On the differential and Walsh spectra of x 2 q+1 over F q 2

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 ( x ) = x 2 q +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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.