def test6(self):
"""Check square free factorization over F5
"""
f5 = PrimeField(5)
p = f5.polynomial(-1, 2, 1, 1)
sqf, factors = factorize(p).square_free()
self.assertTrue(sqf == p)
def testPrimeAndPolynomial(self):
"""Checking correct transtyping int to Polynomial for + and == in F2
"""
f2 = PrimeField(2)
p = f2.polynomial(1, 1, 1)
self.assertTrue(p + 1 == f2.polynomial(0, 1, 1))
self.assertTrue(p.monic(0) + 1 == 0)
def test3(self):
"""Check: irreducibility over F3, medium degree (>2)
"""
f3 = PrimeField(3)
p = f3.polynomial(-1, -1, -1, 1)
self.assertIsIrreducible(p)
p = f3.polynomial(1, 1, 1, 1)
self.assertIsNotIrreducible(p)
def test5(self):
"""Check: printable representation over a prime field
"""
f2 = PrimeField(2)
p = f2.polynomial(1)
p += p.monic(3)
p += p.monic(32)
self.assertEqual(str(p), '1 + X^3 + X^32')
self.assertIsNotIrreducible(p)
def test2(self):
"""Check: irreducibility over F3, low degree (<3)
"""
f3 = PrimeField(3)
p = f3.polynomial(-1, 1, 1)
self.assertIsIrreducible(p)
p = f3.polynomial(1, 1, 1)
self.assertIsNotIrreducible(p)
def test4(self):
"""Parser test for a linear polynomial over the finite field of order 8
"""
expr = '-(1+w+w^2) + X'
f2 = PrimeField(2)
g = f2.polynomial(1, 1, 0, 1)
f8 = FiniteField(2, 3, g, root_symbol='w')
p = symbolic_polynomial(expr, f8)
self.assertTrue(p == f8.polynomial(f8(1, 1, 1), 1))
def test1(self):
"""Parser test over F4
"""
f2 = PrimeField(2)
f4 = FiniteField(2, 2, f2.polynomial(1, 1, 1))
expr = '+(j+1)X + (1+j)X^2 + (j)'
p = symbolic_polynomial(expr, f4)
self.assertTrue(p == f4.polynomial(f4(0, 1), f4(1, 1), f4(1, 1)))
def test3(self):
"""Check distinct degree factorization of a square free equal degree polynomial over F3
"""
f3 = PrimeField(3)
p1 = f3.polynomial(1, 1)
p2 = f3.polynomial(-1, 1)
p = p1 * p2 # square free equal degree
factors = factorize(p).distinct_degree()
self.assertTrue(len(factors) == 1)
self.assertTrue(factors[0].value == p)
def test2(self):
"""Check distinct degree factorization of a non square free polynomial over F2
"""
f2 = PrimeField(2)
p = f2.polynomial(1, 0, 1) # non square free
factors = factorize(p).distinct_degree()
self.assertTrue(all([f.is_irreducible for f in factors]))
self.assertTrue(len(factors) == 2)
self.assertTrue(p == factors[0].value * factors[1].value)
self.assertTrue(factors[0].value == factors[1].value)
def test6(self):
"""Check: irreduciblity over F3, performances for very high degree (>800)
"""
f3 = PrimeField(3)
p0 = f3.polynomial(1, 0, 1, 0, 1, 0, 0, 0, 1)
p1 = f3.polynomial(-1, 0, 0, 0, 1, 0, 0, -1, 1)
p2 = f3.polynomial(1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1)
p = p0 * p1 * p2
self.assertIsNotIrreducible(p)
def testF16gcd(self):
"""Check GCD of polynomials over F16
"""
f2 = PrimeField(2)
p = f2.polynomial(1, 1, 0, 0, 1)
f16 = FiniteField(2, 4, p)
a = symbolic_polynomial('X^6 + X^3 + 1', f16)
b = symbolic_polynomial('X^3 + 1', f16)
self.assertEqual(gcd(a, b), f16.polynomial(1))
def testF9Product(self):
"""Check product of polynomials over F9
"""
f3 = PrimeField(3)
p = f3.polynomial(1, 0, 1)
f9 = FiniteField(3, 2, p)
