Differential spectra of a class of power permutations with Niho exponents

Abstract

Let \begin{document}$ m\geq3 $\end{document} be a positive integer and \begin{document}$ n = 2m $\end{document}. Let \begin{document}$ f(x) = x^{2^m+3} $\end{document} be a power permutation over \begin{document}$ {\mathrm {GF}}(2^n) $\end{document}, which is a monomial with a Niho exponent. In this paper, the differential spectrum of \begin{document}$ f $\end{document} is investigated. It is shown that the differential spectrum of \begin{document}$ f $\end{document} is \begin{document}$ \mathbb S = \{\omega_0 = 2^{2m-1}+2^{2m-3}-1,\omega_2 = 2^{2m-2}+2^{m-1}, \omega_4 = 2^{2m-3}-2^{m-1},\omega_{2^m} = 1\} $\end{document} when \begin{document}$ m $\end{document} is even, and \begin{document}$ \mathbb S = \{\omega_0 = \frac{7\cdot2^{2m-2}+2^m}3, \omega_2 = 3\cdot2^{2m-3}-2^{m-2}-1, \omega_6 = \frac{2^{2m-3}-2^{m-2}}3, \omega_{2^m+2} = 1\} $\end{document} when \begin{document}$ m $\end{document} is odd.