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def _simple_ft(vals, modulus, roots_of_unity):
L = len(roots_of_unity)
o = []
for i in range(L):
last = 0
for j in range(L):
last += vals[j] * roots_of_unity[(i*j)%L]
o.append(last % modulus)
return o
def _fft(vals, modulus, roots_of_unity):
if len(vals) <= 4:
#return vals
return _simple_ft(vals, modulus, roots_of_unity)
L = _fft(vals[::2], modulus, roots_of_unity[::2])
R = _fft(vals[1::2], modulus, roots_of_unity[::2])
o = [0 for i in vals]
for i, (x, y) in enumerate(zip(L, R)):
y_times_root = y*roots_of_unity[i]
o[i] = (x+y_times_root) % modulus
o[i+len(L)] = (x-y_times_root) % modulus
return o
def expand_root_of_unity(root_of_unity, modulus):
# Build up roots of unity
rootz = [1, root_of_unity]
while rootz[-1] != 1:
rootz.append((rootz[-1] * root_of_unity) % modulus)
return rootz
def fft(vals, modulus, root_of_unity, inv=False):
rootz = expand_root_of_unity(root_of_unity, modulus)
# Fill in vals with zeroes if needed
if len(rootz) > len(vals) + 1:
vals = vals + [0] * (len(rootz) - len(vals) - 1)
if inv:
# Inverse FFT
invlen = pow(len(vals), modulus-2, modulus)
return [(x*invlen) % modulus for x in
_fft(vals, modulus, rootz[:0:-1])]
else:
# Regular FFT
return _fft(vals, modulus, rootz[:-1])
# Evaluates f(x) for f in evaluation form
def inv_fft_at_point(vals, modulus, root_of_unity, x):
if len(vals) == 1:
return vals[0]
# 1/2 in the field
half = (modulus + 1)//2
# 1/w
inv_root = pow(root_of_unity, len(vals)-1, modulus)
# f(-x) in evaluation form
f_of_minus_x_vals = vals[len(vals)//2:] + vals[:len(vals)//2]
# e(x) = (f(x) + f(-x)) / 2 in evaluation form
evens = [(f+g) * half % modulus for f,g in zip(vals, f_of_minus_x_vals)]
# o(x) = (f(x) - f(-x)) / 2 in evaluation form
odds = [(f-g) * half % modulus for f,g in zip(vals, f_of_minus_x_vals)]
# e(x^2) + coordinate * x * o(x^2) in evaluation form
comb = [(o * x * inv_root**i + e) % modulus for i, (o, e) in enumerate(zip(odds, evens))]
return inv_fft_at_point(comb[:len(comb)//2], modulus, root_of_unity ** 2 % modulus, x**2 % modulus)
def shift_domain(vals, modulus, root_of_unity, factor):
if len(vals) == 1:
return vals
# 1/2 in the field
half = (modulus + 1)//2
# 1/w
inv_factor = pow(factor, modulus - 2, modulus)
half_length = len(vals)//2
# f(-x) in evaluation form
f_of_minus_x_vals = vals[half_length:] + vals[:half_length]
# e(x) = (f(x) + f(-x)) / 2 in evaluation form
evens = [(f+g) * half % modulus for f,g in zip(vals, f_of_minus_x_vals)]
print('e', evens)
# o(x) = (f(x) - f(-x)) / 2 in evaluation form
odds = [(f-g) * half % modulus for f,g in zip(vals, f_of_minus_x_vals)]
print('o', odds)
shifted_evens = shift_domain(evens[:half_length], modulus, root_of_unity ** 2 % modulus, factor ** 2 % modulus)
print('se', shifted_evens)
shifted_odds = shift_domain(odds[:half_length], modulus, root_of_unity ** 2 % modulus, factor ** 2 % modulus)
print('so', shifted_odds)
return (
[(e + inv_factor * o) % modulus for e, o in zip(shifted_evens, shifted_odds)] +
[(e - inv_factor * o) % modulus for e, o in zip(shifted_evens, shifted_odds)]
)
def shift_poly(poly, modulus, factor):
factor_power = 1
inv_factor = pow(factor, modulus - 2, modulus)
o = []
for p in poly:
o.append(p * factor_power % modulus)
factor_power = factor_power * inv_factor % modulus
return o
def mul_polys(a, b, modulus, root_of_unity):
rootz = [1, root_of_unity]
while rootz[-1] != 1:
rootz.append((rootz[-1] * root_of_unity) % modulus)
if len(rootz) > len(a) + 1:
a = a + [0] * (len(rootz) - len(a) - 1)
if len(rootz) > len(b) + 1:
b = b + [0] * (len(rootz) - len(b) - 1)
x1 = _fft(a, modulus, rootz[:-1])
x2 = _fft(b, modulus, rootz[:-1])
return _fft([(v1*v2)%modulus for v1,v2 in zip(x1,x2)],
modulus, rootz[:0:-1])