The Differential and Boomerang Properties of a Class of Binomials

TL;DR

This paper investigates the differential and boomerang properties of a class of binomial functions over finite fields, providing exact uniformity measures and disproving a conjecture about their APN status.

Contribution

It determines the differential uniformity and boomerang properties of specific binomial functions, and disproves a conjecture regarding their APN nature.

Findings

Determined the differential uniformity of $F_{2,u}$ for all $u$.

Computed the differential spectra and boomerang uniformity of $F_{2, ext{±}1}$.

Disproved the conjecture that infinitely many such functions are APN.

Abstract

Let $q$ be an odd prime power with $q\equiv 3\ ({\rm{mod}}\ 4)$. In this paper, we study the differential and boomerang properties of the function $F_{2,u}(x)=x^2\big(1+u\eta(x)\big)$ over $\mathbb{F}_{q}$, where $u\in\mathbb{F}_{q}^*$ and $\eta$ is the quadratic character of $\mathbb{F}_{q}$. We determine the differential uniformity of $F_{2,u}$ for any $u\in\mathbb{F}_{q}^*$ and determine the differential spectra and boomerang uniformity of the locally-APN functions $F_{2,\pm 1}$, thereby disproving a conjecture proposed in \cite{budaghyan2024arithmetization} which states that there exist infinitely many $q$ and $u$ such that $F_{2,u}$ is an APN function.