def divmod(self, a: 'GaussInt',
b: 'GaussInt') -> Tuple['GaussInt', 'GaussInt']:
"""Returns the quotient and the remainder of the euclidean division of two gaussian integers
"""
p_a = Polynomial([a.x, a.y], IntegerRing())
p_b = Polynomial([b.x, b.y], IntegerRing())
q, r = p_a.long_division_reversed(p_b)
base = self.base_polynomial
q %= base
r %= base
return self.element_from_polynomial(q), self.element_from_polynomial(r)
def polynomial_from_element(self, e: 'FFElement') -> Polynomial:
"""Returns a polynomial whose coefficients are those of the element
e.g. `(1+j+j^2)` would yield `1 + X + X^2`
"""
return Polynomial(e.vector,
self.prime_field,
indeterminate=self.root_symbol)
def polynomial(self,
*args,
indeterminate: Optional[str] = 'X') -> Polynomial:
"""Returns a polynomial built from a list of coefficients."""
coefficients = [c for c in args]
return Polynomial(coefficients,
base_field=self,
indeterminate=indeterminate)
def testStrAndRepr(self):
"""Check representations of a polynomial over a prime field
"""
f5 = self.f5
p1 = f5.polynomial(-2, 1, 2, indeterminate='z')
p2 = Polynomial([-2, 1, 2], f5, indeterminate='z')
self.assertTrue(p1 == p2 == eval(repr(p1)))
self.assertEqual(str(p1), '-2 + z + 2z^2')
def test1(self):
"""Extended checks for monic and non-monic polynomials in Z
"""
p = Polynomial([0, 1, 1], IntegerRing())
self.assertTrue(str(p) == 'X + X^2')
f5 = PrimeField(5)
p = f5.linear_polynomial(f5(-2))
self.assertTrue(str(p) == '2 + X')
p = f5.polynomial(0, -1, 1)
self.assertTrue(str(p) == '-X + X^2')
p = symbolic_polynomial('-2 - X^16', PrimeField(5))
self.assertTrue(str(p) == '-2 - X^16')
def polynomial(self,
*args,
indeterminate: Optional[str] = 'X') -> Polynomial:
"""Returns a polynomial from a list of coefficients. The coefficients are converted to `FFElement`
"""
coefficients = []
for c in args:
if isinstance(c, (tuple, list)):
coefficients.append(self.element(c))
elif isinstance(c, FFElement):
coefficients.append(c)
else:
# shall be an integer or, at least an atom (e.g. a PFElement)
c = [c] + [self.prime_field.zero] * (self.dimension - 1)
coefficients.append(self.element(c))
return Polynomial(coefficients,
base_field=self,
indeterminate=indeterminate)
def __init__(self):
self.root_symbol = 'i'
self.base_polynomial = Polynomial(
[1, 0, 1], IntegerRing()) # X2 + 1 irreducible over Z
def polynomial_from_element(self, e: 'GaussInt') -> Polynomial:
"""Returns the polynomial `yZ + x` from the element `'(x, y)`
"""
return Polynomial([e.x, e.y],
IntegerRing(),
indeterminate=self.root_symbol)
def polynomial(self,
coefficients: Sequence,
indeterminate='z') -> Polynomial:
"""Returns a polynomial from its coefficients
"""
return Polynomial(coefficients, self, indeterminate)
def polynomial(self, *args, indeterminate: str = 'X') -> Polynomial:
"""Returns a polynomial from the arguments
"""
return Polynomial([self.element(c) for c in args],
base_field=self,
indeterminate=indeterminate)
def setUp(self):
self.f25 = FiniteField(5, 2,
Polynomial([1, 1, 1], base_field=PrimeField(5)))
def setUp(self):
self.f27 = FiniteField(
3, 3, Polynomial([-1, -1, 0, 1], base_field=PrimeField(3)))