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 test_no_mod(self):
expected = [1, 1.0/2.0, 1.0/6.0, 0, -1.0/30.0, 0, 1.0/42]
expected = map(lambda x: round(x, 5), expected)
b = Bernoulli(0, 0)
actual = list(b.bernoulli(6))
actual = map(lambda x: round(x, 5), actual)
self.assertEqual(expected, actual)
def test_low_mod(self):
m = ModularSolver(modulus=31,primitive_root=3)
numerators = [1, 1, 1, 0, -1, 0, 1]
denominators = [1, 2, 6, 1, 30, 1, 42]
expected = [m.div(n, d) for n, d in izip(numerators, denominators)]
b = Bernoulli(modulus=31, primitive_root=3)
actual = list(b.bernoulli(6))
self.assertEqual(expected, actual)