On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic

Abstract

In this paper, we investigate the differential and boomerang properties of a class of binomial $F_{r,u}(x)=x^r(1+u\chi (x))$ over the finite field ${\mathbb{F}}_{p^n}$, where $r=\frac{p^n+1}{4}$, $p^n\equiv 3(\mathrm{mod}4)$, and $\chi (x)=x^{\frac{p^n-1}{2}}$ is the quadratic character in ${\mathbb{F}}_{p^n}$. We show that $F_{r,\pm 1}$ is locally-PN with boomerang uniformity $0$ when $p^n\equiv 3(\mathrm{mod}8)$. To the best of our knowledge, it is the second known non-PN function class with boomerang uniformity $0$, and the first such example over odd characteristic fields with $p>3$. Moreover, we show that $F_{r,\pm 1}$ is locally-APN with boomerang uniformity at most $2$ when $p^n\equiv 7(\mathrm{mod}8)$. We also provide complete classifications of the differential and boomerang spectra of $F_{r,\pm 1}$. Furthermore, we thoroughly investigate the differential uniformity of $F_{r,u}$ for $u\in {\mathbb{F}}_{p^n}^{\ast }\setminus \{\pm 1\}$.