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Recent Results on Constructing Boolean Functions with (Potentially) Optimal Algebraic Immunity Based on Decompositions of Finite Fields

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Abstract

Boolean functions with optimal algebraic immunity (OAI functions) are important cryptographic primitives in the design of stream ciphers. During the past decade, a lot of work has been done on constructing such functions, among which mathematics, especially finite fields, play an important role. Notably, the approach based on decompositions of additive or multiplicative groups of finite fields turns out to be a very successful one in constructing OAI functions, where some original ideas are contributed by Tu and Deng (2012), Tang, et al. (2017), and Lou, et al. (2015). Motivated by their pioneering work, the authors and their collaborators have done a series of work, obtaining some more general constructions of OAI functions based on decompositions of finite fields. In this survey article, the authors review our work in this field in the past few years, illustrating the ideas for the step-by-step generalizations of previous constructions and recalling several new observations on a combinatorial conjecture on binary strings known as the Tu-Deng conjecture. In fact, the authors have obtained some variants or more general forms of Tu-Deng conjecture, and the optimal algebraic immunity of certain classes of functions we constructed is based on these conjectures.

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Acknowledgements

The year of 2019 will be the centenary of Professor Wen-tsün Wu’s birth. He is one outstanding Chinese mathematician and passed away on May 7th, 2017. We are writing this paper to commemorate him. As some of Wu’s disciples, we learned a lot from him in the past days, especially got many benefits from the mathematics mechanization initiated by him. Into his 90’s, Prof. Wu showed great interests in cryptography and had worked very hard on such important cryptographic problems as factorization of big integers. Inspired by his spirits, we also carried out some researches in the field of cryptography in recent years, and a part of our results form this paper. We will forever be grateful for the knowledge and spiritual heritage that Prof. Wu had passed on to us.

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhuojun Liu

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

    Baofeng Wu

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  1. Zhuojun Liu
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  2. Baofeng Wu
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Correspondence to Baofeng Wu.

Additional information

This paper was supported by the National Key Research and Development Program of China under Grant No. 2016YFB0800401.

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Liu, Z., Wu, B. Recent Results on Constructing Boolean Functions with (Potentially) Optimal Algebraic Immunity Based on Decompositions of Finite Fields. J Syst Sci Complex 32, 356–374 (2019). https://doi.org/10.1007/s11424-019-8346-2

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  • DOI: https://doi.org/10.1007/s11424-019-8346-2

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