This exercise asks you to verify that the Weil pairing em is well-defined. (a) P

Question

This exercise asks you to verify that the Weil pairing em is well-defined.

(a) Prove that the value of em(P,Q) is independent of the choice of rational functions fP and fQ.

(b) Prove that the value of em(P,Q) is independent of the auxiliary point S. (Hint. Fix the points P and Q and consider the quantity

F(S) = fP (Q + S) fP (S) +fQ(P S) fQ(S) as a function of S.

Compute the divisor of F and use the fact that every nonconstant function on E has at least one zero.) You might also try to prove that the Weil pairing is bilinear, but do not be discouraged if you do not succeed, since the standard proofs use more tools than we have developed in the text.

Explanation / Answer

(a)

Result is true for n=1

Suppose it is correct for n.

M^(n+1) = M*M^n =

t 1    t^n nt^(n-1)

0 t    0 t^n

=

t^(n+1) nt^n+t^n

0 t*t^n

=

t^(n+1) (n+1)t^n

0 t^(n+1)

So this is correct

(b)

d(M^n) =

nt^(n-1)   n(n-1)t^(n-2)

0 nt^(n-1)

= n *

t^(n-1) (n-1)t^(n-2)

0 t^(n-1)

= n*M^(n-1)

(c)

e^M = sum (n>=0)(M^n / n!)

d(e^M) = sum(n>=1) nM^(n-1)/n! = sum(n>=1) M^(n-1)/(n-1)!

d(e^M) = sum(n>=0) M^n/n! (by changing index)

d(e^M) = e^M

(d) since d(e^M)=e^M we need :

e^M =

e^t e^t

0 e^t

B) I let you draw G.

Here we need to compute A^137

Notice A = M(t=7)

So :

A^137 =

7^137 137*7^136

0 7^137

So there is 137*7^136 paths of length 137 from 1->2

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