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Only three classes of Almost Perfect Nonlinear (for short, APN) powerfunctions over odd characteristic finite fields have been investigated in theliterature, and their differential spectra were determined. The differentialuniformity of the power function $F(x)=x^{\frac{p^{n}-3}{2}}$ over the finitefield $F_{p^n}$ of order $p^n$ (where $p$ is an odd prime), was studied byHelleseth and Sandberg in 1997, where $p^n\equiv3\pmod{4}$ is an odd primepower with $p^n>7$. It was shown that $F$ is PN when $p^n=27$, APN when $5$ isa nonsquare in $F_{p^n}$, and differentially $3$-uniform when $5$ is a squarein $F_{p^n}$. In this paper, by investigating some equation systems and certaincharacter sums over $F_{p^n}$, the differential spectrum of $F$ is completelydetermined. 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 ofthe lowest differential uniformity and the nontrivial differential spectrum.Moreover, we examine the extension of the so-called $c$-differential uniformityby investigating the $c$-differential properties of $F$. Specifically, an upperbound of the $c$-differential uniformity of $F$ is given, and its$c$-differential spectrum is considered in the case where $c=-1$. Finally, weemphasize that, throughout our study of the differential spectrum of theconsidered power functions, we provide methods for evaluating sums of specificcharacters with connections to elliptic curves and for determining the numberof solutions of specific systems of equations over finite fields.

On the differential spectrum of a class of APN power functions over odd characteristic finite fields and their $c$-differential properties