The Eta pairing revisited

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.