On a class of APN power functions over odd characteristic finite fields: Their differential spectrum and c-differential properties

The differential spectrum of a cryptographic function is of significant interest for estimating the resistance of the involved vectorial function to some variances of differential cryptanalysis. It is well-known that it is difficult to determine a power function's differential spectrum completely. In the present article, we concentrate on studying the differential and the cdifferential uniformity (for some c is an element of Fpn \ {0, 1}) and their related differential spectrum (resp. c -differential spectrum) of the power functions F(x) = xd over the finite field Fpn of order pn (where p is an odd prime) for d = pn-3 2 . We emphasize that by focusing on the power functions xd with even d over Fpn (p odd), the considered functions are APN, that is, of the lowest differential uniformity and the nontrivial differential spectrum. By investigating some system of equations and specific character sums over Fpn, the differential spectrum of F is completely determined. Moreover, we examine the extension of the so-called c -differential uniformity by investigating the c -differential properties of F. Specifically, an upper bound of the c -differential uniformity of F is given, and its cdifferential spectrum is considered in the case where c = -1. Finally, we highlight that, throughout our study of the differential spectrum of the considered power functions, we provide methods for evaluating sums of specific characters with connections to elliptic curves and determining the number of solutions of specific systems of equations over finite fields. (c) 2024 Elsevier B.V. All rights reserved.