Optimization of the scalar complexity of Chudnovsky$^2$ multiplication algorithms in finite fields

TL;DR

This paper introduces new constructions and optimization strategies for Chudnovsky$^2$ multiplication algorithms in finite fields, significantly reducing scalar complexity and improving existing methods for small extensions.

Contribution

It presents novel constructions and generic optimization strategies for Chudnovsky$^2$ algorithms, with a case study improving the Baum-Shokrollahi method for specific finite fields.

Findings

Reduced scalar complexity in Chudnovsky$^2$ algorithms

Improved multiplication in _{256}/_4

Enhanced elliptic Chudnovsky$^2$ algorithms for small extensions

Abstract

We propose several constructions for the original multiplication algorithm of D.V. and G.V. Chudnovsky in order to improve its scalar complexity. We highlight the set of generic strategies who underlay the optimization of the scalar complexity, according to parameterizable criteria. As an example, we apply this analysis to the construction of type elliptic Chudnovsky$^2$ multiplication algorithms for small extensions. As a case study, we significantly improve the Baum-Shokrollahi construction for multiplication in $\mathbb F_{256}/\mathbb F_4$.