You are asking for help in debugging your own Weil pairing code. Why
not use the Weil pairing code implemented in Sage?
In your example
sage: P.weil_pairing(Q,5)
242
sage: _^5
1
(since P and Q have order 5 the result is a 5th root of unity in the field).
I don't have time to debug your code but you should be able to write
it using genuine polynomials; your var('x') etc give symbolic objects
which are not good to use over a finite field.
John Cremona
On 29 November 2016 at 04:31, Harish SP <[email protected]> wrote:
> I was trying a sage code for weil pairing , and I got the following error
> message:
> '/usr/lib/sagemath/local/bin/sage-python: line 2: 15172 Segmentation fault
> (core dumped) sage -python "$@"' '
>
> Here is the code for weil-pairing:
>
> # Weil 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 weil_pairing(P, Q, S):
> num = eval_miller(P, Q+S)/eval_miller(P, S)
> den = eval_miller(Q, P-S)/eval_miller(Q, -S)
> return (num/den)
>
> e = weil_pairing(P, Q, S)
> print "e(P, Q) =", e
>
> # e^n = 1
> print "e(P, Q)^n =", e^n
>
> P3 = P * 3
> Q4 = Q * 4
> e12 = weil_pairing(P3, Q4, S)
>
> print "[3]P =", P3.xy()
> print "[4]Q =", Q4.xy()
> print "e([3]P, [4]Q) =", e12
> print "e(P, Q)^12 =", e^12
>
>
> Please help me
>
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