Cryptography and Communications

, Volume 9, Issue 5, pp 647–664

# A class of hyper-bent functions and Kloosterman sums

Article

DOI: 10.1007/s12095-016-0207-4

Tang, C. & Qi, Y. Cryptogr. Commun. (2017) 9: 647. doi:10.1007/s12095-016-0207-4

## 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 family $$\sum \limits _{r\in R}\text {Tr}_{1}^{n} (a_{r}x^{r(2^{m}-1)})$$ and Mesnager’s family $$\sum \limits _{r\in R}\text {Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)}) +\text {Tr}_{1}^{2}(bx^{\frac {2^{n}-1}{3}})$$ . In this paper, we generalize these results by considering the following class of Boolean functions over $$\mathbb {F}_{2^{n}}$$:
$$\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i} x^{r(2^{m}-1)+\frac{2^{n}-1}{3}i}) +T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}),$$
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 a characterization of hyper-bentness of these functions in terms of crucial exponential sums. For some special cases, we provide explicit characterizations for some hyper-bent functions in terms of Kloosterman sums and cubic sums. Finally, we explain how our results on binomial, trinomial and quadrinomial hyper-bent functions can be generalized to the general case where the coefficients $$a_{r,i}$$ belong to the whole field $$\mathbb {F}_{2^{n}}$$.

### Keywords

Bent functions Hyper-bent functions Walsh-Hadmard trasform Dickson polynomials Kloosterman sums

06E75 94A60