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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 $p \ge 5$. [ABSTRACT FROM AUTHOR]