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

TL;DR

This paper studies the differential and boomerang properties of a specific class of binomials over finite fields, revealing new instances of functions with optimal cryptographic properties and providing comprehensive spectral classifications.

Contribution

It introduces new classes of functions with zero boomerang uniformity and classifies their spectra, advancing understanding of cryptographically strong functions over finite fields.

Findings

F_{r, ext{±}1} is locally-PN with boomerang uniformity 0 when p^n ≡ 3 mod 8

F_{r, ext{±}1} is locally-APN with boomerang uniformity ≤ 2 when p^n ≡ 7 mod 8

Complete classification of the differential and boomerang spectra for these functions

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 $r = \frac{p^n+1}{4}$, $p^n \equiv 3 \pmod{4}$, and $\chi(x) = x^{\frac{p^n -1}{2}}$ 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 $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$.