  Permutation polynomials over finite fields are fundamental objects as they
are used in various theoretical and practical applications in cryptography,
coding theory, combinatorial design, and related topics. This family of
polynomials constitutes an active research area in which advances are being
made constantly. In particular, constructing infinite classes of permutation
polynomials over finite fields with good differential properties (namely, low)
remains an exciting problem despite much research in this direction for many
years.
  This article exhibits low differentially uniform power permutations over
finite fields of odd characteristic. Specifically, its objective is twofold
concerning the power functions $F(x)=x^{\frac{p^n+3}{2}}$ defined over the
finite field $F_{p^n}$ of order $p^n$, where $p$ is an odd prime, and $n$ is a
positive integer. The first is to complement some former results initiated by
Helleseth and Sandberg in \cite{HS} by solving the open problem left open for
more than twenty years concerning the determination of the differential
spectrum of $F$ when $p^n\equiv3\pmod 4$ and $p\neq 3$. The second is to
determine the exact value of its differential uniformity. Our achievements are
obtained firstly by evaluating some exponential sums over $F_{p^n}$ (which
amounts to evaluating the number of $F_{p^n}$-rational points on some related
curves and secondly by computing the number of solutions in $(F_{p^n})^4$ of a
system of equations presented by Helleseth, Rong, and Sandberg in ["New
families of almost perfect nonlinear power mappings," IEEE Trans. Inform.
Theory, vol. 45. no. 2, 1999], naturally appears while determining the
differential 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.
