Gaps between prime numbers and tensor rank of multiplication in finite fields

TL;DR

This paper establishes improved upper bounds on the complexity of multiplication in finite field extensions by leveraging prime gaps and algebraic curve constructions, updating previous results with recent advances.

Contribution

It combines prime gap estimates with algebraic geometry techniques to enhance bounds on tensor rank of multiplication in finite fields, updating prior unpublished work.

Findings

Provides the best known upper bounds on symmetric bilinear complexity in this context.

Extends results to classical bilinear complexity over Fp and polynomial multiplication.

Discusses open problems related to prime gaps and arithmetic functions.

Abstract

We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal construction of auxiliary divisors for multiplication algorithms by evaluation-interpolation on curves. Most of this material dates back to a 2011 unpublished work of the author, but it still provides the best results on this topic at the present time. Then a few updates are given in order to take recent developments into account, including comparison with a similar work of Ballet and Zykin, generalization to classical bilinear complexity over Fp, and to short multiplication of polynomials, as well as a discussion of open questions on gaps between prime numbers or more generally values of certain arithmetic functions.