def testWeilPairing(self):
# this example was refered to Washington.
e = elliptic.EC([0, 2], 7)
P = [5, 1]
Q = [0, 3]
R = e.WeilPairing(3, P, Q)
self.assertEqual(finitefield.FinitePrimeFieldElement(2, 7), R)
# test case of extension field, characteristic 7
p = 7
r = 11
F = finitefield.FinitePrimeField(p)
PX = uniutil.polynomial({0:3,1:3,2:2,3:1,4:4,5:1,6:1,10:1},F)
Fx = finitefield.FiniteExtendedField(p,PX)
E = elliptic.EC([F.one,-F.one],F)
Ex = elliptic.EC([Fx.one,-Fx.one],Fx)
P = [3,6]
assert E.whetherOn(P)
assert Ex.whetherOn(P)
assert E.mul(11,P) == E.infpoint
Qxcoord = Fx.createElement(6*7**9+7**8+7**6+6*7**3+6*7**2+7+6)
Qycoord = Fx.createElement(3*7**9+6*7**8+4*7**7+2*7**6+5*7**4+5*7**3+7**2+7+3)
Q = [Qxcoord,Qycoord]
assert Ex.whetherOn(Q)
assert Ex.mul(11,Q) == Ex.infpoint
w = Ex.WeilPairing(11, P, Q)
Wp = Fx.createElement(7**9 + 5*7**8 + 4*7**7 + 2*7**5 + 7**4 + 6*7**2)
assert w == Wp
def u_FiniteField(char, modulus=None):
""" Create FiniteField from modulus.
modulus must be a form of PolynomialoverFp.
"""
if type(modulus) is list:
# FIXME: Undefined variable 'FpPolynomial'
poly_Fp = FpPolynomial(char, modulus)
return u_finitefield.FiniteExtendedField(char, poly_Fp)
if type(modulus) is u_finitefield.FinitePrimeFieldPolynomial:
return u_finitefield.FiniteExtendedField(char, modulus)
elif not modulus:
return u_finitefield.FinitePrimeField(char)
def u_FpPolynomial(char, coeffs):
""" Create FinitePrimeFieldPolynomial from coeffs with mapping
FinitePrimeField(char).createElement.
"""
field = u_finitefield.FinitePrimeField(char)
if type(coeffs) is dict:
return u_finitefield.FinitePrimeFieldPolynomial(coeffs, field)
if isinstance(coeffs, list):
coefficients = []
for index in range(len(coeffs)):
coefficients.append([index, coeffs[index]])
return u_finitefield.FinitePrimeFieldPolynomial(coefficients, field)
def testWeilPairingIsFunction(self):
# e2 is isomorphic to Z/256 x Z/256
F_65537 = finitefield.FinitePrimeField(65537)
e2 = elliptic.EC([-1, 0], F_65537)
P1 = list(map(F_65537.createElement, [30840, 53250]))
self.assertFalse(256 % e2.pointorder(P1))
P2 = list(map(F_65537.createElement, [10657, 46245]))
self.assertFalse(256 % e2.pointorder(P2))
weil10 = set(e2.WeilPairing(256, P1, P2) for i in range(10))
# since Weil pairing is a function, the result is always same
self.assertEqual(1, len(weil10))
# Weil pairing is a function E[m]xE[m] -> mu_m
self.assertEqual(e2.basefield.one, weil10.pop()**256)