A note on the Ate pairing

摘要

The Ate pairing has been suggested since it can be computed efficiently on ordinary elliptic curves with small values of the traces of Frobenius t. However, not all pairing-friendly elliptic curves have this property. In this paper, we generalize the Ate pairing and find a series of the variations of the Ate pairing. We show that the shortest Miller loop of the variations of the Ate pairing can possibly be as small as r(1/phi(k)) on some special pairing-friendly curves with large values of Frobenius trace, and hence speed up the pairing computation significantly.