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Designs, Codes and Cryptography

, Volume 52, Issue 3, pp 303–338 | Cite as

Further properties of several classes of Boolean functions with optimum algebraic immunity

  • Claude Carlet
  • Xiangyong ZengEmail author
  • Chunlei Li
  • Lei Hu
Article

Abstract

Based on a method proposed by the first author, several classes of balanced Boolean functions with optimum algebraic immunity are constructed, and they have nonlinearities significantly larger than the previously best known nonlinearity of functions with optimal algebraic immunity. By choosing suitable parameters, the constructed n-variable functions have nonlinearity \({2^{n-1}-{n-1\choose\frac{n}{2}-1}+2{n-2\choose\frac{n}{2}-2}\Big/(n-2)}\) for even \({n\geq 8\,{\rm and}\,2^{n-1}-{n-1\choose\frac{n-1}{2}}+\Delta(n)}\) for odd n, where Δ(n) is a function increasing rapidly with n. The algebraic degrees of some constructed functions are also discussed.

Keywords

Boolean function Algebraic immunity Nonlinearity Balancedness Krawtchouk polynomial 

Mathematics Subject Classification (2000)

94A60 06E30 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Claude Carlet
    • 1
  • Xiangyong Zeng
    • 2
    • 3
    Email author
  • Chunlei Li
    • 2
  • Lei Hu
    • 4
  1. 1.Department of MathematicsLAGA, Universities of Paris 8 and Paris 13 and CNRSSaint-Denis CedexFrance
  2. 2.Faculty of Mathematics and Computer ScienceHubei UniversityWuhanChina
  3. 3.State Key Laboratory of Information SecurityGraduate School of Chinese Academy of SciencesBeijingChina
  4. 4.The Key Laboratory of Mathematics Mechanization, Institute of System SciencesAMSS, Chinese Academy of SciencesBeijingChina

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