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July 6, 2017 10:36
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Tate Pairing Example
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# Copied from: | |
# https://raw.githubusercontent.com/guanzhi/CryptoWithSageMath/master/tate-pairing-example.sage | |
# Tate Pairing Example | |
# Example 5.43 in IMC | |
# E: y^2 = x^3 + 30x + 34 mod 631 | |
p = 631 | |
a = 30 | |
b = 34 | |
E = EllipticCurve(GF(p), [a, b]) | |
print E | |
P = E((36, 60)) | |
Q = E((121, 387)) | |
n = 5 | |
S = E((0, 36)) | |
print "P =", P.xy() | |
print "Q =", Q.xy() | |
print "#P = #Q =", n | |
var('x y') | |
def g(P, Q): | |
(x_P, y_P) = P.xy() | |
(x_Q, y_Q) = Q.xy() | |
if x_P == x_Q and y_P + y_Q == 0: | |
return x - x_P | |
if P == Q: | |
slope = (3 * x_P^2 + a)/(2 * y_P) | |
else: | |
slope = (y_P - y_Q)/(x_P - x_Q) | |
return (y - y_P - slope * (x - x_P))/(x + x_P + x_Q - slope^2) | |
def miller(m, P): | |
m = bin(m)[3:] | |
n = len(m) | |
T = P | |
f = 1 | |
for i in range(n): | |
f = f^2 * g(T, T) | |
T = T + T | |
if int(m[i]) == 1: | |
f = f * g(T, P) | |
T = T + P | |
return f | |
def eval_miller(P, Q): | |
f = miller(n, P) | |
(x1, y1) = Q.xy() | |
return f(x = x1, y = y1) | |
def tate_pairing(P, Q, S): | |
eta = (p - 1)/n | |
num = eval_miller(P, Q+S)/eval_miller(P, S) | |
return (num^eta) | |
# r is \tau | |
r = tate_pairing(P, Q, S) | |
print "r(P, Q) =", r | |
# r^n = 1 | |
print "r(P, Q)^n =", r^n | |
P3 = P * 3 | |
Q4 = Q * 4 | |
r12 = tate_pairing(P3, Q4, S) | |
print "[3]P =", P3.xy() | |
print "[4]Q =", Q4.xy() | |
print "r([3]P, [4]Q) =", r12 | |
print "r(P, Q)^12 =", r^12 |
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