def timeit_shoup_poly_F20_zassenhaus():
gf_factor_sqf(F_20, P_18, method='zassenhaus')
def timeit_shoup_poly_F20_shoup():
gf_factor_sqf(F_20, P_18, method='shoup')
def timeit_shoup_poly_F10_zassenhaus():
gf_factor_sqf(F_10, P_08, method='zassenhaus')
def timeit_shoup_poly_F10_shoup():
gf_factor_sqf(F_10, P_08, method='shoup')
def timeit_gathen_poly_f20_zassenhaus():
gf_factor_sqf(f_20, p_20, method='zassenhaus')
def timeit_gathen_poly_f20_shoup():
gf_factor_sqf(f_20, p_20, method='shoup')
def zzx_zassenhaus(f):
"""Factor square-free polynomials over Z[x]. """
n = zzx_degree(f)
if n == 1:
return [f]
A = zzx_max_norm(f)
b = zzx_LC(f)
B = abs(int(sqrt(n + 1) * 2**n * A * b))
C = (n + 1)**(2 * n) * A**(2 * n - 1)
gamma = int(ceil(2 * log(C, 2)))
prime_max = int(2 * gamma * log(gamma))
for p in xrange(3, prime_max + 1):
if not isprime(p) or b % p == 0:
continue
F = gf_from_int_poly(f, p)
if gf_sqf_p(F, p):
break
l = int(ceil(log(2 * B + 1, p)))
modular = []
for ff in gf_factor_sqf(F, p)[1]:
modular.append(gf_to_int_poly(ff, p))
g = zzx_hensel_lift(p, f, modular, l)
T = set(range(len(g)))
factors, s = [], 1
while 2 * s <= len(T):
for S in subsets(T, s):
G, H = [b], [b]
S = set(S)
for i in S:
G = zzx_mul(G, g[i])
for i in T - S:
H = zzx_mul(H, g[i])
G = zzx_trunc(G, p**l)
H = zzx_trunc(H, p**l)
G_norm = zzx_l1_norm(G)
H_norm = zzx_l1_norm(H)
if G_norm * H_norm <= B:
T = T - S
G = zzx_primitive(G)[1]
f = zzx_primitive(H)[1]
factors.append(G)
b = zzx_LC(f)
break
else:
s += 1
return factors + [f]
def test_gf_factor():
assert gf_factor([], 11) == (0, [])
assert gf_factor([1], 11) == (1, [])
assert gf_factor([1,1], 11) == (1, [([1, 1], 1)])
f, p = [1,0,0,1,0], 2
g = (1, [([1, 0], 1),
([1, 1], 1),
([1, 1, 1], 1)])
assert gf_factor(f, p, method='zassenhaus') == g
assert gf_factor(f, p, method='shoup') == g
g = (1, [[1, 0],
[1, 1],
[1, 1, 1]])
assert gf_factor_sqf(f, p, method='zassenhaus') == g
assert gf_factor_sqf(f, p, method='shoup') == g
assert gf_factor([1, 5, 8, 4], 11) == \
(1, [([1, 1], 1), ([1, 2], 2)])
assert gf_factor([1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11) == \
(1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
assert gf_factor(gf_from_dict({32: 1, 0: 1}, 11), 11) == \
(1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
assert gf_factor(gf_from_dict({32: 8, 0: 5}, 11), 11) == \
(8, [([1, 3], 1),
([1, 8], 1),
([1, 0, 9], 1),
([1, 2, 2], 1),
([1, 9, 2], 1),
([1, 0, 5, 0, 7], 1),
([1, 0, 6, 0, 7], 1),
([1, 0, 0, 0, 1, 0, 0, 0, 6], 1),
([1, 0, 0, 0, 10, 0, 0, 0, 6], 1)])
assert gf_factor(gf_from_dict({63: 8, 0: 5}, 11), 11) == \
(8, [([1, 7], 1),
([1, 4, 5], 1),
([1, 6, 8, 2], 1),
([1, 9, 9, 2], 1),
([1, 0, 0, 9, 0, 0, 4], 1),
([1, 2, 0, 8, 4, 6, 4], 1),
([1, 2, 3, 8, 0, 6, 4], 1),
([1, 2, 6, 0, 8, 4, 4], 1),
