def dup_zz_hensel_lift(p, f, f_list, l, K):
"""
Multifactor Hensel lifting in `Z[x]`.
Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
over `Z[x]` satisfying::
f = lc(f) f_1 ... f_r (mod p)
and a positive integer `l`, returns a list of monic polynomials
`F_1`, `F_2`, ..., `F_r` satisfying::
f = lc(f) F_1 ... F_r (mod p**l)
F_i = f_i (mod p), i = 1..r
References
==========
1. [Gathen99]_
"""
r = len(f_list)
lc = dup_LC(f, K)
if r == 1:
F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
return [ dup_trunc(F, p**l, K) ]
m = p
k = r // 2
d = int(_ceil(_log(l, 2)))
g = gf_from_int_poly([lc], p)
for f_i in f_list[:k]:
g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)
h = gf_from_int_poly(f_list[k], p)
for f_i in f_list[k + 1:]:
h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)
s, t, _ = gf_gcdex(g, h, p, K)
g = gf_to_int_poly(g, p)
h = gf_to_int_poly(h, p)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
for _ in range(1, d + 1):
(g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2
return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
+ dup_zz_hensel_lift(p, h, f_list[k:], l, K)
def dup_zz_hensel_lift(p, f, f_list, l, K):
"""
Multifactor Hensel lifting in `Z[x]`.
Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
over `Z[x]` satisfying::
f = lc(f) f_1 ... f_r (mod p)
and a positive integer `l`, returns a list of monic polynomials
`F_1`, `F_2`, ..., `F_r` satisfying::
f = lc(f) F_1 ... F_r (mod p**l)
F_i = f_i (mod p), i = 1..r
References
==========
1. [Gathen99]_
"""
r = len(f_list)
lc = dup_LC(f, K)
if r == 1:
F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
return [dup_trunc(F, p**l, K)]
m = p
k = r // 2
d = int(_ceil(_log(l, 2)))
g = gf_from_int_poly([lc], p)
for f_i in f_list[:k]:
g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)
h = gf_from_int_poly(f_list[k], p)
for f_i in f_list[k + 1:]:
h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)
s, t, _ = gf_gcdex(g, h, p, K)
g = gf_to_int_poly(g, p)
h = gf_to_int_poly(h, p)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
for _ in range(1, d + 1):
(g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2
return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
+ dup_zz_hensel_lift(p, h, f_list[k:], l, K)
def test_gf_berlekamp():
f = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11)
Q = [[1, 0, 0, 0, 0, 0], [3, 5, 8, 8, 6, 5], [3, 6, 6, 1, 10, 0],
[9, 4, 10, 3, 7, 9], [7, 8, 10, 0, 0, 8], [8, 10, 7, 8, 10, 8]]
V = [[1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0], [0, 0, 7, 9, 0, 1]]
assert gf_Qmatrix(f, 11, ZZ) == Q
assert gf_Qbasis(Q, 11, ZZ) == V
assert gf_berlekamp(f, 11, ZZ) == \
[[1, 1], [1, 5, 3], [1, 2, 3, 4]]
f = [1, 0, 1, 0, 10, 10, 8, 2, 8]
Q = [[1, 0, 0, 0, 0, 0, 0, 0], [2, 1, 7, 11, 10, 12, 5, 11],
[3, 6, 4, 3, 0, 4, 7, 2], [4, 3, 6, 5, 1, 6, 2, 3],
[2, 11, 8, 8, 3, 1, 3, 11], [6, 11, 8, 6, 2, 7, 10, 9],
[5, 11, 7, 10, 0, 11, 7, 12], [3, 3, 12, 5, 0, 11, 9, 12]]
V = [[1, 0, 0, 0, 0, 0, 0, 0], [0, 5, 5, 0, 9, 5, 1, 0],
[0, 9, 11, 9, 10, 12, 0, 1]]
assert gf_Qmatrix(f, 13, ZZ) == Q
assert gf_Qbasis(Q, 13, ZZ) == V
assert gf_berlekamp(f, 13, ZZ) == \
[[1, 3], [1, 8, 4, 12], [1, 2, 3, 4, 6]]
def dup_zz_diophantine(F, m, p, K):
"""Wang/EEZ: Solve univariate Diophantine equations. """
