Some New Optimal Pairings

The Ate pairing can be computed efficiently on ordinary elliptic curves with small value of the trace of Frobenius t. The Ate pairing generalizes the Ate pairing, and can possibly shorten the Miller loop to be as small as r 1/φ(k) on some special pairing-friendly curves with large value of Frobenius t. However, not all pairing-friendly curves have this property. In this paper, we generalize the Ate_i pairing further. By our method, we can shorten the the Miller loop to be nearly r 1/φ(k) on some pairing-friendly curves, while the Ate pairing can not reach.