Multiplication in finite fields with Chudnovsky-type algorithms on the projective line

TL;DR

This paper introduces a recursive polynomial construction for multiplication in finite fields, achieving quasi-linear complexity, with deterministic and polynomial-time algorithm construction, based on Chudnovsky's method on the projective line.

Contribution

It presents a new recursive polynomial generic construction for finite field multiplication algorithms with quasi-linear complexity and deterministic polynomial-time construction methods.

Findings

Quasi-linear bilinear complexity in extension degree n.

Deterministic construction of algorithms in polynomial time.

Asymptotic bounds for the complexity of the construction.

Abstract

We propose a Recursive Polynomial Generic Construction (RPGC) of multiplication algorithms in any finite field $\mathbb{F}_{q^n}$ based on the method of D.V. and G.V. Chudnovsky specialized on the projective line. They are usual polynomial interpolation algorithms in small extensions and the Karatsuba algorithm is seen as a particular case of this construction. Using an explicit family of such algorithms, we show that their bilinear complexity is quasi-linear with respect to the extension degree n, and we give a uniform bound for this complexity. We also prove that the construction of these algorithms is deterministic and can be done in polynomial time. We give an asymptotic bound for the complexity of their construction.