Bent and Hyper-bent Functions over a Field of $2^l$ Elements

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $P=\mathbb{F}_q$ with $q=2^l$ elements, $l>1$. Any such function is identified with a function $F:Q\to P$, where $P<Q=\mathbb{F}_qn$. The latter has a reduced trace representation $F=\mathrm{tr}^Q_P(\Phi)$, where $\Phi(x)$ is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case $l=1$ to the case $l>1$ is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case $l=1$ these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [Youssef, A. M. and Gong, G., Lect. Notes Comp. Sci., vol. 2045, Berlin: Springer, 2001, pp. 406–419]; we indicate a set of parameters $q$ and $n$ for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics.