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Two infinite classes of rotation symmetric bent functions with simple representation

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Abstract

In the literature, few n-variable rotation symmetric bent functions have been constructed. In this paper, we present two infinite classes of rotation symmetric bent functions on \({\mathbb {F}}_2^{n}\) of the two forms:

  1. (i)

    \(f(x)=\sum _{i=0}^{m-1}x_ix_{i+m} + {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})\),

  2. (ii)

    \(f_t(x)= \sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_{i}x_{i+t})+ \sum _{i=0}^{m-1}x_ix_{i+m}+ {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})\),

where \(n=2m\), \({\upgamma }(X_0,X_1,\ldots , X_{m-1})\) is any rotation symmetric polynomial, and \(m/\textit{gcd}(m,t)\) is odd. The class (i) of rotation symmetric bent functions has algebraic degree ranging from 2 to m and the other class (ii) has algebraic degree ranging from 3 to m. Moreover, the two classes of rotation symmetric bent functions are disjoint.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11401480, 11531002, 11371138). 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 and LQ16A010005). The work of Z. Zhou and C. Fan was supported by the Sichuan Provincial Youth Science and Technology Fund under Grant 2016JQ0004, and in part by National Cryptography Development Fund under Grant MMJJ20170119.

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Authors and Affiliations

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

    Chunming Tang

  2. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Yanfeng Qi

  3. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

    Zhengchun Zhou & Cuiling Fan

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  1. Chunming Tang
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Correspondence to Chunming Tang.

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Tang, C., Qi, Y., Zhou, Z. et al. Two infinite classes of rotation symmetric bent functions with simple representation. AAECC 29, 197–208 (2018). https://doi.org/10.1007/s00200-017-0337-8

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