Two Classes of Power Mappings with Boomerang Uniformity 2

TL;DR

This paper investigates the boomerang uniformity of two specific power mappings over finite fields, revealing that both have a boomerang uniformity of 2 under certain conditions, which is significant for cryptographic applications.

Contribution

The paper identifies two classes of power mappings with boomerang uniformity 2 and analyzes their differential properties over finite fields.

Findings

Both mappings have boomerang uniformity 2 under certain conditions.

The mappings are characterized by their differential properties.

Results contribute to cryptographic function design.

Abstract

Let $q$ be an odd prime power. Let $F_1(x)=x^{d_1}$ and $F_2(x)=x^{d_2}$ be power mappings over $\mathrm{GF}(q^2)$, where $d_1=q-1$ and $d_2=d_1+\frac{q^2-1}{2}=\frac{(q-1)(q+3)}{2}$. In this paper, we study the the boomerang uniformity of $F_1$ and $F_2$ via their differential properties. It is shown that, the boomerang uniformity of $F_i$ ($i=1,2$) is 2 with some conditions on $q$.