The elliptic net algorithm revisited

Abstract

Efficient implementation of pairings is a fundamental ingredient in pairing-based cryptographic protocols. The Elliptic Net algorithm is an alternative method to Miller’s algorithm for computing the (Optimal) Ate pairing in polynomial time. In this paper, we utilize several tricks to speed up the Elliptic Net algorithm. Firstly, we eliminate the inversion in the improved Elliptic Net algorithm, which allows for further improvements under certain circumstances. Second, we apply lazy reduction to the Elliptic Net algorithm for a better performance. Finally, we propose a new derivation of the formulas for computing the (Optimal) Ate pairing on the twisted curves. In addition, we provide implementations of all versions of the Elliptic Net algorithm on personal computers based on RELIC toolkit. Our implementations indicate that on this research line the Elliptic Net algorithm is about \(80\%\) faster than the previous fastest ones on the twisted 381-bit BLS12 curve and \(71.5\%\) faster on the twisted 676-bit KSS18 curve on 64-bit platforms, respectively.