Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

  • 42 Accesses

  • 9 Citations

Abstract

This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler's upper bound of algebraic degree. Also a class of 1-resilient functions on any number n > 2 of variables with at least sub-optimal algebraic immunity is provided.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    Carlet C, Dalai D K, Gupta K C, Maitra S. Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction. IEEE Transactions on Information Theory, 2006, 52(7): 3105–3121.

  2. [2]

    Tu Z, Deng Y. A class of 1-resilient function with high nonlinearity and algebraic immunity. Cryptography ePrint Archive, Report 2010/179, 2010, http://eprint.iacr.org/.

  3. [3]

    Le Bars J M, Viola A. Equivalence classes of Boolean functions for first-order correlation. IEEE Transactions on Information Theory, 2010, 56(3): 1247–1261.

  4. [4]

    Wang Q, Peng J, Kan H, Xue X. Constructions of cryptographically significant Boolean functions using primitive polynomials. IEEE Transactions on Information Theory, 2010, 56(6): 3048–3053.

  5. [5]

    Sarkar S, Maitra S. Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros. Des. Codes Cryptogr., 2008, 49: 95–103.

  6. [6]

    Siegenthaler T. Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, 1984, 30(5): 776–780.

  7. [7]

    Xiao G Z, Massey J L. A spectral characterization of correlation-immune combining functions. IEEE Transactions on Information Theory, 1988, 34(3): 569–571.

  8. [8]

    Meier W, Staffelbach O. Nonlinearity criteria for cryptographic functions. In Proc. Advances in Cryptology — EUROCRYPT’89, Houthalen, Belgium, April 10–13, 1990, pp.549-562.

  9. [9]

    MacWilliams F J, Sloane N J A. The Theory of Error-Correcting Codes. Amsterdam: North-Holland Publishing Co., The Netherlands, 1977.

  10. [10]

    Carlet C, Feng K. An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity. In Proc. Advances in Cryptology-ASIACRYPT, Melbourne, Australia, 2008, pp.425-440.

Download references

Author information

Correspondence to Sen-Shan Pan.

Additional information

This work is supported by the National Natural Science Foundations of China under Grant Nos. 60903200, 61003299.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

(PDF 76 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pan, S., Fu, X. & Zhang, W. Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity. J. Comput. Sci. Technol. 26, 269–275 (2011). https://doi.org/10.1007/s11390-011-9433-6

Download citation

Keywords

  • stream ciphers
  • Boolean functions
  • 1-resilient
  • algebraic immunity
  • algebraic degree