class Faulhaber(ModularSolver):
def __init__(self, max_degree, modulus, primitive_root, *args, **kwargs):
super(Faulhaber, self).__init__(modulus=modulus, primitive_root=primitive_root, *args, **kwargs)
self._max_degree = max_degree
self._bernoulli = Bernoulli(modulus=modulus, primitive_root=primitive_root)
self._bn = list(self._bernoulli.bernoulli(max_degree))
self._choose = Choose(modulus=modulus, primitive_root=primitive_root)
def base_polynomial(self, p, scalar=1):
# returns a Polynomial P (of degree p + 1) such that P(x) == scalar * sum(i^p for i in range(1, x + 1))
if p + 1 > self._max_degree:
raise ValueError
# compute p(x) = sum(i^p for i in range(1, n + 1))
coef = list(repeat(0, p + 2))
for i in range(p + 1):
coef[p + 1 - i] = self.mult(self._choose.choose(p + 1, i), self._bn[i])
poly = Polynomial(coef, modulus=self._modulus, primitive_root=self._primitive_root)
poly.scalarmult(self.div(scalar, p + 1))
return poly
def general_polynomial(self, poly):
# returns a Polynomial P such that P(x) == sum(poly(i) for i in range(1, x + 1))
p = Polynomial([], modulus=self._modulus, primitive_root=self._primitive_root)
for i, c in enumerate(poly._coef):
if c:
p.add_polynomial(self.base_polynomial(i, c))
return p
def __init__(self, max_degree, modulus, primitive_root, *args, **kwargs):
super(Faulhaber, self).__init__(modulus=modulus, primitive_root=primitive_root, *args, **kwargs)
self._max_degree = max_degree
self._bernoulli = Bernoulli(modulus=modulus, primitive_root=primitive_root)
self._bn = list(self._bernoulli.bernoulli(max_degree))
self._choose = Choose(modulus=modulus, primitive_root=primitive_root)
def test_mod_31(self):
c = Choose(31, 3)
self.assertEqual(6, c.choose(4, 2))
self.assertEqual(6 * 5 * 4 / (3 * 2 * 1) % 31, c.choose(6, 3))