questions to weil pairing

Hi all,

I started reading about some crypto-schemes based on bilinear pairings.
I think, that I understand the basic stuff, but in one scheme, I started
to analyze today, they use some "strange" relations. It would be really
cool, if you could help me understanding, what they are doing there.

So let e: G_1 x G_1 -> G_2 be an admissible bilinear pairing. Let G_1
and G_2 be 2 groups of prime order q. Let Q be a generator of G_1.
First, 2 basic questions:
(i) Is the pairing symmetric? So is the following correct, P \in G_1:
e(P,Q)=e(Q,P)?
(ii) Let a,b,c \in Z_q and Q_a =aQ, Q_b =bQ then we have:
e(P,Q_a)^c * e(P,Q_b)^c
=e(P,Q)^{ac} * e(P,Q)^{bc}
=e(P,Q)^{ac+bc}
=e(P,Q)^{c(a+b)}
=e(P,(a+b)Q)^{c}
=e(P,Q_a*Q_b)^{c}
I suppose this is correct. Is my suggestion correct?

Now, the authers use something, what seems strange to me. The use the
following relation:
e(P,Q_a)^{-c} * e(P,Q_b)^{c} = e(Q_a*Q_b,P)^c
Is there any reason, why this could be correct?

Thanks for your help,
Andreas
Received on Thu Sep 29 21:52:24 2005