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Construction and enumeration of Boolean functions with maximum algebraic immunity

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Abstract

Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also given. For a given function p(x) with deg(p(x)) < , we give a method to construct functions in the form p(x)+q(x) which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q(x) has degree no less than .

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Authors and Affiliations

  1. School of Information Science and Engineering, Shandong Normal University, Jinan, 250014, China

    WenYing Zhang

  2. State Key Lab of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing, 100080, China

    WenYing Zhang & ChuanKun Wu

  3. No. 2 Middle School Attached to Shandong Normal University, Jinan, 250014, China

    XiangZhong Liu

Authors
  1. WenYing Zhang
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  2. ChuanKun Wu
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  3. XiangZhong Liu
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Corresponding author

Correspondence to WenYing Zhang.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 60673068), and the Natural Science Foundation of Shandong Province (Grant Nos. Y2007G16, Y2008G01)

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Zhang, W., Wu, C. & Liu, X. Construction and enumeration of Boolean functions with maximum algebraic immunity. Sci. China Ser. F-Inf. Sci. 52, 32–40 (2009). https://doi.org/10.1007/s11432-009-0007-y

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  • DOI: https://doi.org/10.1007/s11432-009-0007-y

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