International Association
for Cryptologic Research

This paper is devoted to the characterization of hyper-bent functions.

Several classes of hyper-bent functions have been studied, such as

Charpin and Gong\'s $\\sum\\limits_{r\\in R}\\mathrm{Tr}_{1}^{n}

(a_{r}x^{r(2^m-1)})$ and Mesnager\'s $\\sum\\limits_{r\\in R}\\mathrm{Tr}_{1}^{n}(a_{r}x^{r(2^m-1)})

+\\mathrm{Tr}_{1}^{2}(bx^{\\frac{2^n-1}{3}})$, where $R$ is a set of representations of the cyclotomic

cosets modulo $2^m+1$ of full size $n$ and $a_{r}\\in \\mathbb{F}_{2^m}$.

In this paper, we generalize their results and consider a class of Boolean functions of the form $\\sum_{r\\in R}\\sum_{i=0}^{2}Tr^n_1(a_{r,i}x^{r(2^m-1)+\\frac{2^n-1}{3}i})

+Tr^2_1(bx^{\\frac{2^n-1}{3}})$, where $n=2m$, $m$ is odd, $b\\in\\mathbb{F}_4$, and $a_{r,i}\\in \\mathbb{F}_{2^n}$.

With the restriction of $a_{r,i}\\in \\mathbb{F}_{2^m}$, we present the characterization of hyper-bentness of these functions with character sums. Further, we reformulate this characterization in terms of the number of points on

hyper-elliptic curves. For some special cases, with the help of Kloosterman sums and cubic sums, we determine the characterization for some hyper-bent functions including functions with four, six and ten traces terms. Evaluations of Kloosterman sums at three general points are used in the characterization. Actually, our results can generalized to the general

case: $a_{r,i}\\in \\mathbb{F}_{2^n}$. And we explain this for characterizing binomial, trinomial and quadrinomial hyper-bent functions.