Weil sums of binomials: properties, applications, and open problems

TL;DR

This survey reviews Weil sums of binomials over finite fields, their properties, applications in cryptography and sequence analysis, and discusses open problems and recent research progress.

Contribution

It provides a comprehensive overview of Weil sums of binomials, highlighting their properties, applications, and open research questions in finite field theory.

Findings

Characterization of Weil sum spectra

Bounds on Weil sums established

Open problems in spectrum analysis identified

Abstract

We present a survey on Weil sums in which an additive character of a finite field $F$ is applied to a binomial whose individual terms (monomials) become permutations of $F$ when regarded as functions. Then we indicate how these Weil sums are used in applications, especially how they characterize the nonlinearity of power permutations and the correlation of linear recursive sequences over finite fields. In these applications, one is interested in the spectrum of Weil sum values that are obtained as the coefficients in the binomial are varied. We review the basic properties of such spectra, and then give a survey of current topics of research: Archimedean and non-Archimedean bounds on the sums, the number of values in the spectrum, and the presence or absence of zero in the spectrum. We indicate some important open problems and discuss progress that has been made on them.