On the differential and Walsh spectra of x2q+1 over Fq2

Let q be an odd prime power and let $F_{q^2}$ be the finite field with $q^2$ elements. In this paper, we first present a method to determine the differential spectrum of the power function $F\left(x\right)=x^{2q+1}$ on $F_{q^2}$, which provides an alternative proof of the results established by Man et al. (2022) [23]. When the characteristic of $F_{q^2}$ is 3, we also determine the value distribution of the Walsh spectrum of F, showing that it is 4-valued, and use the obtained result to determine the weight distribution of a 4-weight cyclic code.