Differential spectrum of a class of APN power functions

APN power functions are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. Let p be an odd prime and n be a positive integer. Let \(F(x)=x^d\) be a power function over \({\mathbb {F}}_{p^n}\), where \(d=\frac{3p^n-1}{4}\) when \(p^n\equiv 3\pmod 8\) and \(d=\frac{p^n+1}{4}\) when \(p^n\equiv 7\pmod 8\). When \(p^n>7\), F is an APN function, which is proved by Helleseth et al. (IEEE Trans Inform Theory 45(2):475–485, 1999). In this paper, we study the differential spectrum of F. By investigating some system of equations, the number of solutions of certain system of equations and consequently the differential spectrum of F can be expressed by quadratic character sums over \({\mathbb {F}}_{p^n}\). By the theory of elliptic curves over finite fields, the differential spectrum of F can be investigated by a given p. It is the fourth infinite family of APN power functions with nontrivial differential spectrum.