The Differential and Boomerang Properties of a Class of Binomials

Abstract

Let $q$ be an odd prime power with $q\equiv 3(\mathrm{mod}4)$. In this paper, we study the differential and boomerang properties of the function $F_{2,u}(x)=x^2(1+u\eta (x))$ over ${\mathbb{F}}_q$, where $u\in {\mathbb{F}}_q^{\ast }$ and $\eta$ is the quadratic character of ${\mathbb{F}}_q$. We determine the differential uniformity of $F_{2,u}$ for any $u\in {\mathbb{F}}_q^{\ast }$ 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.