On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic
Namhun Koo, Soonhak Kwon
TL;DR
This work analyzes the differential and boomerang properties of the binomial form $F_{r,u}(x)=x^r\bigl(1+u\chi(x)\bigr)$ over the odd-characteristic finite field $\mathbb{F}_{p^n}$ with $r=\frac{p^n+1}{4}$ and $p^n\equiv3\pmod4$, where $\chi$ is the quadratic character. It derives exact differential and boomerang spectra and identifies how the parameter $u$ and the congruence class of $p^n$ mod $8$ govern PN/APN behavior and permutation properties. Notably, $F_{r,\pm1}$ is locally-PN with boomerang uniformity $0$ when $p^n\equiv3\pmod8$ (the second non-PN class with zero boomerang uniformity and the first over odd characteristic with $p>3$) and locally-APN with boomerang uniformity at most $2$ when $p^n\equiv7\pmod8$. The paper also classifies the differential uniformity for $u\neq\pm1$, showing a dichotomy: $4$-uniform or $5$-uniform permutations under explicit character conditions, and provides detailed spectra results that enhance the understanding of low-differential- and low-boomerang-uniformity binomials beyond previously known families.
Abstract
In this paper, we investigate the differential and boomerang properties of a class of binomial $F_{r,u}(x) = x^r(1 + uχ(x))$ over the finite field $\mathbb{F}_{p^n}$, where $r = \frac{p^n+1}{4}$, $p^n \equiv 3 \pmod{4}$, and $χ(x) = x^{\frac{p^n -1}{2}}$ is the quadratic character in $\mathbb{F}_{p^n}$. We show that $F_{r,\pm1}$ is locally-PN with boomerang uniformity $0$ when $p^n \equiv 3 \pmod{8}$. 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 $F_{r,\pm1}$ is locally-APN with boomerang uniformity at most $2$ when $p^n \equiv 7 \pmod{8}$. We also provide complete classifications of the differential and boomerang spectra of $F_{r,\pm1}$. Furthermore, we thoroughly investigate the differential uniformity of $F_{r,u}$ for $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$.