if len(F) == 2:
a, b = F
f = gf_from_int_poly(a, p)
g = gf_from_int_poly(b, p)
s, t, G = gf_gcdex(g, f, p, K)
s = gf_lshift(s, m, K)
t = gf_lshift(t, m, K)
q, s = gf_div(s, f, p, K)
t = gf_add_mul(t, q, g, p, K)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
result = [s, t]
else:
G = [F[-1]]
for f in reversed(F[1:-1]):
G.insert(0, dup_mul(f, G[0], K))
S, T = [], [[1]]
for f, g in zip(F, G):
t, s = dmp_zz_diophantine([g, f], T[-1], [], 0, p, 1, K)
T.append(t)
S.append(s)
result, S = [], S + [T[-1]]
for s, f in zip(S, F):
s = gf_from_int_poly(s, p)
f = gf_from_int_poly(f, p)
r = gf_rem(gf_lshift(s, m, K), f, p, K)
s = gf_to_int_poly(r, p)
result.append(s)
return result
def dup_zz_diophantine(F, m, p, K):
"""Wang/EEZ: Solve univariate Diophantine equations. """
if len(F) == 2:
a, b = F
f = gf_from_int_poly(a, p)
g = gf_from_int_poly(b, p)
s, t, G = gf_gcdex(g, f, p, K)
s = gf_lshift(s, m, K)
t = gf_lshift(t, m, K)
q, s = gf_div(s, f, p, K)
t = gf_add_mul(t, q, g, p, K)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
result = [s, t]
else:
G = [F[-1]]
for f in reversed(F[1:-1]):
G.insert(0, dup_mul(f, G[0], K))
S, T = [], [[1]]
for f, g in zip(F, G):
t, s = dmp_zz_diophantine([g, f], T[-1], [], 0, p, 1, K)
T.append(t)
S.append(s)
result, S = [], S + [T[-1]]
for s, f in zip(S, F):
s = gf_from_int_poly(s, p)
f = gf_from_int_poly(f, p)
r = gf_rem(gf_lshift(s, m, K), f, p, K)
s = gf_to_int_poly(r, p)
result.append(s)
return result
def applyModulus(self):
coeffs = gf_from_int_poly(
self.coeffs[::-1], self.p
)[::
-1] # Yep, reversing the list twice is faster than applying the modulus manually
#print "after:", self.coeffs
for i in range(len(coeffs))[::-1]:
if coeffs[i] != 0:
self.coeffs = coeffs[:i + 1]
return
#self.coeffs = [self.coeffs[0]] # In this case, we don't care if c[0] == 0, we need at least one element.
if len(coeffs) == 0:
self.coeffs = [0]
def test_gf_berlekamp():
f = gf_from_int_poly([1,-3,1,-3,-1,-3,1], 11)
Q = [[1, 0, 0, 0, 0, 0],
[3, 5, 8, 8, 6, 5],
[3, 6, 6, 1,10, 0],
[9, 4,10, 3, 7, 9],
[7, 8,10, 0, 0, 8],
[8,10, 7, 8,10, 8]]
V = [[1, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 1, 0],
[0, 0, 7, 9, 0, 1]]
assert gf_Qmatrix(f, 11, ZZ) == Q
assert gf_Qbasis(Q, 11, ZZ) == V
assert gf_berlekamp(f, 11, ZZ) == \
[[1, 1], [1, 5, 3], [1, 2, 3, 4]]
f = [1,0,1,0,10,10,8,2,8]
Q = [[1, 0, 0, 0, 0, 0, 0, 0],
[2, 1, 7,11,10,12, 5,11],
[3, 6, 4, 3, 0, 4, 7, 2],
[4, 3, 6, 5, 1, 6, 2, 3],
[2,11, 8, 8, 3, 1, 3,11],
[6,11, 8, 6, 2, 7,10, 9],
[5,11, 7,10, 0,11, 7,12],
[3, 3,12, 5, 0,11, 9,12]]
V = [[1, 0, 0, 0, 0, 0, 0, 0],
[0, 5, 5, 0, 9, 5, 1, 0],
[0, 9,11, 9,10,12, 0, 1]]
assert gf_Qmatrix(f, 13, ZZ) == Q
assert gf_Qbasis(Q, 13, ZZ) == V
assert gf_berlekamp(f, 13, ZZ) == \
[[1, 3], [1, 8, 4, 12], [1, 2, 3, 4, 6]]
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1)) * 2**n * A * b))
C = int((n + 1)**(2 * n) * A**(2 * n - 1))
gamma = int(_ceil(2 * _log(C, 2)))
bound = int(2 * gamma * _log(gamma))
for p in xrange(3, bound + 1):
if not isprime(p) or b % p == 0:
continue
p = K.convert(p)
F = gf_from_int_poly(f, p)
if gf_sqf_p(F, p, K):
break
l = int(_ceil(_log(2 * B + 1, p)))
modular = []
for ff in gf_factor_sqf(F, p, K)[1]:
modular.append(gf_to_int_poly(ff, p))
g = dup_zz_hensel_lift(p, f, modular, l, K)
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 = dup_mul(G, g[i], K)
for i in T - S:
H = dup_mul(H, g[i], K)
G = dup_trunc(G, p**l, K)
H = dup_trunc(H, p**l, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm * H_norm <= B:
T = T - S
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
fc = f[-1]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1)) * 2**n * A * b))
C = int((n + 1)**(2 * n) * A**(2 * n - 1))
gamma = int(_ceil(2 * _log(C, 2)))
bound = int(2 * gamma * _log(gamma))
a = []
# choose a prime number `p` such that `f` be square free in Z_p
# if there are many factors in Z_p, choose among a few different `p`
# the one with fewer factors
for px in xrange(3, bound + 1):
if not isprime(px) or b % px == 0:
continue
px = K.convert(px)
F = gf_from_int_poly(f, px)
if not gf_sqf_p(F, px, K):
continue
fsqfx = gf_factor_sqf(F, px, K)[1]
a.append((px, fsqfx))
if len(fsqfx) < 15 or len(a) > 4:
break
p, fsqf = min(a, key=lambda x: len(x[1]))
l = int(_ceil(_log(2 * B + 1, p)))
modular = [gf_to_int_poly(ff, p) for ff in fsqf]
g = dup_zz_hensel_lift(p, f, modular, l, K)
sorted_T = range(len(g))
T = set(sorted_T)
factors, s = [], 1
pl = p**l
while 2 * s <= len(T):
for S in subsets(sorted_T, s):
# lift the constant coefficient of the product `G` of the factors
# in the subset `S`; if it is does not divide `fc`, `G` does
# not divide the input polynomial
if b == 1:
q = 1
for i in S:
q = q * g[i][-1]
q = q % pl
if not _test_pl(fc, q, pl):
continue
else:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
G1 = dup_primitive(G, K)[1]
q = G1[-1]
if q and fc % q != 0:
continue
H = [b]
S = set(S)
T_S = T - S
if b == 1:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
for i in T_S:
H = dup_mul(H, g[i], K)
H = dup_trunc(H, pl, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm * H_norm <= B:
T = T_S
sorted_T = [i for i in sorted_T if i not in S]
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n+1))*2**n*A*b))
C = int((n+1)**(2*n)*A**(2*n-1))
gamma = int(ceil(2*log(C, 2)))
bound = int(2*gamma*log(gamma))
for p in xrange(3, bound+1):
if not isprime(p) or b % p == 0:
continue
p = K.convert(p)
F = gf_from_int_poly(f, p)
if gf_sqf_p(F, p, K):
break
l = int(ceil(log(2*B + 1, p)))
modular = []
for ff in gf_factor_sqf(F, p, K)[1]:
modular.append(gf_to_int_poly(ff, p))
g = dup_zz_hensel_lift(p, f, modular, l, K)
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 = dup_mul(G, g[i], K)
for i in T-S:
H = dup_mul(H, g[i], K)
G = dup_trunc(G, p**l, K)
H = dup_trunc(H, p**l, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm*H_norm <= B:
T = T - S
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
def test_gf_factor():
assert gf_factor([], 11, ZZ) == (0, [])
assert gf_factor([1], 11, ZZ) == (1, [])
assert gf_factor([1,1], 11, ZZ) == (1, [([1, 1], 1)])
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
f, p = [1,0,0,1,0], 2
g = (1, [([1, 0], 1),
([1, 1], 1),
([1, 1, 1], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 0],
[1, 1],
[1, 1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
f, p = gf_from_int_poly([1,-3,1,-3,-1,-3,1], 11), 11
g = (1, [([1, 1], 1),
([1, 5, 3], 1),
([1, 2, 3, 4], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 5, 8, 4], 11
