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_irreducible_p():
assert gf_irred_p_ben_or([7], 11, ZZ) is True
assert gf_irred_p_ben_or([7, 3], 11, ZZ) is True
assert gf_irred_p_ben_or([7, 3, 1], 11, ZZ) is False
assert gf_irred_p_rabin([7], 11, ZZ) is True
assert gf_irred_p_rabin([7, 3], 11, ZZ) is True
assert gf_irred_p_rabin([7, 3, 1], 11, ZZ) is False
assert gf_irred_p_ben_or([2, 3, 4, 5, 6], 13, ZZ) is False
assert gf_irred_p_ben_or([2, 3, 4, 5, 8], 13, ZZ) is True
config.setup('GF_IRRED_METHOD', 'ben-or')
assert gf_irreducible_p([7], 11, ZZ) is True
assert gf_irreducible_p([7, 3], 11, ZZ) is True
assert gf_irreducible_p([7, 3, 1], 11, ZZ) is False
config.setup('GF_IRRED_METHOD', 'rabin')
assert gf_irreducible_p([7], 11, ZZ) is True
assert gf_irreducible_p([7, 3], 11, ZZ) is True
assert gf_irreducible_p([7, 3, 1], 11, ZZ) is False
config.setup('GF_IRRED_METHOD', 'other')
pytest.raises(KeyError, lambda: gf_irreducible_p([7], 11, ZZ))
config.setup('GF_IRRED_METHOD')
f = [1, 9, 9, 13, 16, 15, 6, 7, 7, 7, 10]
g = [1, 7, 16, 7, 15, 13, 13, 11, 16, 10, 9]
h = gf_mul(f, g, 17, ZZ)
assert gf_irred_p_ben_or(f, 17, ZZ) is True
assert gf_irred_p_ben_or(g, 17, ZZ) is True
assert gf_irred_p_ben_or(h, 17, ZZ) is False
assert gf_irred_p_rabin(f, 17, ZZ) is True
assert gf_irred_p_rabin(g, 17, ZZ) is True
assert gf_irred_p_rabin(h, 17, ZZ) is False
def test_gf_irreducible_p():
assert gf_irred_p_ben_or(ZZ.map([7]), 11, ZZ) is True
assert gf_irred_p_ben_or(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irred_p_ben_or(ZZ.map([7, 3, 1]), 11, ZZ) is False
assert gf_irred_p_rabin(ZZ.map([7]), 11, ZZ) is True
assert gf_irred_p_rabin(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irred_p_rabin(ZZ.map([7, 3, 1]), 11, ZZ) is False
assert gf_irred_p_ben_or(ZZ.map([2, 3, 4, 5, 6]), 13, ZZ) is False
assert gf_irred_p_ben_or(ZZ.map([2, 3, 4, 5, 8]), 13, ZZ) is True
config.setup('GF_IRRED_METHOD', 'ben-or')
assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False
config.setup('GF_IRRED_METHOD', 'rabin')
assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False
config.setup('GF_IRRED_METHOD', 'other')
pytest.raises(KeyError, lambda: gf_irreducible_p([7], 11, ZZ))
config.setup('GF_IRRED_METHOD')
f = ZZ.map([1, 9, 9, 13, 16, 15, 6, 7, 7, 7, 10])
g = ZZ.map([1, 7, 16, 7, 15, 13, 13, 11, 16, 10, 9])
h = gf_mul(f, g, 17, ZZ)
assert gf_irred_p_ben_or(f, 17, ZZ) is True
assert gf_irred_p_ben_or(g, 17, ZZ) is True
assert gf_irred_p_ben_or(h, 17, ZZ) is False
assert gf_irred_p_rabin(f, 17, ZZ) is True
assert gf_irred_p_rabin(g, 17, ZZ) is True
assert gf_irred_p_rabin(h, 17, ZZ) is False
def test_gf_irreducible_p():
assert gf_irred_p_ben_or([7], 11, ZZ) is True
assert gf_irred_p_ben_or([7, 3], 11, ZZ) is True
assert gf_irred_p_ben_or([7, 3, 1], 11, ZZ) is False
assert gf_irred_p_rabin([7], 11, ZZ) is True
assert gf_irred_p_rabin([7, 3], 11, ZZ) is True
assert gf_irred_p_rabin([7, 3, 1], 11, ZZ) is False
