**A new class of hyper-bent functions and Kloosterman sums**

*Chunming Tang, Yanfeng Qi*

**Abstract: **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.

**Category / Keywords: **Bent function, hyper-bent functions, Walsh-Hadamard transform, Dickson polynomial, Kloosterman sums

**Date: **received 26 Dec 2013

**Contact author: **tangchunmingmath at 163 com

**Available format(s): **PDF | BibTeX Citation

**Version: **20131229:115417 (All versions of this report)

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