0j, 0.5+2.59807621j, 0.5-2.59807621j]))
- all(self.assertAlmostEqual(a, b) for (a, b) in zip(dft([3, 1, 4, 1]), [9+0j, -1+0j, 5+0j, -1+0j]))
+ 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(
- dft([3, 1, 4, 1, 5]),
+ 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)
+ b = complex_dft(coefficients)
+ all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))
+
+ def test_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])
+ )
diff --git a/src/shamira/tests/test_gf256.py b/src/shamira/tests/test_gf256.py
--- a/src/shamira/tests/test_gf256.py
+++ b/src/shamira/tests/test_gf256.py
@@ -22,6 +22,26 @@ class TestGF256(TestCase):
for b in range(256):
self.assertEqual(_gfmul(a, b), gfmul(a, b))
+ def test_gfpow(self):
+ self.assertEqual(gfpow(0, 0), 1)
+
+ for i in range(1, 256):
+ self.assertEqual(gfpow(i, 0), 1)
+ self.assertEqual(gfpow(i, 1), i)
+ self.assertEqual(gfpow(0, i), 0)
+ self.assertEqual(gfpow(1, i), 1)
+ self.assertEqual(gfpow(i, 256), i)
+ self.assertEqual(gfpow(i, 2), gfmul(i, i))
+
+ random.seed(1918)
+ for i in range(256):
+ j = random.randint(2, 255)
+ k = random.randint(3, 255)
+ y = 1
+ for m in range(k):
+ y = gfmul(y, j)
+ self.assertEqual(gfpow(j, k), y)
+
def test_evaluate(self):
for x in range(256):
(a0, a1, a2, a3) = (x, x>>1, x>>2, x>>3)