def powq(self):
X = Fp2(Fp(curve.Fra), Fp(curve.Frb))
X2 = Fp2(Fp(curve.Fra), Fp(curve.Frb))
X2.sqr()
self.a = self.a.powq()
self.b = self.b.powq().muls(X)
self.c = self.c.powq().muls(X2)
return self
def __init__(self, a=None, b=None):
if b is None:
if a is None:
self.a = Fp2()
self.b = Fp2()
else:
self.a = a.copy()
self.b = Fp2()
else:
self.a = a.copy()
self.b = b.copy()
def fromBytes(self, W):
FS = curve.EFS
typ = W[0]
xa = big.from_bytes(W[1:FS + 1])
xb = big.from_bytes(W[FS + 1:2 * FS + 1])
x = Fp2(Fp(xa), Fp(xb))
if typ == 0x04:
ya = big.from_bytes(W[2 * FS + 1:3 * FS + 1])
yb = big.from_bytes(W[3 * FS + 1:4 * FS + 1])
y = Fp2(Fp(ya), Fp(yb))
return self.set(x, y)
else:
return self.setx(x, typ & 1)
def rand(self):
r = Fp2()
r.rand()
self.a = r.copy()
r.rand()
self.b = r.copy()
return self
def affine(self):
if self.isinf() or self.z.isone():
return self
iz = self.z.inverse()
self.x *= iz
self.y *= iz
self.z = Fp2(Fp(1))
return self
def set(self, x, y):
mx = x.copy()
my = y.copy()
if my * my != RHS(mx):
return False
self.x = mx
self.y = my
self.z = Fp2(Fp(1))
return True
def setx(self, x, s):
mx = x.copy()
rhs = RHS(mx)
if rhs.qr() != 1:
return False
rhs.sqrt()
if rhs.sign() != s:
rhs = -rhs
self.x = mx
self.y = rhs
self.z = Fp2(Fp(1))
return True
def unpack(T, Qx, Qy):
aa, bb = T.get()
aa = aa.muls(Qy)
a = Fp4(aa, bb)
t = Fp2(Qx)
if curve.SexticTwist == D_TYPE:
b = Fp4(t)
c = Fp4()
else:
b = Fp4()
c = Fp4(t).times_i()
return Fp12(a, b, c)
def frobenius(self):
X = Fp2(Fp(curve.Fra), Fp(curve.Frb))
if curve.SexticTwist == M_TYPE:
X = X.inverse()
X2 = X.copy()
X2.sqr()
# self.affine()
self.x = self.x.conj()
self.y = self.y.conj()
self.z = self.z.conj()
self.x *= X2
self.y *= X2
self.y *= X
return self
def fromBytes(self, E):
FS = curve.EFS
a = Fp4(Fp2(Fp(big.from_bytes(E[0:FS])),
Fp(big.from_bytes(E[FS:2 * FS]))),
Fp2(Fp(big.from_bytes(E[2 * FS:3 * FS])),
Fp(big.from_bytes(E[3 * FS:4 * FS]))))
b = Fp4(Fp2(Fp(big.from_bytes(E[4 * FS:5 * FS])),
Fp(big.from_bytes(E[5 * FS:6 * FS]))),
Fp2(Fp(big.from_bytes(E[6 * FS:7 * FS])),
Fp(big.from_bytes(E[7 * FS:8 * FS]))))
c = Fp4(Fp2(Fp(big.from_bytes(E[8 * FS:9 * FS])),
Fp(big.from_bytes(E[9 * FS:10 * FS]))),
Fp2(Fp(big.from_bytes(E[10 * FS:11 * FS])),
Fp(big.from_bytes(E[11 * FS:12 * FS]))))
self.set(a, b, c)
def __init__(self):
self.x = Fp2()
self.y = Fp2(Fp(1))
self.z = Fp2()
def generator():
P = ECp2()
P.set(Fp2(Fp(curve.Pxa), Fp(curve.Pxb)), Fp2(Fp(curve.Pya), Fp(curve.Pyb)))
return P
def powq(self):
X = Fp2(Fp(curve.Fra), Fp(curve.Frb))
X3 = X * X * X
return Fp4(self.a.conj(), self.b.conj() * X3)
def get(self): # return tuple Fp2 x and Fp2 y
W = self.copy()
if (W.isinf()):
return (Fp2(), Fp2())
W.affine()
return (W.x, W.y)
def one():
return Fp12(Fp4(Fp2(Fp(1))))
def zero():
return Fp12(Fp4(Fp2(Fp(0))))
def RHS(x):
if curve.SexticTwist == D_TYPE:
return x * x * x + Fp2(ECp.B).divQNR() # on the sextic twist
else:
return x * x * x + Fp2(ECp.B).mulQNR() # on the sextic twist