Explicit characterization of two classes of regular bent functions

Original Paper
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Abstract

This paper considers two classes of p-ary functions studied by Li et al. (IEEE Trans Inf Theory 59(3):1818–1831, 2013). The first class of p-ary functions is of the form
$$\begin{aligned} f(x)=Tr^n_1\left( a x^{l(q-1)}+b x^{\left( l+\frac{q+1}{2}\right) (q-1)}\right) +\epsilon x^{\frac{q^2-1}{2}}. \end{aligned}$$
Another class of p-ary functions is of the form
$$\begin{aligned} f(x)={\left\{ \begin{array}{ll} \sum ^{q-1}_{i=0} Tr^n_1(a x^{(ri+s)(q-1)})+\epsilon x^{\frac{q^2-1}{2}},&{} x\ne 0,\\ f(0),&{} x=0. \end{array}\right. } \end{aligned}$$
We generalize Li et al.’s results, give necessary conditions for two classes of bent functions, and present more explicit characterization of these regular bent functions for different cases.

Keywords

Regular bent function p-ary function Walsh transform Kloosterman sums 

Mathematics Subject Classification

06E75 94A60 11T23 

Notes

Acknowledgements

The authors are very grateful to the anonymous reviewers and Prof. Teo Mora for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11401480, 11531002, 11701129). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

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

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