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 sub(self, x, y):
return gf_sub(x, y, self.p, ZZ)
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 sub(self, x: list, y: list) -> list:
return gf_sub(x, y, self.p, ZZ)
