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一类代数免疫度达到最优的布尔函数的构造
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Supported by the National Natural Science Foundation of China under Grant No.60872025 (国家自然科学基金)


Construction of Boolean Functions with Maximum Algebraic Immunity
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    摘要:

    给出了一种具有最优代数免疫度的偶数元布尔函数的构造,同时还给出了一种具有最优代数免疫度的平衡旋转对称偶数元布尔函数的构造.在构造过程中用到了线性代数和组合计数中的有关结论,这些函数对代数攻击均有很强的抵抗能力.构造的平衡旋转对称布尔函数还可用在Hash算法的轮函数中,增加了算法的安全性.

    Abstract:

    This paper presents a construction of Boolean functions with the maximum algebraic immunity on even number of variables. It also gives a construction of balanced rotation symmetric Boolean functions with the maximum algebraic immunity on even number of variables. This paper uses some results of linear algebra and enumerative combinatorics in the constructions. These functions have strong resistance against algebraic attacks. The balanced rotation symmetric Boolean functions constructed can also be used in the construction of safer hashing functions.

    参考文献
    [1] St?nic? P, Maitra S. Rotation symmetric Boolean functions-count and cryptographic properties. Discrete Applied Mathematics, 2008,156(10):1567?1580. [doi: 10.1016/j.dam.2007.04.029]
    [2] Filiol E, Fontaine C. Highly nonlinear balanced Boolean functions with a good correlation immunity. In: Nyberg K, ed. Advances in Cryptology—EUROCRYPT’98. LNCS 1403, Heidelberg: Springer-Verlag, 1998. 475?488.
    [3] Carlet C. Algebraic immunity for cryptographically significant Boolean functions: analysis and construction. IEEE Trans. on Information Theory, 2006,52(7):3105?3121. [doi: 10.1109/TIT.2006.876253]
    [4] Li N, Qi WF. Symmetric Boolean functions depending on an odd number of variables with maximum algebraic immunity. IEEE Trans. on Information Theory, 2006,52(5):2271?2273. [doi: 10.1109/TIT.2006.872977]
    [5] Li N, Qi WF. Construction and count of Boolean functions of an odd number of variables with maximum algebraic immunity. Science in China (Series F: Information Science), 2007,50(3):307?317 (in Chinese with English abstract). [doi: 10.1007/ s11432-007-0027-4]
    [6] Sarkar S, Maitra S. Construction of rotation symmetric Boolean functions with maximum algebraic immunity an odd number of variables. In: Boztas S, Lu HF, eds. Proc. of the 17th Symp. on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS 4851, Heidelberg: Springer-Verlag, 2007. 271?280.
    [7] Carlet C, Zeng XY, Li CL, Hu L. Further properties of several classes of Boolean functions with optimum algebraic immunity. Designs, Codes and Cryptography, 2009,52(3):303?338. [doi: 10.1007/s10623-009-9284-0]
    [8] Lobanov MS. Exact relation between nonlinearity and algebraic immunity. Discrete Mathematics and Applications, 2006,16(5): 453?460. [doi: 10.1515/156939206779238418]
    [9] Kisa?anin B. Mathematical Problems and Proofs: Combinatorics, Number Theory, and Geometry. New York: Plenum Press, 1998. 67?68, 97?98.
    附中文参考文献: [5] 李娜,戚文峰.具有最优代数免疫的奇数元Boole函数.中国科学(E辑:科学信息),2007,50(3):307?317.
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孟 强,陈鲁生,符方伟.一类代数免疫度达到最优的布尔函数的构造.软件学报,2010,21(7):1758-1767

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  • 收稿日期:2009-04-14
  • 最后修改日期:2009-08-26
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