Beta Weil pairing revisited

Abstract

In this paper, we extend the $$\beta $$ -Weil pairing, initially introduced in the setting of ordinary elliptic curves with even embedding degree by Aranha et al. [2], to ordinary elliptic curves of any embedding degree. We also propose a new optimal pairing which is the product of some rational functions with the same Miller loop and having a simple final exponentiation. The new pairing is appropriated for using the multi-pairing technique for an efficient implementation. We focus our computation at high security level. Exploiting the fact that the $$\beta $$ -Weil pairing is suitable for parallel execution, we first show that calculating the extended $$\beta $$ -Weil pairing over pairing-friendly elliptic curves with embedding degree 27 is more efficient than calculating the optimal ate pairing. Finally we show that calculating our new pairing over Barreto–Lynn–Scott curves with embedding degree 12 (BLS12) and pairing-friendly elliptic curves with embedding degree 15 is more efficient than calculating the optimal ate pairing.