The differential spectrum and boomerang spectrum of a class of locally-APN functions

TL;DR

This paper analyzes the differential and boomerang spectra of a class of power functions over finite fields, revealing their local-APN properties and uniformity, and generalizing previous specific cases to an infinite class.

Contribution

It determines the differential and boomerang spectra of power mappings over finite fields, extending prior results to a broader class and establishing their boomerang uniformity.

Findings

Differential spectrum of the power mapping is determined.

The power mapping is shown to be locally-APN.

Boomerang uniformity is 4 for certain parameters and 2 otherwise.

Abstract

In this paper, we study the boomerang spectrum of the power mapping $F(x)=x^{k(q-1)}$ over ${\mathbb F}_{q^2}$, where $q=p^m$, $p$ is a prime, $m$ is a positive integer and $\gcd(k,q+1)=1$. We first determine the differential spectrum of $F(x)$ and show that $F(x)$ is locally-APN. This extends a result of [IEEE Trans. Inf. Theory 57(12):8127-8137, 2011] from $(p,k)=(2,1)$ to general $(p,k)$. We then determine the boomerang spectrum of $F(x)$ by making use of its differential spectrum, which shows that the boomerang uniformity of $F(x)$ is 4 if $p=2$ and $m$ is odd and otherwise it is 2. Our results not only generalize the results in [Des. Codes Cryptogr. 89:2627-2636, 2021] and [arXiv:2203.00485, 2022] but also extend the example $x^{45}$ over ${\mathbb F}_{2^8}$ in [Des. Codes Cryptogr. 89:2627-2636, 2021] into an infinite class of power mappings with boomerang uniformity 2.