assert gf_irred_p_ben_or([2, 3, 4, 5, 6], 13, ZZ) is False
assert gf_irred_p_ben_or([2, 3, 4, 5, 8], 13, ZZ) is True
with config.using(gf_irred_method='ben-or'):
assert gf_irreducible_p([7], 11, ZZ) is True
assert gf_irreducible_p([7, 3], 11, ZZ) is True
assert gf_irreducible_p([7, 3, 1], 11, ZZ) is False
with config.using(gf_irred_method='rabin'):
assert gf_irreducible_p([7], 11, ZZ) is True
assert gf_irreducible_p([7, 3], 11, ZZ) is True
assert gf_irreducible_p([7, 3, 1], 11, ZZ) is False
with config.using(gf_irred_method='other'):
pytest.raises(KeyError, lambda: gf_irreducible_p([7], 11, ZZ))
f = [1, 9, 9, 13, 16, 15, 6, 7, 7, 7, 10]
g = [1, 7, 16, 7, 15, 13, 13, 11, 16, 10, 9]
h = gf_mul(f, g, 17, ZZ)
assert gf_irred_p_ben_or(f, 17, ZZ) is True
assert gf_irred_p_ben_or(g, 17, ZZ) is True
assert gf_irred_p_ben_or(h, 17, ZZ) is False
assert gf_irred_p_rabin(f, 17, ZZ) is True
assert gf_irred_p_rabin(g, 17, ZZ) is True
assert gf_irred_p_rabin(h, 17, ZZ) is False
def test_gf_arith():
assert gf_neg([], 11, ZZ) == []
assert gf_neg([1], 11, ZZ) == [10]
assert gf_neg([1, 2, 3], 11, ZZ) == [10, 9, 8]
assert gf_add_ground([], 0, 11, ZZ) == []
assert gf_sub_ground([], 0, 11, ZZ) == []
assert gf_add_ground([], 3, 11, ZZ) == [3]
assert gf_sub_ground([], 3, 11, ZZ) == [8]
assert gf_add_ground([1], 3, 11, ZZ) == [4]
assert gf_sub_ground([1], 3, 11, ZZ) == [9]
assert gf_add_ground([8], 3, 11, ZZ) == []
assert gf_sub_ground([3], 3, 11, ZZ) == []
assert gf_add_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 6]
assert gf_sub_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 0]
assert gf_mul_ground([], 0, 11, ZZ) == []
assert gf_mul_ground([], 1, 11, ZZ) == []
assert gf_mul_ground([1], 0, 11, ZZ) == []
assert gf_mul_ground([1], 1, 11, ZZ) == [1]
assert gf_mul_ground([1, 2, 3], 0, 11, ZZ) == []
assert gf_mul_ground([1, 2, 3], 1, 11, ZZ) == [1, 2, 3]
assert gf_mul_ground([1, 2, 3], 7, 11, ZZ) == [7, 3, 10]
assert gf_add([], [], 11, ZZ) == []
assert gf_add([1], [], 11, ZZ) == [1]
assert gf_add([], [1], 11, ZZ) == [1]
assert gf_add([1], [1], 11, ZZ) == [2]
assert gf_add([1], [2], 11, ZZ) == [3]
assert gf_add([1, 2], [1], 11, ZZ) == [1, 3]
assert gf_add([1], [1, 2], 11, ZZ) == [1, 3]
assert gf_add([1, 2, 3], [8, 9, 10], 11, ZZ) == [9, 0, 2]
assert gf_sub([], [], 11, ZZ) == []
assert gf_sub([1], [], 11, ZZ) == [1]
assert gf_sub([], [1], 11, ZZ) == [10]
assert gf_sub([1], [1], 11, ZZ) == []
assert gf_sub([1], [2], 11, ZZ) == [10]
assert gf_sub([1, 2], [1], 11, ZZ) == [1, 1]
assert gf_sub([1], [1, 2], 11, ZZ) == [10, 10]
assert gf_sub([3, 2, 1], [8, 9, 10], 11, ZZ) == [6, 4, 2]
assert gf_add_mul([1, 5, 6], [7, 3], [8, 0, 6, 1], 11,
ZZ) == [1, 2, 10, 8, 9]
assert gf_sub_mul([1, 5, 6], [7, 3], [8, 0, 6, 1], 11,
ZZ) == [10, 9, 3, 2, 3]
assert gf_mul([], [], 11, ZZ) == []
assert gf_mul([], [1], 11, ZZ) == []
assert gf_mul([1], [], 11, ZZ) == []
assert gf_mul([1], [1], 11, ZZ) == [1]
