def calculate_lagrange_polynomials(x_values):
"""
Given the x_values for evaluating some polynomials, it computes part of the lagrange polynomials
required to interpolate a polynomial over this domain.
"""
lagrange_polynomials = []
monomials = [
Polynomial.monomial(1, FieldElement.one()) - Polynomial.monomial(0, x)
for x in x_values
]
numerator = prod(monomials)
for j in tqdm(range(len(x_values))):
# In the denominator, we have:
# (x_j-x_0)(x_j-x_1)...(x_j-x_{j-1})(x_j-x_{j+1})...(x_j-x_{len(X)-1})
denominator = prod(
[x_values[j] - x for i, x in enumerate(x_values) if i != j])
# Numerator is a bit more complicated, since we need to compute a poly multiplication here.
# Similarly to the denominator, we have:
# (x-x_0)(x-x_1)...(x-x_{j-1})(x-x_{j+1})...(x-x_{len(X)-1})
cur_poly, _ = numerator.qdiv(monomials[j].scalar_mul(denominator))
lagrange_polynomials.append(cur_poly)
return lagrange_polynomials
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 X(cls):
"""
Returns the polynomial x.
"""
return cls([FieldElement.zero(), FieldElement.one()])
def gen_linear_term(point):
"""
Generates the polynomial (x-p) for a given point p.
"""
return Polynomial([FieldElement.zero() - point, FieldElement.one()])