An in-depth study of the power function $x^{q+2}$ over the finite field $\mathbb{F}_{q^2}$: the differential, boomerang, and Walsh spectra, with an application to coding theory

Let $q = p^m$, where $p$ is an odd prime number and $m$ is a positive integer. In this paper, we examine the finite field $\mathbb{F}_{q^2}$, which consists of $q^2$ elements. We first present an alternative method to determine the differential spectrum of the power function $f(x) = x^{q+2}$ on $\mathbb{F}_{q^2}$, incorporating several key simplifications. This methodology provides a new proof of the results established by Man, Xia, Li, and Helleseth in Finite Fields and Their Applications 84 (2022), 102100, which not only completely determine the differential spectrum of $f$ but also facilitate the analysis of its boomerang uniformity.Specifically, we determine the boomerang uniformity of $f$ for the cases where $q \equiv 1$ or $3$ (mod $6$), with the exception of the scenario where $p = 5$ and $m$ is even. Furthermore, for $p = 3$, we investigate the value distribution of the Walsh spectrum of $f$, demonstrating that it takes on only four distinct values. Using this result, we derive the weight distribution of a ternary cyclic code with four Hamming weights. The article integrates refined mathematical techniques from algebraic number theory and the theory of finite fields, employing several ingredients, such as exponential sums, to explore the cryptographic analysis of functions over finite fields. They can be used to explore the differential/boomerang uniformity across a wider range of functions.