# The Eta Pairing Revisited

Florian Hess, Nigel Smart, Fre Vercauteren, The Eta Pairing Revisited. *IEEE Transactions on Information Theory*, 52(10), pp. 4595–4602. October 2006. No electronic version available. External information
## Abstract

In this paper we simplify and extend the Eta pairing, originally
discovered in the setting of supersingular curves by Barreto et al.,
to ordinary curves. Furthermore, we show that by swapping the
arguments of the Eta pairing, one obtains a very efficient algorithm
resulting in a speed-up of a factor of around six
over the usual Tate pairing, in the case of curves
which have large security parameters, complex multiplication
by an order of $\Q(\sqrt{-3})$,
and when the trace of Frobenius is chosen to be suitably small.
Other, more minor savings are obtained for more general curves.

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