Sequences, Bent Functions and Jacobsthal sums

The $p$-ary function $f(x)$ mapping $\mathrm{G}\mathrm{F}(p^{4k})$ to $\mathrm{G}\mathrm{F}(p)$ and given by $f(x)={\mathrm{T}\mathrm{r}}_{4k}(a{x}^d+b{x}^2)$ with $a,b\in \mathrm{G}\mathrm{F}(p^{4k})$ and $d=p^{3k}+p^{2k}-p^k+1$ is studied with the respect to its exponential sum. In the case when either $a^{p^k(p^k+1)}\neq b^{p^k+1}$or$a^2=b^d$with$b\neq 0$, this sum is shown to be three-valued and the values are determined. For the remaining cases, the value of the exponential sum is expressed using Jacobsthal sums of order $p^k+1$. Finding the values and the distribution of those sums is a long-lasting open problem.