The Elliptic Net Algorithm Revisited

TL;DR

This paper enhances the Elliptic Net algorithm for pairing computations by eliminating inverse operations, applying lazy reduction, and deriving new formulas, resulting in significant speedups on specific elliptic curves.

Contribution

It introduces methods to accelerate the Elliptic Net algorithm, including inverse elimination, lazy reduction, and new formulas for optimal pairings on twisted curves.

Findings

Achieves up to 80% speedup on twisted 381-bit BLS12 curve.

Achieves up to 71.5% speedup on twisted 676-bit KSS18 curve.

Demonstrates significant performance improvements over previous algorithms.

Abstract

Pairings have been widely used since their introduction to cryptography. They can be applied to identity-based encryption, tripartite Diffie-Hellman key agreement, blockchain and other cryptographic schemes. The Acceleration of pairing computations is crucial for these cryptographic schemes or protocols. In this paper, we will focus on the Elliptic Net algorithm which can compute pairings in polynomial time, but it requires more storage than Miller's algorithm. We use several methods to speed up the Elliptic Net algorithm. Firstly, we eliminate the inverse operation in the improved Elliptic Net algorithm. In some circumstance, this finding can achieve further improvements. Secondly, we apply lazy reduction technique to the Elliptic Net algorithm, which helps us achieve a faster implementation. Finally, we propose a new derivation of the formulas for the computation of the Optimal Ate pairing on the twisted curve. Results show that the Elliptic Net algorithm can be significantly accelerated especially on the twisted curve. The algorithm can be $80\%$ faster than the previous ones on the twisted 381-bit BLS12 curve and $71.5\%$ faster on the twisted 676-bit KSS18 curve respectively.