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polynomials_generic.py
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polynomials_generic.py
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import random
class PolynomialRing:
"""
Initialise the polynomial ring:
R = GF(q) / (X^n + 1)
"""
def __init__(self, q, n):
self.q = q
self.n = n
self.element = Polynomial
def gen(self):
"""
Return the generator `x` of the polynomial ring
"""
return self([0, 1])
def random_element(self):
"""
Compute a random element of the polynomial ring with coefficients in the
canonical range: ``[0, q-1]``
"""
coefficients = [random.randint(0, self.q - 1) for _ in range(self.n)]
return self(coefficients)
def __call__(self, coefficients):
if isinstance(coefficients, int):
return self.element(self, [coefficients])
if not isinstance(coefficients, list):
raise TypeError(
f"Polynomials should be constructed from a list of integers, of length at most d = {self.n}"
)
return self.element(self, coefficients)
def __repr__(self):
return f"Univariate Polynomial Ring in x over Finite Field of size {self.q} with modulus x^{self.n} + 1"
class Polynomial:
def __init__(self, parent, coefficients):
self.parent = parent
self.coeffs = self._parse_coefficients(coefficients)
def is_zero(self):
"""
Return if polynomial is zero: f = 0
"""
return all(c == 0 for c in self.coeffs)
def is_constant(self):
"""
Return if polynomial is constant: f = c
"""
return all(c == 0 for c in self.coeffs[1:])
def _parse_coefficients(self, coefficients):
"""
Helper function which right pads with zeros
to allow polynomial construction as
f = R([1,1,1])
"""
l = len(coefficients)
if l > self.parent.n:
raise ValueError(
f"Coefficients describe polynomial of degree greater than maximum degree {self.parent.n}"
)
elif l < self.parent.n:
coefficients = coefficients + [0 for _ in range(self.parent.n - l)]
return coefficients
def reduce_coefficients(self):
"""
Reduce all coefficients modulo q
"""
self.coeffs = [c % self.parent.q for c in self.coeffs]
return self
def _add_mod_q(self, x, y):
"""
add two coefficients modulo q
"""
return (x + y) % self.parent.q
def _sub_mod_q(self, x, y):
"""
sub two coefficients modulo q
"""
return (x - y) % self.parent.q
def _schoolbook_multiplication(self, other):
"""
Naive implementation of polynomial multiplication
suitible for all R_q = F_1[X]/(X^n + 1)
"""
n = self.parent.n
a = self.coeffs
b = other.coeffs
new_coeffs = [0 for _ in range(n)]
for i in range(n):
for j in range(0, n - i):
new_coeffs[i + j] += a[i] * b[j]
for j in range(1, n):
for i in range(n - j, n):
new_coeffs[i + j - n] -= a[i] * b[j]
return [c % self.parent.q for c in new_coeffs]
def __neg__(self):
"""
Returns -f, by negating all coefficients
"""
neg_coeffs = [(-x % self.parent.q) for x in self.coeffs]
return self.parent(neg_coeffs)
def _add_(self, other):
if isinstance(other, type(self)):
new_coeffs = [
self._add_mod_q(x, y)
for x, y in zip(self.coeffs, other.coeffs)
]
elif isinstance(other, int):
new_coeffs = self.coeffs.copy()
new_coeffs[0] = self._add_mod_q(new_coeffs[0], other)
else:
raise NotImplementedError(
"Polynomials can only be added to each other"
)
return new_coeffs
def __add__(self, other):
new_coeffs = self._add_(other)
return self.parent(new_coeffs)
def __radd__(self, other):
return self.__add__(other)
def __iadd__(self, other):
self = self + other
return self
def _sub_(self, other):
if isinstance(other, type(self)):
new_coeffs = [
self._sub_mod_q(x, y)
for x, y in zip(self.coeffs, other.coeffs)
]
elif isinstance(other, int):
new_coeffs = self.coeffs.copy()
new_coeffs[0] = self._sub_mod_q(new_coeffs[0], other)
else:
raise NotImplementedError(
"Polynomials can only be subtracted from each other"
)
return new_coeffs
def __sub__(self, other):
new_coeffs = self._sub_(other)
return self.parent(new_coeffs)
def __rsub__(self, other):
return -self.__sub__(other)
def __isub__(self, other):
self = self - other
return self
def __mul__(self, other):
if isinstance(other, type(self)):
new_coeffs = self._schoolbook_multiplication(other)
elif isinstance(other, int):
new_coeffs = [(c * other) % self.parent.q for c in self.coeffs]
else:
raise NotImplementedError(
"Polynomials can only be multiplied by each other, or scaled by integers"
)
return self.parent(new_coeffs)
def __rmul__(self, other):
return self.__mul__(other)
def __imul__(self, other):
self = self * other
return self
def __pow__(self, n):
if not isinstance(n, int):
raise TypeError(
"Exponentiation of a polynomial must be done using an integer."
)
# Deal with negative scalar multiplication
if n < 0:
raise ValueError(
"Negative powers are not supported for elements of a Polynomial Ring"
)
f = self
g = self.parent(1)
while n > 0:
if n % 2 == 1:
g = g * f
f = f * f
n = n // 2
return g
def __eq__(self, other):
if isinstance(other, type(self)):
return self.coeffs == other.coeffs
elif isinstance(other, int):
if (
self.is_constant()
and (other % self.parent.q) == self.coeffs[0]
):
return True
return False
def __getitem__(self, idx):
return self.coeffs[idx]
def __repr__(self):
if self.is_zero():
return "0"
info = []
for i, c in enumerate(self.coeffs):
if c != 0:
if i == 0:
info.append(f"{c}")
elif i == 1:
if c == 1:
info.append("x")
else:
info.append(f"{c}*x")
else:
if c == 1:
info.append(f"x^{i}")
else:
info.append(f"{c}*x^{i}")
return " + ".join(info)
def __str__(self):
return self.__repr__()