Differential Spectrum of Some Power Functions With Low Differential Uniformity

Abstract

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 $k|n$, the differential spectrum of the power function $x^{\frac{p^n+1}{p^k+1}+\frac{p^n-1}{2}}$ in ${\mathbb{F}}_{p^n}$ is also derived. From their differential spectrums, the differential uniformities of these two power functions are determined. We also find some new power functions having low differential uniformity.