Differential Spectrum of Some Power Functions With Low Differential Uniformity

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 $p\equiv 3\bmod 4$ 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.