Explicit formulas for efficient multiplication in F_{3^{6m}}

TL;DR

This paper introduces explicit formulas and a new method based on FFT to reduce the number of multiplications in F_{3^{6m}} fields, enhancing efficiency for pairing-based cryptography.

Contribution

It presents a novel approach using FFT and explicit formulas to decrease F_{3^{6m}} multiplications from 18 to 15, improving computational efficiency.

Findings

Reduction of multiplications from 18 to 15 in F_{3^{6m}}

Implementation results demonstrate improved efficiency

Explicit formulas facilitate faster computations

Abstract

Efficient computation of the Tate pairing is an important part of pairing-based cryptography. Recently with the introduction of the Duursma-Lee method special attention has been given to the fields of characteristic 3. Especially multiplication in F_{3^{6m}}, where m is prime, is an important operation in the above method. In this paper we propose a new method to reduce the number of F_{3^m} multiplications for multiplication in F_{3^{6m}} from 18 in recent implementations to 15. The method is based on the fast Fourier tranmsform and explicit formulas are given. The execution times of our software implementations for F_{3^{6m}} show the efficiency of our results.