Designs, Codes and Cryptography

, Volume 76, Issue 2, pp 279–305 | Cite as

Two constructions of balanced Boolean functions with optimal algebraic immunity, high nonlinearity and good behavior against fast algebraic attacks

  • Jiao Li
  • Claude Carlet
  • Xiangyong Zeng
  • Chunlei Li
  • Lei Hu
  • Jinyong Shan
Article

Abstract

In this paper, two constructions of Boolean functions with optimal algebraic immunity are proposed. They generalize previous ones respectively given by Rizomiliotis (IEEE Trans Inf Theory 56:4014–4024, 2010) and Zeng et al. (IEEE Trans Inf Theory 57:6310–6320, 2011) and some new functions with desired properties are obtained. The functions constructed in this paper can be balanced and have optimal algebraic degree. Further, a new lower bound on the nonlinearity of the proposed functions is established, and as a special case, it gives a new lower bound on the nonlinearity of the Carlet-Feng functions, which is slightly better than the best previously known ones. For \(n\le 19\), the numerical results reveal that among the constructed functions in this paper, there always exist some functions with nonlinearity higher than or equal to that of the Carlet-Feng functions. These functions are also checked to have good behavior against fast algebraic attacks at least for small numbers of input variables.

Keywords

Algebraic immunity Boolean function Balance Algebraic degree Nonlinearity Fast algebraic attack 

Mathematics Subject Classification

06E30 94C10 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jiao Li
    • 1
  • Claude Carlet
    • 3
  • Xiangyong Zeng
    • 1
    • 2
  • Chunlei Li
    • 4
  • Lei Hu
    • 5
    • 6
  • Jinyong Shan
    • 5
  1. 1.Faculty of Mathematics and StatisticsHubei UniversityWuhan China
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijing China
  3. 3.LAGAUniversities of Paris 8 and Paris 13 and CNRSSaint-Denis CedexFrance
  4. 4.Department of InformaticsUniversity of BergenBergenNorway
  5. 5.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijing China
  6. 6.Beijing Center for Mathematics and Information Interdisciplinary SciencesBeijing China

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