Refinements of Miller's Algorithm over Weierstrass Curves Revisited

TL;DR

This paper extends Miller's algorithm for pairing computations on elliptic curves, eliminating all vertical lines to achieve about 25% faster performance on general curves.

Contribution

It introduces a method to remove all vertical lines in Miller's algorithm for general elliptic curves, improving efficiency beyond previous refinements.

Findings

Achieves approximately 25% faster pairing computations.

Applicable to all pairing-friendly elliptic curves.

Extends previous methods to general elliptic curves.

Abstract

In 1986 Victor Miller described an algorithm for computing the Weil pairing in his unpublished manuscript. This algorithm has then become the core of all pairing-based cryptosystems. Many improvements of the algorithm have been presented. Most of them involve a choice of elliptic curves of a \emph{special} forms to exploit a possible twist during Tate pairing computation. Other improvements involve a reduction of the number of iterations in the Miller's algorithm. For the generic case, Blake, Murty and Xu proposed three refinements to Miller's algorithm over Weierstrass curves. Though their refinements which only reduce the total number of vertical lines in Miller's algorithm, did not give an efficient computation as other optimizations, but they can be applied for computing \emph{both} of Weil and Tate pairings on \emph{all} pairing-friendly elliptic curves. In this paper we extend the Blake-Murty-Xu's method and show how to perform an elimination of all vertical lines in Miller's algorithm during Weil/Tate pairings computation on \emph{general} elliptic curves. Experimental results show that our algorithm is faster about 25% in comparison with the original Miller's algorithm.