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A new lower bound on the second-order nonlinearity of a class of monomial bent functions

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Abstract

The second-order nonlinearity can provide knowledge on classes of Boolean functions used in symmetric-key cryptosystems, coding theory, and Gowers norm. It is well-known that bent functions possess the highest nonlinearity on even number of variables and so it will be of great interest to investigate the lower bound on the second-order nonlinearity of such functions. In 2008, Canteaut et al. (Finite Fields Appl. 14(1), 221–241, 2) found a class of monomial bent functions on n = 6r variables and proved that their derivatives have nonlinearities either 2n− 1 − 24r− 1 or 2n− 1 − 25r− 1. In this paper, we completely determine the distributions of the nonlinearities of the derivatives of this class of bent functions. Further, we present a new lower bound on the second-order nonlinearity of this class of bent functions, which is better than the previous one.

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Acknowledgments

The authors would like to thank the anonymous reviewers and the Associate Editor for their valuable suggestions and comments that improved the quality of this paper. The work of the first author was supported by the National Natural Science Foundation of China (grants 61872435 and 61602394), the work of the second author was supported by the National Natural Science Foundation of China (grant 11801468), the work of the third author was supported by the National Cryptography Development Fund under Grant MMJJ20170119.

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Authors and Affiliations

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

    Deng Tang, Haode Yan & Zhengchun Zhou

  2. Center for Cyber Security, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Xiaosong Zhang

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  1. Deng Tang
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  2. Haode Yan
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  3. Zhengchun Zhou
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  4. Xiaosong Zhang
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Correspondence to Haode Yan.

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Tang, D., Yan, H., Zhou, Z. et al. A new lower bound on the second-order nonlinearity of a class of monomial bent functions. Cryptogr. Commun. 12, 77–83 (2020). https://doi.org/10.1007/s12095-019-00360-y

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  • DOI: https://doi.org/10.1007/s12095-019-00360-y

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