Some New Optimal Pairings

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