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Cryptographic properties of nested functions and algebraic immunity of the Boolean function in Hitag2 stream cipher

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Abstract

Hitag2 is a widely applied lightweight stream cipher with a traditional structure containing linear shift feedback and nonlinear filtering. It uses a Boolean function of 20 variables as its nonlinear filter. For easy implementation, this function is constructed by a two-layer composition of one 5-variable Boolean function and five 4-variable Boolean functions. In this paper, the concept of nested function is extracted from the construction of the two-layer Boolean function in Hitag2. Then we study some general properties of nested functions, such as balancedness, algebraic degree, Walsh spectra and algebraic immunity. We prove that the Walsh spectra of a nested function can be split into a product of the Walsh spectra of its subfunctions and generating function when the subfunctions are all balanced. Moreover, two upper bounds on algebraic immunity of nested functions are proposed. By using a hybrid approach of logical reasoning and computer computation, we obtain the precise value of the algebraic immunity of the filter function used in Hitag2, which is equal to 6.

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Acknowledgment

The authors would like to thank the anonymous reviewers for their helpful comments and suggestions. Their suggestions on considering other cryptographic properties of nested functions enrich the study of the paper and their comments on our method greatly improve the quality of the paper. The work of J. Shan and L. Hu was supported by the National Key Basic Research Program of China (2013CB834203), the National Natural Science Foundation of China (NSFC) (Grant 61070172), the Strategic Priority Research Program of Chinese Academy of Sciences under Grant XDA06010702. The work of X. Zeng was supported by the NSFC (Grant 61170257) and the External Science and Technology Cooperation Program of Hubei Province under Grant 2012IHA01402

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Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Jinyong Shan, Lei Hu & Xiangyong Zeng

  2. Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100048, China

    Lei Hu

  3. Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, Hubei, China

    Xiangyong Zeng

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  1. Jinyong Shan
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  2. Lei Hu
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  3. Xiangyong Zeng
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Correspondence to Jinyong Shan.

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Shan, J., Hu, L. & Zeng, X. Cryptographic properties of nested functions and algebraic immunity of the Boolean function in Hitag2 stream cipher. Cryptogr. Commun. 6, 233–254 (2014). https://doi.org/10.1007/s12095-014-0099-0

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  • DOI: https://doi.org/10.1007/s12095-014-0099-0

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