assert gf_mul([5], [7], 11, ZZ) == [2]
assert gf_mul([3, 0, 0, 6, 1, 2], [4, 0, 1, 0], 11,
ZZ) == [1, 0, 3, 2, 4, 3, 1, 2, 0]
assert gf_mul([4, 0, 1, 0], [3, 0, 0, 6, 1, 2], 11,
ZZ) == [1, 0, 3, 2, 4, 3, 1, 2, 0]
assert gf_mul([2, 0, 0, 1, 7], [2, 0, 0, 1, 7], 11,
ZZ) == [4, 0, 0, 4, 6, 0, 1, 3, 5]
assert gf_sqr([], 11, ZZ) == []
assert gf_sqr([2], 11, ZZ) == [4]
assert gf_sqr([1, 2], 11, ZZ) == [1, 4, 4]
assert gf_sqr([2, 0, 0, 1, 7], 11, ZZ) == [4, 0, 0, 4, 6, 0, 1, 3, 5]
def test_gf_arith():
assert gf_neg([], 11, ZZ) == []
assert gf_neg([1], 11, ZZ) == [10]
assert gf_neg([1, 2, 3], 11, ZZ) == [10, 9, 8]
assert gf_add_ground([], 0, 11, ZZ) == []
assert gf_sub_ground([], 0, 11, ZZ) == []
assert gf_add_ground([], 3, 11, ZZ) == [3]
assert gf_sub_ground([], 3, 11, ZZ) == [8]
assert gf_add_ground([1], 3, 11, ZZ) == [4]
assert gf_sub_ground([1], 3, 11, ZZ) == [9]
assert gf_add_ground([8], 3, 11, ZZ) == []
assert gf_sub_ground([3], 3, 11, ZZ) == []
assert gf_add_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 6]
assert gf_sub_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 0]
assert gf_mul_ground([], 0, 11, ZZ) == []
assert gf_mul_ground([], 1, 11, ZZ) == []
assert gf_mul_ground([1], 0, 11, ZZ) == []
assert gf_mul_ground([1], 1, 11, ZZ) == [1]
assert gf_mul_ground([1, 2, 3], 0, 11, ZZ) == []
assert gf_mul_ground([1, 2, 3], 1, 11, ZZ) == [1, 2, 3]
assert gf_mul_ground([1, 2, 3], 7, 11, ZZ) == [7, 3, 10]
assert gf_add([], [], 11, ZZ) == []
assert gf_add([1], [], 11, ZZ) == [1]
assert gf_add([], [1], 11, ZZ) == [1]
assert gf_add([1], [1], 11, ZZ) == [2]
assert gf_add([1], [2], 11, ZZ) == [3]
assert gf_add([1, 2], [1], 11, ZZ) == [1, 3]
assert gf_add([1], [1, 2], 11, ZZ) == [1, 3]
assert gf_add([1, 2, 3], [8, 9, 10], 11, ZZ) == [9, 0, 2]
assert gf_sub([], [], 11, ZZ) == []
assert gf_sub([1], [], 11, ZZ) == [1]
assert gf_sub([], [1], 11, ZZ) == [10]
assert gf_sub([1], [1], 11, ZZ) == []
assert gf_sub([1], [2], 11, ZZ) == [10]
assert gf_sub([1, 2], [1], 11, ZZ) == [1, 1]
assert gf_sub([1], [1, 2], 11, ZZ) == [10, 10]
assert gf_sub([3, 2, 1], [8, 9, 10], 11, ZZ) == [6, 4, 2]
assert gf_add_mul(
[1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [1, 2, 10, 8, 9]
assert gf_sub_mul(
[1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [10, 9, 3, 2, 3]
assert gf_mul([], [], 11, ZZ) == []
assert gf_mul([], [1], 11, ZZ) == []
assert gf_mul([1], [], 11, ZZ) == []
assert gf_mul([1], [1], 11, ZZ) == [1]
assert gf_mul([5], [7], 11, ZZ) == [2]
assert gf_mul([3, 0, 0, 6, 1, 2], [4, 0, 1, 0], 11, ZZ) == [1, 0,
3, 2, 4, 3, 1, 2, 0]
assert gf_mul([4, 0, 1, 0], [3, 0, 0, 6, 1, 2], 11, ZZ) == [1, 0,
3, 2, 4, 3, 1, 2, 0]
assert gf_mul([2, 0, 0, 1, 7], [2, 0, 0, 1, 7], 11, ZZ) == [4, 0,
0, 4, 6, 0, 1, 3, 5]
assert gf_sqr([], 11, ZZ) == []
assert gf_sqr([2], 11, ZZ) == [4]
assert gf_sqr([1, 2], 11, ZZ) == [1, 4, 4]
assert gf_sqr([2, 0, 0, 1, 7], 11, ZZ) == [4, 0, 0, 4, 6, 0, 1, 3, 5]