g = (1, [([1, 1], 1), ([1, 2], 2)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11
g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11
g = (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)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: 8, 0: 5}, 11, ZZ), 11
g = (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)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({63: 8, 0: 5}, 11, ZZ), 11
g = (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)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
assert gf_sqf_p(f, p, ZZ) == 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)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 22730, 68144],
[1, 81553, 77449, 86810, 4724],
[1, 86276, 56779, 14859, 31575],
[1, 15347, 95022, 84569, 94508, 92335]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == 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 = ZZ(nextprime(int((2**4 * pi).evalf())))
f = [1, 2, 5, 26, 41, 39, 38]
assert gf_sqf_p(f, p, ZZ) == True
g = (1, [([1, 44, 26], 1),
([1, 11, 25, 18, 30], 1)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 44, 26],
[1, 11, 25, 18, 30]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'other')
raises(KeyError, lambda: gf_factor([1,1], 11, ZZ))
config.setup('GF_FACTOR_METHOD')
def test_gf_from_to_int_poly():
assert gf_from_int_poly([1,0,7,2,20], 5) == [1,0,2,2,0]
assert gf_to_int_poly([1,0,4,2,3], 5) == [1,0,-1,2,-2]
assert gf_to_int_poly([10], 11, symmetric=True) == [-1]
assert gf_to_int_poly([10], 11, symmetric=False) == [10]
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
fc = f[-1]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1))*2**n*A*b))
C = int((n + 1)**(2*n)*A**(2*n - 1))
gamma = int(_ceil(2*_log(C, 2)))
bound = int(2*gamma*_log(gamma))
a = []
# choose a prime number `p` such that `f` be square free in Z_p
# if there are many factors in Z_p, choose among a few different `p`
# the one with fewer factors
for px in range(3, bound + 1):
if not isprime(px) or b % px == 0:
continue
px = K.convert(px)
F = gf_from_int_poly(f, px)
if not gf_sqf_p(F, px, K):
continue
fsqfx = gf_factor_sqf(F, px, K)[1]
a.append((px, fsqfx))
if len(fsqfx) < 15 or len(a) > 4:
break
p, fsqf = min(a, key=lambda x: len(x[1]))
l = int(_ceil(_log(2*B + 1, p)))
modular = [gf_to_int_poly(ff, p) for ff in fsqf]
g = dup_zz_hensel_lift(p, f, modular, l, K)
sorted_T = range(len(g))
T = set(sorted_T)
factors, s = [], 1
pl = p**l
while 2*s <= len(T):
for S in subsets(sorted_T, s):
# lift the constant coefficient of the product `G` of the factors
# in the subset `S`; if it is does not divide `fc`, `G` does
# not divide the input polynomial
if b == 1:
q = 1
for i in S:
q = q*g[i][-1]
q = q % pl
if not _test_pl(fc, q, pl):
continue
else:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
G = dup_primitive(G, K)[1]
q = G[-1]
if q and fc % q != 0:
continue
H = [b]
S = set(S)
T_S = T - S
if b == 1:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
for i in T_S:
H = dup_mul(H, g[i], K)
H = dup_trunc(H, pl, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm*H_norm <= B:
T = T_S
sorted_T = [i for i in sorted_T if i not in S]
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]
