New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

TL;DR

This paper introduces new uniform and asymptotic upper bounds on the tensor rank of multiplication in finite field extensions, improving understanding of computational complexity in finite field arithmetic.

Contribution

It provides novel uniform bounds applicable to all finite fields and derives improved asymptotic bounds using Shimura curves with optimal rational place ratios.

Findings

New uniform upper bounds for tensor rank in finite fields

Enhanced asymptotic bounds leveraging Shimura curves

Bounds applicable to all prime power finite fields

Abstract

We obtain new uniform upper bounds for the (non necessarily symmetric) tensor rank of the multiplication in the extensions of the finite fields $\F_q$ for any prime or prime power $q\geq2$; moreover these uniform bounds lead to new asymptotic bounds as well. In addition, we also give purely asymptotic bounds which are substantially better by using a family of Shimura curves defined over $\F_q$, with an optimal ratio of $\F_{q^t}$-rational places to their genus where $q^t$ is a square.