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A Note on Semi-bent and Hyper-bent Boolean Functions

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Information Security and Cryptology (Inscrypt 2013)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8567))

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Abstract

Semi-bent and hyper-bent funcitons as two classes of Boolean functions with low Walsh transform, are applied in cryptography and commnunications. This paper considers a new class of semi-bent quadratic Boolean function and a generalization of a new class of hyper-bent Boolean functions. The new class of semi-bent quadratic Boolean function of the form \(f(x)=\sum _{i=1}^{\lfloor \frac{m-1}{2}\rfloor }Tr^n_1(c_ix^{1+4^{i}}) (c_i\in \mathbb {F}_4\),\(n=2m)\) is simply characterized and enumerated. Then we present the characterization of a generalization of a new class of hyper-bent Boolean functions of the form \(f^{(r)}_{a,b}:=\mathrm {Tr}_{1}^{n}(ax^{r(2^m-1)}) +\mathrm {Tr}_{1}^{4}(bx^{\frac{2^n-1}{5}})\), where \(n=2m\), \(m\equiv 2\pmod 4\), \(a\in \mathbb {F}_{2^m}\) and \(b\in \mathbb {F}_{16}\).

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Acknowledgements

The authors would like to thank anonymous reviewers for their helpful advice and comments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10990011, 11401480, 61272499), and Science and Technology on Information Assurance Laboratory (Grant No. KJ-11-02). Yanfeng Qi acknowledges support from Aisino Corporation Inc.

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Authors and Affiliations

  1. School of Mathematics and Information, China West Normal University, Nanchong, 637002, Sichuan, China

    Chunming Tang

  2. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Chunming Tang, Yu Lou, Yanfeng Qi & Maozhi Xu

  3. Aisino Corporation Inc., Beijing, 100195, China

    Yanfeng Qi & Baoan Guo

Authors
  1. Chunming Tang
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  2. Yu Lou
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  3. Yanfeng Qi
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  4. Maozhi Xu
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  5. Baoan Guo
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Corresponding author

Correspondence to Yanfeng Qi .

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Editors and Affiliations

  1. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

  2. University of Texas, San Antonio, Texas, USA

    Shouhuai Xu

  3. Columbia University, New York, New York, USA

    Moti Yung

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Tang, C., Lou, Y., Qi, Y., Xu, M., Guo, B. (2014). A Note on Semi-bent and Hyper-bent Boolean Functions. In: Lin, D., Xu, S., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2013. Lecture Notes in Computer Science(), vol 8567. Springer, Cham. https://doi.org/10.1007/978-3-319-12087-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-12087-4_1

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  • Online ISBN: 978-3-319-12087-4

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