def is_polynomial_irreducible(f, p):
"""
Rabin's polynomial irreducibility test over finite fields.
"""
gf_degree = len(f) - 1
if gf_degree <= 1:
return True
monic_field = galois_field_monic(f, p)
compare_polynomial = [1, 0]
indices = {gf_degree // i for i in factorint(gf_degree)}
monomial_base = gf_frobenius_monomial_base(monic_field, p, ZZ)
h = monomial_base[1]
for i in range(1, gf_degree):
if i in indices:
g = gf_sub(h, compare_polynomial, p, ZZ)
if gf_gcd(monic_field, g, p, ZZ) != [1]:
return False
h = gf_frobenius_map(h, monic_field, monomial_base, p, ZZ)
return h == compare_polynomial
def test_gf_frobenius_map():
f = ZZ.map([2, 0, 1, 0, 2, 2, 0, 2, 2, 2])
g = ZZ.map([1, 1, 0, 2, 0, 1, 0, 2, 0, 1])
p = 3
b = gf_frobenius_monomial_base(g, p, ZZ)
h = gf_frobenius_map(f, g, b, p, ZZ)
h1 = gf_pow_mod(f, p, g, p, ZZ)
assert h == h1
def test_gf_frobenius_map():
f = ZZ.map([2, 0, 1, 0, 2, 2, 0, 2, 2, 2])
g = ZZ.map([1,1,0,2,0,1,0,2,0,1])
p = 3
n = 4
b = gf_frobenius_monomial_base(g, p, ZZ)
h = gf_frobenius_map(f, g, b, p, ZZ)
h1 = gf_pow_mod(f, p, g, p, ZZ)
assert h == h1