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Regular p-ary bent functions with five terms and Kloosterman sums

  • Chunming TangEmail author
  • Yanfeng Qi
  • Dongmei Huang
Article
  • 9 Downloads
Part of the following topical collections:
  1. Special Issue on Sequences and Their Applications

Abstract

Kloosterman sums are vital in the study of bent functions, including regular p-ary bent functions. In this paper, a congruence property for Kloosterman sums is presented first and is used to prove the nonexistence of a class of p-ary bent functions. Further, this paper considers p-ary functions of the form \(f(x)= \text {Tr}^{n}_{1}(a_{1}x^{r_{1}(q-1)})+\text {Tr}^{n}_{1}\left (c_{1}x^{r_{1}(q-1)+\frac {q^{2}-1}{2}}\right ) +\text {Tr}^{n}_{1}\left (a_{2}x^{r_{2}(q-1)}\right )+\text {Tr}^{n}_{1}\left (c_{2}x^{r_{2}(q-1)+\frac {q^{2}-1}{2}}\right ) +bx^{\frac {q^{2}-1}{2}}\). We use Kloosterman sums in the characterization of this class of p-ary bent functions. Finally, an open problem of Jia et al. (IEEE Trans Inf. Theory 58(9): 6054–6063, 2012) is solved and we prove the nonexistence for a class of regular p-ary bent functions.

Keywords

Regular bent functions Walsh transformation Kloosterman sums p-ary functions Congruence 

Mathematics Subject Classification (2010)

06E75 94A60 11T23 

Notes

Acknowledgments

We would like to thank the anonymous reviewers and Prof. Claude Carlet for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11871058, 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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Authors and Affiliations

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

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