a = symbolic_polynomial('-1 + X - (1+j)X^2', f9)
b = symbolic_polynomial('1 + X^2 + (j)X', f9)
self.assertTrue(
a * b == f9.polynomial(-1, f9(1, -1), 1, f9(-1, -1), f9(-1, -1)))
def testF25Repr(self):
"""Check equivalence of symbolic representation and printable representation of a polynomial over F25
"""
f5 = PrimeField(5)
f25 = FiniteField(5, 2, f5.polynomial(2, 0, 1))
p = symbolic_polynomial('(1+j)X^2 + (j)X^5 + X + 2X^3 + (2+j)', f25)
self.assertEqual(
p,
eval(repr(f25.polynomial(f25(2, 1), 1, f25(1, 1), 2, 0, f25(0,
1)))))
def test1(self):
"""Check square free factorization of an irreducible polynomial over F2
"""
f2 = PrimeField(2)
p1 = f2.polynomial(1, 0, 1, 1)
sqf, factors = factorize(p1).square_free()
self.assertTrue(len(factors) == 0)
self.assertTrue(sqf.is_irreducible)
self.assertTrue(sqf == p1)
def test5(self):
"""Check distinct degree factorization of a equal degree polynomial over F5
"""
f5 = PrimeField(5)
p1 = f5.polynomial(2, 0, 1)
p2 = f5.polynomial(1, 1, 1)
p = p1 * p2
factors = factorize(p).distinct_degree()
self.assertTrue(len(factors) == 1)
self.assertTrue(factors[0].value == p)
def test1(self):
"""Check distinct degree factorization over F2
"""
f2 = PrimeField(2)
p0 = f2.polynomial(1, 1)
p1 = f2.polynomial(1, 1, 1)
p = p0 * p1
factors = factorize(p).distinct_degree()
self.assertTrue(all([f.is_irreducible for f in factors]))
self.assertTrue(len(factors) == 2)
self.assertTrue(p == factors[0].value * factors[1].value)
def test2(self):
"""Check full factorization over F5
"""
f5 = PrimeField(5)
p = f5.polynomial(1, 2, 0, 2, -1, 1)
try:
factors, c = factorize(p).cantor_zassenhaus()
self.assertTrue(c == f5(1))
self.assertTrue(factorize(p).factors_product(factors) == p)
except RuntimeError as e:
self.skipTest(f'Non-deterministic factorization failed: {e}')
def test8(self):
"""Parser check of internal state automaton
'(j+)X' is an invalid expression
"""
expr = '(j+)X'
f2 = PrimeField(2)
p = f2.polynomial(1, 1, 1)
f4 = FiniteField(2, 2, p)
with self.assertRaises(SyntaxError):
symbolic_polynomial(expr, f4)
def test5(self):
"""Check equal degree factorization over F2: parameter mismatch
"""
f2 = PrimeField(2)
p = f2.polynomial(1, 1, 1)
# bad parameters
with self.assertRaises(RuntimeError) as a_exc:
_ = factorize(p).equal_degree(2, 1)
self.assertEqual(str(a_exc.exception),
'unable to find 2 degree 1 factors for 1 + X + X^2')
def test2(self):
"""Check square free factorization over F2
"""
f2 = PrimeField(2)
p1 = f2.polynomial(1, 0, 1, 0, 1)
sqf, factors = factorize(p1).square_free()
self.assertTrue(sqf.is_unit)
self.assertTrue(len(factors) == 1)
self.assertTrue(factors[0].multiplicity == 2)
self.assertTrue(factors[0].value == f2.polynomial(1, 1, 1))
def test6(self):
"""Check distinct degree factorization over F5
"""
f5 = PrimeField(5)
p1 = f5.polynomial(2, 0, 1)
p2 = f5.polynomial(1, 1, 0, 1)
p = p1 * p2
factors = factorize(p).distinct_degree()
self.assertTrue(len(factors) == 2)
self.assertTrue(factors[0].max_degree == factors[0].value.degree)
self.assertTrue(factors[1].max_degree == factors[1].value.degree)
def testF4Str(self):
"""Check printable representation of a polynomial over F4
