def Weil_2(P, Q, R, S, r):
zz = P.ec.field
pl = poly(zz)
mil = millersF(P.ec, pl)
f_QS = mil.mfunc(P, r, Q + S) * ((mil.line(P, R)[Q + S] * mil.vertical(P + R)[Q + S]) ** (-r))
f_S = mil.mfunc(P, r, S) * ((mil.line(P, R)[S] * mil.vertical(P + R)[S]) ** (-r))
g_R = mil.mfunc(Q, r, R) * ((mil.line(Q, S)[R] * mil.vertical(Q + S)[R]) ** (-r))
g_PR = mil.mfunc(Q, r, P + R) * ((mil.line(Q, S)[P + R] * mil.vertical(Q + S)[P + R]) ** (-r))
pairing = f_QS * g_R / (f_S * g_PR)
return pairing
def Weil_1(P, Q, R, S, r):
zz = P.ec.field
pl = poly(zz)
mil = millersF(P.ec, pl)
fp = mil.mfunc_slow(P, r)
fq = mil.mfunc_slow(Q, r)
f = fp * (mil.line(P, R) * mil.vertical(P + R)) ** (-r)
g = fq * (mil.line(Q, S) * mil.vertical(Q + S)) ** (-r)
pairing = f[Q + S] * g[R] / (f[S] * g[P + R])
return pairing
def Weil(P, Q, R, S, r):
zz = ec.field
pl = poly(zz)
mil = millersF(P.ec, pl)
fp = mil.mfunc_slow(P, r)
fq = mil.mfunc_slow(Q, r)
f = fp * (mil.line(P, R) * mil.vertical(P + R))**(-r)
g = fq * (mil.line(Q, S) * mil.vertical(Q + S))**(-r)
pairing = f.apply(Q + S) * g.apply(R) / (f.apply(S) * g.apply(P + R))
return pairing
from ecs import * from finField import finField from millersF import millersF from pairings import Weil_1, Weil_2 from poly2 import poly from tortion import tortion ec = beginners5_3_1() zzz = ec.field p = poly(zzz) ff = finField(p.of([zzz.of(1), zzz.of(0), zzz.of(zzz.N - 4), zzz.of(0), zzz.of(5)])) P = ec.of(zzz.of(45), zzz.of(23)) qx = p.of([zzz.of(31), zzz.of(0), zzz.of(29)]) qy = p.of([zzz.of(35), zzz.of(0), zzz.of(11), zzz.of(0)]) # Q = tor_points[6] Q = ec.of(ff.of(qx), ff.of(qy)) it = ec.all() it.__next__() R = it.__next__() S = it.__next__() # R = ec.of(zzz.of(0), zzz.of(11)) # S = ec.of(zzz.of(0), zzz.of(12)) r = 17 print(tortion.k(r, zzz.N))
from matrix import matrix
from zzn import zzn
from finField import finField
from poly2 import poly
Bit = zzn(2)
bitPoly = poly(Bit)
GF256 = finField(bitPoly.irredusable(8).__next__())
p = poly(GF256)
# convert int to field element
def f(x):
l = []
while (x != 0):
(x, b) = divmod(x, 2)
if b:
l.append(Bit.one)
else:
l.append(Bit.zero)
if not l:
l.append(Bit.zero)
l.reverse()
return GF256.of(bitPoly.of(l))
k = 2
n = 6
e = (n - k) // 2
a = [f(1), f(2), f(3), f(4), f(5), f(6)]
def poly(self):
p = poly(self.field)
return p.of([self.field.one, self.field.zero, self.a, self.b])