A direct method for calculating the differential spectrum of an APN power mapping

Let n be a positive integer, p be an odd prime, \(d=\frac{p^n+1}{4}+\frac{p^n-1}{2}\) $d=\frac{p^n+1}{4}+\frac{p^n-1}{2}$ if \(p^n \equiv 3 ~(\mathrm{mod ~8})\) $p^n\equiv 3(\mathrm{mod}8)$ and \(d=\frac{p^n+1}{4}\) $d=\frac{p^n+1}{4}$ if \(p^n \equiv 7 ~(\mathrm{mod ~8})\) $p^n\equiv 7(\mathrm{mod}8)$ . When \(p^n>7\) $p^n>7$ , the power mapping \(x^d\) $x^d$ from \(\mathbb {F}_{p^n}\) ${\mathbb{F}}_{p^n}$ to \(\mathbb {F}_{p^n}\) ${\mathbb{F}}_{p^n}$ was proved to be almost perfect nonlinear by Helleseth, Rong and Sandberg in IEEE Trans. Inform. Theory, 45(2): 475-485, 1999. By regarding the components in the differential spectrum as unknowns and establishing a system of linear equations concerning them, Tan and Yan completely determined the differential spectrum of this power mapping in Des. Codes Cryptogr., 91(8): 2755-2768, 2023. In this paper, we directly characterize the conditions on \(b\in \mathbb {F}_{p^n}\) $b\in {\mathbb{F}}_{p^n}$ under which the derivative equation \((x+1)^d-x^d=b\) ${(x+1)}^d-x^d=b$ has exactly i solution(s) for \(i=0,1,2\) $i=0,1,2$ , respectively. Then, utilizing the theory of character sums, the number of those b’s in each case is determined and thus the differential spectrum of \(x^d\) $x^d$ is obtained. Our method provides more information about the derivative equation of \(x^d\) $x^d$ , which can be used to describe the DDT of this APN power function.