On the tensor rank of multiplication in finite extensions of finite fields

TL;DR

This paper surveys existing results on the tensor rank of multiplication in finite fields and introduces new asymptotic and non-asymptotic upper bounds, advancing understanding of computational complexity in finite field arithmetic.

Contribution

It provides a comprehensive survey and establishes novel upper bounds for the tensor rank of multiplication in finite fields, both asymptotically and non-asymptotically.

Findings

New asymptotic upper bounds for tensor rank

New non-asymptotic upper bounds for tensor rank

Enhanced understanding of multiplication complexity in finite fields

Abstract

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite fields and we establish new asymptotical and not asymptotical upper bounds about it.