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Permutation polynomials over finite fields are fundamental objects as theyare used in various theoretical and practical applications in cryptography,coding theory, combinatorial design, and related topics. This family ofpolynomials constitutes an active research area in which advances are beingmade constantly. In particular, constructing infinite classes of permutationpolynomials over finite fields with good differential properties (namely, low)remains an exciting problem despite much research in this direction for manyyears.  This article exhibits low differentially uniform power permutations overfinite fields of odd characteristic. Specifically, its objective is twofoldconcerning the power functions $F(x)=x^{\frac{p^n+3}{2}}$ defined over thefinite field $F_{p^n}$ of order $p^n$, where $p$ is an odd prime, and $n$ is apositive integer. The first is to complement some former results initiated byHelleseth and Sandberg in \cite{HS} by solving the open problem left open formore than twenty years concerning the determination of the differentialspectrum of $F$ when $p^n\equiv3\pmod 4$ and $p\neq 3$. The second is todetermine the exact value of its differential uniformity. Our achievements areobtained firstly by evaluating some exponential sums over $F_{p^n}$ (whichamounts to evaluating the number of $F_{p^n}$-rational points on some relatedcurves and secondly by computing the number of solutions in $(F_{p^n})^4$ of asystem of equations presented by Helleseth, Rong, and Sandberg in ["Newfamilies of almost perfect nonlinear power mappings," IEEE Trans. Inform.Theory, vol. 45. no. 2, 1999], naturally appears while determining thedifferential spectrum of $F$. We show that in the considered case($p^n\equiv3\pmod 4$ and $p\neq 3$), $F$ is an APN power permutation when$p^n=11$, and a differentially $4$-uniform power permutation otherwise.

Two low differentially uniform power permutations over odd characteristic finite fields: APN and differentially $4$-uniform functions