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The answer can be more simplest ! By definition a pairing e, is a non degenerate bilinear form with image in the group of N-root of Unity. Then $\forall P, Q \in E(\bar{K}): e(P,Q)^N = 1$ $E(\bar{K})$ is the torsion group of the elliptic curve E over the field K.

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I think that there are some confusion about the parameters of computing Pairing, and specially in the definition of group order. Pairing which make use of elliptic curves, were firstly intoduced by Dan Boneh from his first seminal "Identity Based Encryption from Weil Pairing; SIAM J. of Computing 2001. Many other sheemes were introduced since, and make use ...

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DrLecter's comment should really be an answer, but one point which bears emphasis is that it is not sufficient for crypto purposes to say "a (cyclic) group of prime order $p$", because even though mathematically all groups of order $p$ are essentially the same group (i.e., they are isomorphic), we can't really say the same thing in crypto. When doing crypto ...

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