def qdiv(self, other):
"""
Returns q, r the quotient and remainder polynomials respectively, such that
f = q * g + r, where deg(r) < deg(g).
* Assert that g is not the zero polynomial.
"""
other = Polynomial.typecast(other)
pol2 = trim_trailing_zeros(other.poly)
assert pol2, 'Dividing by zero polynomial.'
pol1 = trim_trailing_zeros(self.poly)
if not pol1:
return [], []
rem = pol1
deg_dif = len(rem) - len(pol2)
quotient = [FieldElement.zero()] * (deg_dif + 1)
g_msc_inv = pol2[-1].inverse()
while deg_dif >= 0:
tmp = rem[-1] * g_msc_inv
quotient[deg_dif] = quotient[deg_dif] + tmp
last_non_zero = deg_dif - 1
for i, coef in enumerate(pol2, deg_dif):
rem[i] = rem[i] - (tmp * coef)
if rem[i] != FieldElement.zero():
last_non_zero = i
# Eliminate trailing zeroes (i.e. make r end with its last non-zero coefficient).
rem = rem[:last_non_zero + 1]
deg_dif = len(rem) - len(pol2)
return Polynomial(trim_trailing_zeros(quotient)), Polynomial(rem)
def get_nth_degree_coefficient(self, n):
"""
Returns the coefficient of x**n
"""
if n > self.degree():
return FieldElement.zero()
else:
return self.poly[n]
def random_polynomial(degree):
"""
Returns a random polynomial of a prescribed degree which is NOT the zero polynomial.
"""
leading = FieldElement.random_element(
exclude_elements=[FieldElement.zero()])
p = [FieldElement.random_element() for i in range(degree)] + [leading]
return Polynomial(p)
def test_field_div():
for _ in range(100):
t = FieldElement.random_element(exclude_elements=[FieldElement.zero()])
t_inv = FieldElement.one() / t
assert t_inv == t.inverse()
assert t_inv * t == FieldElement.one()
def __init__(self, coefficients, var='x'):
# Internally storing the coefficients in self.poly, least-significant (i.e. free term)
# first, so $9 - 3x^2 + 19x^5$ is represented internally by the list [9, 0, -3, 0, 0, 19].
# Note that coefficients is copied, so the caller may freely modify the given argument.
self.poly = remove_trailing_elements(coefficients, FieldElement.zero())
self.var = var
def X(cls):
"""
Returns the polynomial x.
"""
return cls([FieldElement.zero(), FieldElement.one()])
def trim_trailing_zeros(p):
"""
Removes zeros from the end of a list.
"""
return remove_trailing_elements(p, FieldElement.zero())
def gen_linear_term(point):
"""
Generates the polynomial (x-p) for a given point p.
"""
return Polynomial([FieldElement.zero() - point, FieldElement.one()])
def monomial(degree, coefficient):
"""
Constructs the monomial coefficient * x**degree.
"""
return Polynomial([FieldElement.zero()] * degree + [coefficient])
def __sub__(self, other):
other = Polynomial.typecast(other)
return Polynomial(
two_lists_tuple_operation(self.poly, other.poly, operator.sub,
FieldElement.zero()))