def test4(self):
"""Check full factorization over F3, non monic
"""
f3 = PrimeField(3)
p = symbolic_polynomial('-X^3 - X^2 + X + 1', f3)
factors, c = factorize(p).cantor_zassenhaus()
self.assertTrue(factorize(p).factors_product(factors) * c == p)
def test5(self):
"""Check full factorization over F7, non monic
"""
f7 = PrimeField(7)
p = symbolic_polynomial('-3X^3 - 2X^2 + X + 3', f7)
factors, c = factorize(p).cantor_zassenhaus()
self.assertTrue(factorize(p).factors_product(factors) * c == p)
def test3(self):
"""Check full factorization over F2, square of degree 8
"""
f2 = PrimeField(2)
p = symbolic_polynomial('1 + X^2 + X^4 + X^8', f2)
factors, c = factorize(p).cantor_zassenhaus()
self.assertTrue(c == 1)
self.assertTrue(len(factors) == 2)
self.assertTrue(factorize(p).factors_product(factors) == p)
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 test1(self):
"""Check full factorization over F3
"""
f3 = PrimeField(3)
c0 = f3.polynomial(1, 1) # irred deg 1
c1 = f3.polynomial(0, 0, 1) # trivial square
c2 = f3.polynomial(-1, 1)**3
c3 = f3.polynomial(-1, 1, 1)
p = c0 * c1 * c2 * c3
factors, c = factorize(p).cantor_zassenhaus()
self.assertTrue(c == 1)
self.assertTrue(len(factors) == 4)
self.assertEqual(factorize(p).factors_product(factors), p)
def testFiniteFieldF4(self):
"""Check full Cantor-Zassenhaus factorization over F4
"""
f4 = finite_field(4)
p1 = f4.polynomial(1, f4(1, 1), 1)
p2 = f4.polynomial(1, f4(1, 1), 0, 1)
p3 = f4.polynomial(f4(0, 1), 1)**2
p4 = f4.polynomial(f4(0, 1), 1, f4(0, 1), 0, f4(1, 1)) / f4(1, 1)
ps = [p1, p2, p3, p4]
for p in ps:
with self.subTest(f'Testing factorizing {str(p)}'):
factors, c = factorize(p).cantor_zassenhaus()
self.assertTrue(p == factorize(p).factors_product(factors) * c)
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 testFiniteFieldF9(self):
"""Check full Cantor-Zassenhaus factorization over F9
"""
f9 = finite_field(9)
p1 = f9.polynomial(1, f9(1, 1), 1)
p2 = f9.polynomial(1, f9(1, 1), 0, 1)
p3 = f9.polynomial(f9(0, 1), 1)**2
p4 = f9.polynomial(f9(0, 1), 1, f9(0, 1), 0, f9(1, 1)) / f9(1, 1)
p5 = f9.polynomial(f9(-1, 1), f9(1, -1), 1)**3
ps = [p1, p2, p3, p4, p5]
for p in ps:
with self.subTest(f'Testing factorizing {str(p)}'):
factors, c = factorize(p).cantor_zassenhaus()
self.assertTrue(p == factorize(p).factors_product(factors) * c)
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 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 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 test7(self):
"""Check square free factorization of a quadratic polynomial over F2
"""
f5 = PrimeField(5)
p = f5.polynomial(1, 1)**2 * f5.polynomial(2, 1)
sqf, factors = factorize(p).square_free()
self.assertTrue(sqf == f5.polynomial(2, 1))
self.assertTrue(len(factors) == 1)
fact = factors[0]
self.assertTrue(fact.multiplicity == 2)
self.assertTrue(fact.value == f5.polynomial(1, 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 test8(self):
"""Check square free factorization of a cubic polynomial over F3
"""
f3 = PrimeField(3)
p = f3.polynomial(1, 1)**3 * f3.polynomial(-1, 1)
sqf, factors = factorize(p).square_free()
self.assertTrue(sqf == f3.polynomial(-1, 1))
self.assertTrue(len(factors) == 1)
fact = factors[0]
self.assertTrue(fact.multiplicity == 3)
self.assertTrue(fact.value == f3.polynomial(1, 1))
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 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 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 test9(self):
"""Check square free factorization of a polynomial of degree 9 over F3
"""
f3 = PrimeField(3)
p = f3.polynomial(-1, 1)**3 * f3.polynomial(1, 1)**2 * f3.polynomial(