"""
f2 = PrimeField(2)
p_gen = f2.polynomial(1, 1, 1)
f4 = FiniteField(2, 2, p_gen)
p = f4.polynomial(f4(0, 1), f4(1, 0), f4(1, 1))
self.assertTrue(str(p) == '(j) + X + (1+j)X^2')
p *= f4(0, 1)
self.assertTrue(str(p) == '(1+j) + (j)X + X^2')
def testPrimeAndFinite(self):
"""Checking correct transtyping int, PFElement to FFElement for + and - in F9
"""
f3 = PrimeField(3)
f9 = FiniteField(3, 2, f3.polynomial(-1, 1, 1))
self.assertTrue(f9(1, 0) + 1 == f3(-1))
self.assertTrue(1 + f3(1) == f9(-1, 0))
self.assertTrue(f9(-1, 0) == -1)
self.assertTrue(3 * f9(-1, 0) == f9(-1, 0) * 3)
self.assertTrue(f3(1) * f9(1, 1) == f3(1) + f9(0, 1))
def test1(self):
"""Check return type of power in F4 and F2
"""
f2 = PrimeField(2)
f4 = FiniteField(2, 2, f2.polynomial(1, 1, 1))
b = f4(0, 1)**0
self.assertIsInstance(b, FFElement)
self.assertTrue(b == 1)
b = f2(1)**0
self.assertIsInstance(b, PFElement)
self.assertTrue(b == 1)
def test4(self):
"""Check distinct degree factorization over F3
"""
f3 = PrimeField(3)
p1 = f3.polynomial(1, 1)
p2 = f3.polynomial(-1, 1, 1)
p = p1 * p2
factors = factorize(p).distinct_degree()
self.assertTrue(len(factors) == 2)
self.assertTrue(all([f.is_irreducible for f in factors]))
self.assertTrue(factors[0].value * factors[1].value == p)
self.assertTrue(factors[0].value.degree != factors[1].value.degree)
def test4(self):
"""Check: irreducibility over F3, high degree (>6)
"""
f3 = PrimeField(3)
p1 = f3.polynomial(1, 0, 1)
p2 = f3.polynomial(-1, 1, 1)
p3 = f3.polynomial(-1, -1, 1)
p = p1 * p2 * p3
self.assertIsNotIrreducible(p)
p = p**4
self.assertIsNotIrreducible(p)
def test3(self):
"""Check square free factorization of a quadratic polynomial over F2
"""
f2 = PrimeField(2)
p1 = f2.polynomial(1, 1, 1, 1, 1, 1)
sqf, factors = factorize(p1).square_free()
self.assertTrue(sqf == f2.polynomial(1, 1))
self.assertTrue(len(factors) == 1)
self.assertTrue(factors[0].value.is_irreducible)
self.assertTrue(factors[0].multiplicity == 2)
self.assertTrue(factors[0].value**2 * sqf == p1)
def test5(self):
"""Check square free factorization over F3
"""
f3 = PrimeField(3)
p0 = f3.polynomial(1, -1, 0, 0, -1, 0, 0, 1, 1)
p1 = f3.polynomial(-1, 1, -1, -1, 1)
self.assertTrue(p0.is_irreducible and p1.is_irreducible)
p = p0 * p1
sqf, factors = factorize(p).square_free()
self.assertTrue(len(factors) == 0)
self.assertTrue(sqf == p)
def testF8Add(self):
"""Check addition of polynomials over F8
"""
f2 = PrimeField(2)
p = f2.polynomial(1, 1, 0, 1)
f8 = FiniteField(2, 3, p, root_symbol='w')
a = symbolic_polynomial('1 + X + (1+w^2)X^3', f8)
b = symbolic_polynomial('(1+w)X^2 + X^3', f8)
self.assertEqual(b.trailing, FFElement(f8, (1, 1, 0)))
self.assertTrue(a + b == f8.polynomial(1, 1, f8(1, 1, 0), f8(0, 0, 1)))
def test6(self):
"""Check equal degree factorization over F2: non square free polynomial
"""
f2 = PrimeField(2)
p = f2.polynomial(1, 1)
p **= 3
# non square-free polynomial
with self.assertRaises(RuntimeError) as a_exc:
_ = factorize(p).equal_degree(3, 1)
self.assertEqual(
str(a_exc.exception),
'unable to find 3 degree 1 factors for 1 + X + X^2 + X^3')