Cryptography and Communications

, Volume 9, Issue 5, pp 647–664

A class of hyper-bent functions and Kloosterman sums

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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 

Mathematics Subject Classification (2010)

06E75 94A60 

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematics and InformationChina West Normal UniversitySichuanChina
  2. 2.School of ScienceHangzhou Dianzi UniversityHangzhouChina

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