Differential uniformity properties of some classes of permutation polynomials

TL;DR

This paper investigates the $c$-differential uniformity of permutation polynomials, providing new classes with low uniformity by analyzing Weil sums and finite field equations, advancing cryptographic function design.

Contribution

It introduces new classes of permutation polynomials with low $c$-differential uniformity using novel analytical techniques.

Findings

Identified permutation polynomials with low $c$-differential uniformity.

Developed methods involving Weil sums and finite field equations.

Contributed to the mathematical understanding of $c$-differential properties.

Abstract

The notion of $c$-differential uniformity has recently received a lot of attention since its proposal~\cite{Ellingsen}, and recently a characterization of perfect $c$-nonlinear functions in terms of difference sets in some quasigroups was obtained in~\cite{AMS22}. Independent of their applications as a measure for certain statistical biases, the construction of functions, especially permutations, with low $c$-differential uniformity is an interesting mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low $c$-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest.