  Let $n$ be a positive integer and $p$ a prime. The power mapping $x^{p^n-3}$
over $\mathbb{F}_{p^n}$ has desirable differential properties, and its
differential spectra for $p=2,\,3$ have been determined. In this paper, for any
odd prime $p$, by investigating certain quadratic character sums and some
equations over $\mathbb{F}_{p^n}$, we determine the differential spectrum of
$x^{p^n-3}$ with a unified approach. The obtained result shows that for any
given odd prime $p$, the differential spectrum can be expressed explicitly in
terms of $n$. Compared with previous results, a special elliptic curve over
$\mathbb{F}_{p}$ plays an important role in our computation for the general
case $p \ge 5$.
