The Differential Spectrum of the Power Mapping \(x^{p^n-3}\)

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 \(p \ge 5\).