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 $(\mathcal{C}^{(2)}_{j_1}-1)\cap(\mathcal{C}^{(2)}_{j_2}-2)\cap(\mathcal{C}^{(2)}_{j_3}-3) \cap(\mathcal{C}^{(2)}_{j_4}-4)$ for $j_1,j_2,j_3,j_4\in\{0,1\}$, where $\mathcal{C}^{(2)}_0, \mathcal{C}^{(2)}_1$ 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 $(\mathcal{C}^{(2)}_{i_1}-1)\cap(\mathcal{C}^{(2)}_{i_2}-2)\cap(\mathcal{C}^{(2)}_{i_3}-3)$, 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.