On the Differential Properties of the Power Mapping $x^{p^m+2}$

Let $m$ be a positive integer and $p$ a prime. In this paper, we investigate the differential properties of the power mapping $x^{pm+2}$ over $\mathbb{F}_{p^n}$, where $n=2m$ or $n=2m-1$. For the case $n=2m$, by transforming the derivative equation of $x^{pm+2}$ and studying some related equations, we completely determine the differential spectrum of this power mapping. For the case $n=2m-1$, the derivative equation can be transformed to a polynomial of degree $p+3$. The problem is more difficult and we obtain partial results about the differential spectrum of $x^{pm+2}$.