Differential spectrum of some power functions in odd prime characteristic

Abstract

Upper bound on ${\mathrm{\Delta }}_f$ of the power function $x^{\frac{p^k+1}{2}}$ in ${\mathbb{F}}_{p^n}$ (Helleseth et al. (1999) [7]) is not tight, for example $p=5$, $n=3$, and $k=2$, which is the motivation of this work. In this paper, for an odd prime p, the differential spectrum of the power function $x^{\frac{p^k+1}{2}}$ in ${\mathbb{F}}_{p^n}$ is calculated. For an odd prime p such that $p\equiv 3\mathrm{mod}4$ and odd n with $m|n$, the differential spectrum of the power function $x^{\frac{p^n+1}{p^m+1}+\frac{p^n-1}{2}}$ in ${\mathbb{F}}_{p^n}$ is also derived. We also find some new power functions which are differentially 4 and 6-uniform.