  Only three classes of Almost Perfect Nonlinear (for short, APN) power
functions over odd characteristic finite fields have been investigated in the
literature, and their differential spectra were determined. The differential
uniformity of the power function $F(x)=x^{\frac{p^{n}-3}{2}}$ over the finite
field $F_{p^n}$ of order $p^n$ (where $p$ is an odd prime), was studied by
Helleseth and Sandberg in 1997, where $p^n\equiv3\pmod{4}$ is an odd prime
power with $p^n>7$. It was shown that $F$ is PN when $p^n=27$, APN when $5$ is
a nonsquare in $F_{p^n}$, and differentially $3$-uniform when $5$ is a square
in $F_{p^n}$. In this paper, by investigating some equation systems and certain
character sums over $F_{p^n}$, the differential spectrum of $F$ is completely
determined. We focusing on the power functions $x^d$ with even $d$ over
$F_{p^n}$ ($p$ odd), the power functions $F$ we consider are APN which are of
the lowest differential uniformity and the nontrivial differential spectrum.
Moreover, we examine the extension of the so-called $c$-differential uniformity
by investigating the $c$-differential properties of $F$. Specifically, an upper
bound of the $c$-differential uniformity of $F$ is given, and its
$c$-differential spectrum is considered in the case where $c=-1$. Finally, we
emphasize that, throughout our study of the differential spectrum of the
considered power functions, we provide methods for evaluating sums of specific
characters with connections to elliptic curves and for determining the number
of solutions of specific systems of equations over finite fields.
