def test_GFPoly():
from sympy.polynomials.fast import gfpoly
p = 5
IntMod5Poly = gfpoly.GFPolyFactory(p)
IntMod5 = IntMod5Poly.coeff_type
f = IntMod5Poly.from_int_dict({2: 1, 0: 1})
g = IntMod5Poly.from_int_dict({3: 2, 1: 1, 0: 2})
assert f[0] == IntMod5(1)
assert f[1] == IntMod5(0)
assert +f == f
assert -f == IntMod5Poly.from_int_dict({2: -1, 0: -1})
assert f.scale(IntMod5(2), 3) == IntMod5Poly.from_int_dict({5: 2, 3: 2})
assert f + g == IntMod5Poly.from_int_dict({3: 2, 2: 1, 1: 1, 0: 3})
assert f - g == IntMod5Poly.from_int_dict({3: -2, 2: 1, 1: -1, 0: -1})
assert f * g == IntMod5Poly.from_int_dict({5: 2, 3: 3, 2: 2, 1: 1, 0: 2})
assert f**5 == IntMod5Poly.from_int_dict({10: 1, 0: 1})
assert (f * g).diff() == IntMod5Poly.from_int_dict({2: 4, 1: 4, 0: 1})
assert g.monic() \
== (IntMod5(2), IntMod5Poly.from_int_dict({3:1, 1:-2, 0:1}))
assert g.monic()[1].to_sym_int_dict() == {3: 1, 1: -2, 0: 1}
q, r = gfpoly.div(f, g)
assert (not q) and (r == f)
q, r = gfpoly.div(g, f)
assert q == IntMod5Poly.from_int_dict({1: 2})
assert r == IntMod5Poly.from_int_dict({1: 4, 0: 2})
assert gfpoly.gcd(f, g) == IntMod5Poly.from_int_dict({1: 1, 0: 3})
assert gfpoly.gcd(f**3, f**4) == f**3
h, s, t = gfpoly.xgcd(f, g)
assert h == IntMod5Poly.from_int_dict({1: 1, 0: 3})
assert s == IntMod5Poly.from_int_dict({1: 2})
assert t == IntMod5Poly.from_int_dict({0: 4})
assert h == (s * f + t * g)
assert gfpoly.truncate(f, 5) == f
assert gfpoly.truncate(f, 1) == IntMod5Poly.from_int_dict({0: 1})
assert gfpoly.pow_mod(f, 3, IntMod5Poly.from_int_dict({2:1})) \
== gfpoly.truncate(f**3, 2)
p = 3
IntMod3Poly = gfpoly.GFPolyFactory(p)
f = IntMod3Poly.from_int_dict({1:1}) \
* IntMod3Poly.from_int_dict({1:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2})
a = gfpoly.distinct_degree_factor(f)
assert [g.to_sym_int_dict() for g in a] \
== [{1: 1, 2: 1}, {0: -1, 1: 1, 3: 1, 4: 1}]
b = gfpoly.equal_degree_factor(a[0], 1)
assert sorted([g.to_sym_int_dict() for g in b]) \
== sorted([{1:1}, {1:1, 0:1}])
c = gfpoly.equal_degree_factor(a[1], 2)
assert sorted([g.to_sym_int_dict() for g in c]) \
== sorted([{2:1, 0:1}, {2:1, 1:1, 0:-1}])
r = gfpoly.factor_sqf(f)
assert r[0] == IntMod3Poly.coeff_type(1)
r = r[1:]
assert len(r) == 4
assert IntMod3Poly.from_int_dict({1: 1}) in r
assert IntMod3Poly.from_int_dict({1: 1, 0: 1}) in r
assert IntMod3Poly.from_int_dict({2: 1, 0: 1}) in r
assert IntMod3Poly.from_int_dict({2: 1, 1: 1, 0: 2}) in r
f = IntMod3Poly.from_int_dict({1:1}) \
* IntMod3Poly.from_int_dict({1:1, 0:1}) \
* IntMod3Poly.from_int_dict({1:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2}) \
* IntMod3Poly.from_int_dict({1:1, 0:1}) \
* IntMod3Poly.from_int_dict({1:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2}) \
* IntMod3Poly.from_int_dict({1:1, 0:1}) \
* IntMod3Poly.from_int_dict({1:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 0:1}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2}) \
* IntMod3Poly.from_int_dict({2:1, 1:1, 0:2})
r = gfpoly.factor(f)
assert r[0] == IntMod3Poly.coeff_type(1)
test_dict = {}
for item in r[1:]:
test_dict[item[1]] = item[0].to_sym_int_dict()
assert test_dict == {
1: {
1: 1
},
3: {
2: 1,
0: 1
},
6: {
1: 1,
0: 1
},
7: {
2: 1,
1: 1,
0: -1
}
}
def zassenhaus(f):
"""Factors a square-free primitive polynomial.
Returns a list of the unique factors.
"""
def subsets(M, k):
"""Generates all k-subsets of M."""
def recursion(result, M, k):
if k == 0:
yield result
else:
for i, result2 in enumerate(M[0 : len(M) + 1 - k]):
for el in recursion(result + [result2], M[i + 1:], k - 1):
yield el
for i, result in enumerate(M[0 : len(M) + 1 - k]):
for el in recursion([result], M[i + 1:], k - 1):
yield el
n = f.degree
if n == 1:
return [f]
A = max([abs(c) for c in f.coeffs.itervalues()])
b = f[n]
B = int(math.sqrt(n+1)*2**n*A*b)
C = (n+1)**(2*n)*A**(2*n-1)
gamma = int(math.ceil(2*math.log(C, 2)))
prime_border = int(2*gamma*math.log(gamma)) + 1
# Choose a prime.
while True:
p = ntheory.generate.randprime(2, prime_border)
if b % p:
poly_type = gfpoly.GFPolyFactory(p)
ff = poly_type.from_int_dict(f.coeffs)
gg = gfpoly.gcd(ff, ff.diff())
if gg.degree == 0:
break
l = int(math.ceil(math.log(2*B + 1, p)))
# Modular factorization.
mod_factors = gfpoly.factor_sqf(ff)
bb = mod_factors[0] # Leading coefficient mod p.
h = [IntPoly(hh.to_sym_int_dict()) for hh in mod_factors[1:]]
# Hensel lifting.
g = multi_hensel_lift(p, f, h, l)
# Factor combination and trial division.
G = []
T = range(len(g))
s = 1
while 2*s <= len(T):
for S in subsets(T, s):
gg = IntPoly({0:b})
for i in S:
gg *= g[i]
gg = gg.mod_int(p**l, symmetric=True)
hh = IntPoly({0:b})
for i in [i for i in T if i not in S]: # T \ S
hh *= g[i]
hh = hh.mod_int(p**l, symmetric=True)
gg_norm = sum([abs(c) for c in gg.coeffs.itervalues()])
hh_norm = sum([abs(c) for c in hh.coeffs.itervalues()])
if gg_norm*hh_norm <= B: # Found divisor
T = [i for i in T if i not in S] # T \ S
G.append(gg.primitive()[1])
f = hh.primitive()[1]
b = f[f.degree]
break
else: # No factors of degree s
s += 1
G.append(f)
return G
