Paper 2022/610

On the Differential Spectrum of a Differentially $3$-Uniform Power Function

Tingting Pang, Nian Li, and Xiangyong Zeng

Abstract

In this paper, we investigate the cardinality, denoted by $(j_1,j_2,j_3,j_4{)}_2$, of the intersection of $({\mathcal{C}}_{j_1}^{(2)}-1)\cap ({\mathcal{C}}_{j_2}^{(2)}-2)\cap ({\mathcal{C}}_{j_3}^{(2)}-3)\cap ({\mathcal{C}}_{j_4}^{(2)}-4)$ for $j_1,j_2,j_3,j_4\in \{0,1\}$, where ${\mathcal{C}}_0^{(2)},{\mathcal{C}}_1^{(2)}$ are the cyclotomic classes of order two over the finite field ${\mathbb{F}}_{p^n}$, $p$ is an odd prime and $n$ is a positive integer. By making most use of the results on cyclotomic classes of orders two and four as well as the cardinality of the intersection $({\mathcal{C}}_{i_1}^{(2)}-1)\cap ({\mathcal{C}}_{i_2}^{(2)}-2)\cap ({\mathcal{C}}_{i_3}^{(2)}-3)$, we compute the values of $(j_1,j_2,j_3,j_4{)}_2$ in the case of $p=5$, where $i_1,i_2,i_3\in \{0,1\}$. As a consequence, the power function $x^{\frac{5^n-1}{2}+2}$ over ${\mathbb{F}}_{5^n}$ is shown to be differentially $3$-uniform and its differential spectrum is also completely determined.