def zzx_hensel_lift(p, f, f_list, l):
"""Multifactor Hensel lifting.
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
For more details on the implemented algorithm refer to:
[1] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
First Edition, Cambridge University Press, 1999, pp. 424
"""
r = len(f_list)
lc = zzx_LC(f)
if r == 1:
F = zzx_mul_term(f, igcdex(lc, p**l)[0], 0)
return [zzx_trunc(F, p**l)]
m = p
k = int(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)
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)
s, t, _ = gf_gcdex(g, h, p)
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 = zzx_hensel_step(m, f, g, h, s, t), m**2
return zzx_hensel_lift(p, g, f_list[:k], l) \
+ zzx_hensel_lift(p, h, f_list[k:], l)
def zzx_hensel_lift(p, f, f_list, l):
"""Multifactor Hensel lifting.
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
For more details on the implemented algorithm refer to:
[1] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
First Edition, Cambridge University Press, 1999, pp. 424
"""
r = len(f_list)
lc = zzx_LC(f)
if r == 1:
F = zzx_mul_term(f, igcdex(lc, p**l)[0], 0)
return [ zzx_trunc(F, p**l) ]
m = p
k = int(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)
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)
s, t, _ = gf_gcdex(g, h, p)
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 = zzx_hensel_step(m, f, g, h, s, t), m**2
return zzx_hensel_lift(p, g, f_list[:k], l) \
+ zzx_hensel_lift(p, h, f_list[k:], l)
def test_gf_euclidean():
assert gf_gcd([], [], 11) == []
assert gf_gcd([2], [], 11) == [1]
assert gf_gcd([], [2], 11) == [1]
assert gf_gcd([2], [2], 11) == [1]
assert gf_gcd([], [1,0], 11) == [1,0]
assert gf_gcd([1,0], [], 11) == [1,0]
assert gf_gcd([3,0], [3,0], 11) == [1,0]
assert gf_gcd([1,8,7], [1,7,1,7], 11) == [1,7]
assert gf_gcdex([], [], 11) == ([1], [], [])
assert gf_gcdex([2], [], 11) == ([6], [], [1])
assert gf_gcdex([], [2], 11) == ([], [6], [1])
assert gf_gcdex([2], [2], 11) == ([], [6], [1])
assert gf_gcdex([], [3,0], 11) == ([], [4], [1,0])
assert gf_gcdex([3,0], [], 11) == ([4], [], [1,0])
assert gf_gcdex([3,0], [3,0], 11) == ([], [4], [1,0])
assert gf_gcdex([1,8,7], [1,7,1,7], 11) == ([5,6], [6], [1,7])