-1, 1, 1)
sqf, factors = factorize(p).square_free()
self.assertTrue(sqf == f3.polynomial(-1, 1, 1))
self.assertTrue(len(factors) == 2)
for fct in factors:
self.assertTrue(
fct.multiplicity == 2 and fct.value == f3.polynomial(1, 1)
or fct.multiplicity == 3 and fct.value == f3.polynomial(-1, 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 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 test2(self):
"""Check equal degree factorization over F2
(non-deterministic, might be skipped)
"""
f2 = PrimeField(2)
p = f2.polynomial(0, 1, 1)
try:
factors = factorize(p).equal_degree(nb_factors=2, max_degree=1)
self.assertTrue(len(factors) == 2)
self.assertTrue(all([f.value.degree == 1 for f in factors]))
self.assertTrue(factors[0].value * factors[1].value == p)
except RuntimeError as e:
self.skipTest(f'Non-deterministic factorization failed: {e}')
def test4(self):
"""Check equal degree factorization over F3: parameter mismatch
"""
f3 = PrimeField(3)
p1 = f3.polynomial(-1, 1, 1)
p2 = f3.polynomial(1, 0, 1)
p = p1 * p2
with self.assertRaises(AssertionError) as a_exc:
_ = factorize(p).equal_degree(nb_factors=3, max_degree=2)
self.assertEqual(
str(a_exc.exception),
f'Degree of {str(p)} mismatches input parameters 3, 2')
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')
def test4(self):
"""Check square free factorization of a cubic polynomial over F2
"""
f2 = PrimeField(2)
p0 = f2.polynomial(1, 1, 1)
p0 **= 3
p0 *= f2.polynomial(1, 1)
p1 = p0.copy
sqf, factors = factorize(p1).square_free()
self.assertTrue(sqf == f2.polynomial(1, 1))
self.assertTrue(sqf.is_irreducible)
self.assertTrue(len(factors) == 1)
self.assertTrue(factors[0].value == f2.polynomial(1, 1, 1)
and factors[0].multiplicity == 3)
def test3(self):
"""Check equal degree factorization over F3
(non-deterministic, might be skipped)
"""
f3 = PrimeField(3)
p1 = f3.polynomial(-1, 1, 1)
p2 = f3.polynomial(1, 0, 1)
p = p1 * p2
try:
factors = factorize(p).equal_degree(nb_factors=2, max_degree=2)
self.assertTrue(len(factors) == 2)
self.assertTrue(all([f.value.degree == 2 for f in factors]))
self.assertTrue(factors[0].value * factors[1].value == p)
except RuntimeError as e:
self.skipTest(f'Non-deterministic factorization failed: {e}')
def test7(self):
"""Check equal degree factorization over F3: non equal degree
Raises most of the time but due to the non-deterministic nature of the Cantor-Zassenhaus algorithm,
it might find a factor anyway
"""
f3 = PrimeField(3)
p1 = f3.polynomial(1, 1)
p2 = f3.polynomial(1, -1, 0, 1)
p = p1 * p2
try:
factors = factorize(p).equal_degree(2, 2)
self.assertTrue(len(factors) == 2)
self.assertTrue(factors[0].value.degree +
factors[1].value.degree == p.degree)
except RuntimeError as exc:
self.assertEqual(
str(exc),
'unable to find 2 degree 2 factors for 1 - X^2 + X^3 + X^4')
def test7(self):
"""Check distinct degree factorization over F5 with:
- two irreducible factors of the same degree
- one irreducible factor of distinct degree
"""
f5 = PrimeField(5)
p1 = f5.polynomial(2, 0, 1)
p2 = f5.polynomial(1, -1, 1)
p3 = f5.polynomial(1, 1, 0, 1)
p = p1 * p2 * p3
factors = factorize(p).distinct_degree()
self.assertTrue(len(factors) == 2)
fdeg4 = [f for f in factors if f.max_degree == 2][0]
fdeg3 = [f for f in factors if f.max_degree == 3][0]
self.assertTrue(fdeg4.value.degree == 2 * fdeg4.max_degree == 4)
self.assertTrue(fdeg3.value.degree == fdeg3.max_degree == 3)
self.assertTrue(fdeg4.value == p1 * p2)
self.assertTrue(fdeg3.value == p3)
def test1(self):
f7 = PrimeField(7)
f = factorize(f7.polynomial(0))
self.assertEqual(f.factors_product([]), f7.zero)
