On The Effective Construction of Asymmetric Chudnovsky Multiplication Algorithms in Finite Fields Without Derivated Evaluation

TL;DR

This paper presents an effective asymmetric Chudnovsky multiplication algorithm in finite fields that avoids derivated evaluation, improving efficiency and providing concrete examples over various finite fields.

Contribution

It introduces a practical asymmetric multiplication algorithm in finite fields without derivated evaluation, expanding on recent theoretical generalizations.

Findings

Algorithm applied to fields like F_{16^{13}} with rational places

Extended to fields like F_{4^{13}} using higher degree places

Demonstrated efficiency over multiple finite field extensions

Abstract

The Chudnovsky and Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear whith respect to the degree of the extension. Recently, Randriambololona has generalized the method, allowing asymmetry in the interpolation procedure and leading to new upper bounds on the bilinear complexity. We describe the effective algorithm of this asymmetric method, without derivated evaluation. Finally, we give examples with the finite field $\F_{16^{13}}$ using only rational places, $\F_{4^{13}}$ using also places of degree two and $\F_{2^{13}}$ using also places of degree four.