([1, 3, 3, 1, 6, 8, 4], 1),
([1, 5, 6, 0, 8, 6, 4], 1),
([1, 6, 2, 7, 9, 8, 4], 1),
([1, 10, 4, 7, 10, 7, 4], 1),
([1, 10, 10, 1, 4, 9, 4], 1)])
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
p = nextprime(int((2**15 * pi).evalf()))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p)
assert gf_sqf_p(f, p) == True
g = (1, [([1, 22730, 68144], 1),
([1, 81553, 77449, 86810, 4724], 1),
([1, 86276, 56779, 14859, 31575], 1),
([1, 15347, 95022, 84569, 94508, 92335], 1)])
assert gf_factor(f, p, method='zassenhaus') == g
assert gf_factor(f, p, method='shoup') == g
g = (1, [[1, 22730, 68144],
[1, 81553, 77449, 86810, 4724],
[1, 86276, 56779, 14859, 31575],
[1, 15347, 95022, 84569, 94508, 92335]])
assert gf_factor_sqf(f, p, method='zassenhaus') == g
assert gf_factor_sqf(f, p, method='shoup') == g
# Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
# (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
p = nextprime(int((2**4 * pi).evalf()))
f = [1, 2, 5, 26, 41, 39, 38] # deg(f) = 6
assert gf_sqf_p(f, p) == True
g = (1, [([1, 44, 26], 1),
([1, 11, 25, 18, 30], 1)])
assert gf_factor(f, p, method='zassenhaus') == g
assert gf_factor(f, p, method='shoup') == g
g = (1, [[1, 44, 26],
[1, 11, 25, 18, 30]])
assert gf_factor_sqf(f, p, method='zassenhaus') == g
assert gf_factor_sqf(f, p, method='shoup') == g
def timeit_shoup_poly_F20_zassenhaus():
gf_factor_sqf(F_20, P_18, method='zassenhaus')
def timeit_shoup_poly_F20_shoup():
gf_factor_sqf(F_20, P_18, method='shoup')
def timeit_shoup_poly_F10_shoup():
gf_factor_sqf(F_10, P_08, method='shoup')
def timeit_shoup_poly_F10_zassenhaus():
gf_factor_sqf(F_10, P_08, method='zassenhaus')
def timeit_gathen_poly_f20_shoup():
gf_factor_sqf(f_20, p_20, method='shoup')
def timeit_gathen_poly_f20_zassenhaus():
gf_factor_sqf(f_20, p_20, method='zassenhaus')
def zzx_zassenhaus(f):
"""Factor square-free polynomials over Z[x]. """
n = zzx_degree(f)
if n == 1:
return [f]
A = zzx_max_norm(f)
b = zzx_LC(f)
B = abs(int(sqrt(n+1)*2**n*A*b))
C = (n+1)**(2*n)*A**(2*n-1)
gamma = int(ceil(2*log(C, 2)))
prime_max = int(2*gamma*log(gamma))
for p in xrange(3, prime_max+1):
if not isprime(p) or b % p == 0:
continue
F = gf_from_int_poly(f, p)
if gf_sqf_p(F, p):
break
l = int(ceil(log(2*B + 1, p)))
modular = []
for ff in gf_factor_sqf(F, p)[1]:
modular.append(gf_to_int_poly(ff, p))
g = zzx_hensel_lift(p, f, modular, l)
T = set(range(len(g)))
factors, s = [], 1
while 2*s <= len(T):
for S in subsets(T, s):
G, H = [b], [b]
S = set(S)
for i in S:
G = zzx_mul(G, g[i])
for i in T-S:
H = zzx_mul(H, g[i])
G = zzx_trunc(G, p**l)
H = zzx_trunc(H, p**l)
G_norm = zzx_l1_norm(G)
H_norm = zzx_l1_norm(H)
if G_norm*H_norm <= B:
T = T - S
G = zzx_primitive(G)[1]
f = zzx_primitive(H)[1]
factors.append(G)
b = zzx_LC(f)
break
else:
s += 1
return factors + [f]
