def untwist(point: AffinePoint, ec=default_ec) -> AffinePoint:
"""
Given a point on G2 on the twisted curve, this converts it's
coordinates back from Fq2 to Fq12. See Craig Costello book, look
up twists.
"""
f = Fq12.one(ec.q)
wsq = Fq12(ec.q, f.root, Fq6.zero(ec.q))
wcu = Fq12(ec.q, Fq6.zero(ec.q), f.root)
return AffinePoint(point.x / wsq, point.y / wcu, False, ec)
def twist(point: AffinePoint, ec=default_ec_twist) -> AffinePoint:
"""
Given an untwisted point, this converts it's
coordinates to a point on the twisted curve. See Craig Costello
book, look up twists.
"""
f = Fq12.one(ec.q)
wsq = Fq12(ec.q, f.root, Fq6.zero(ec.q))
wcu = Fq12(ec.q, Fq6.zero(ec.q), f.root)
new_x = point.x * wsq
new_y = point.y * wcu
return AffinePoint(new_x, new_y, False, ec)
def untwist(P):
q = P.X.q
root = Fq6(Fq2.zero(q), Fq2.one(q), Fq2.zero(q))
zero = Fq6.zero(q)
omega2 = Fq12(root, zero)
omega3 = Fq12(zero, root)
return EC.from_affine(omega2.inverse() * P.x, omega3.inverse() * P.y)
def twist(P):
q = P.X.q
root = Fq6(Fq2.zero(q), Fq2.one(q), Fq2.zero(q))
zero = Fq6.zero(q)
omega2 = Fq12(root, zero)
omega3 = Fq12(zero, root)
c0 = omega2 * P.x
c1 = omega3 * P.y
return TwistedEC.from_affine(c0.c0.c0, c1.c0.c0)
def test_fields():
a = Fq(17, 30)
b = Fq(17, -18)
c = Fq2(17, a, b)
d = Fq2(17, a + a, -5)
e = c * d
f = e * d
assert f != e
e_sq = e * e
e_sqrt = e_sq.modsqrt()
assert pow(e_sqrt, 2) == e_sq
a2 = Fq(
172487123095712930573140951348,
3012492130751239573498573249085723940848571098237509182375,
)
b2 = Fq(172487123095712930573140951348, 3432984572394572309458723045723849)
c2 = Fq2(172487123095712930573140951348, a2, b2)
assert b2 != c2
g = Fq6(17, c, d, d * d * c)
h = Fq6(17, a + a * c, c * b * a, b * b * d * 21)
i = Fq12(17, g, h)
assert ~(~i) == i
assert (~(i.root)) * i.root == Fq6.one(17)
x = Fq12(17, Fq6.zero(17), i.root)
assert (~x) * x == Fq12.one(17)
j = Fq6(17, a + a * c, Fq2.zero(17), Fq2.zero(17))
j2 = Fq6(17, a + a * c, Fq2.zero(17), Fq2.one(17))
assert j == (a + a * c)
assert j2 != (a + a * c)
assert j != j2
# Test frob_coeffs
one = Fq(default_ec.q, 1)
two = one + one
a = Fq2(default_ec.q, two, two)
b = Fq6(default_ec.q, a, a, a)
c = Fq12(default_ec.q, b, b)
for base in (a, b, c):
for expo in range(1, base.extension):
assert base.qi_power(expo) == pow(base, pow(default_ec.q, expo))
# * uses curve impl from curve_ops
# * only supports BLS12-381
# * Miller loop implementation avoids computing inversions
#
# (C) 2019 Riad S. Wahby <*****@*****.**>
from functools import reduce
from operator import mul
from consts import p, ell_u, k_final
from curve_ops import from_jacobian, point_double, point_add, to_coZ
from fields import Fq, Fq2, Fq6, Fq12
# constants for untwisting
ut_root = Fq12.one(p).root
ut_wsq_inv = ~Fq12(p, ut_root, Fq6.zero(p))
ut_wcu_inv = ~Fq12(p, Fq6.zero(p), ut_root)
del ut_root
def _untwist(R):
assert all(isinstance(pp, Fq2) for pp in R)
(x, y, z) = R
return (x * ut_wsq_inv, y * ut_wcu_inv, z)
def _double_eval(R, P):
(xP, yP) = P
(xR, yR, zR) = _untwist(R)
zR3 = pow(zR, 3)