Skip to main content
SpringerLink
Log in

Some lower bounds on the algebraic immunity of functions given by their trace forms

  • Information Protection
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

The algebraic immunity of a Boolean function is a parameter that characterizes the possibility to bound this function from above or below by a nonconstant Boolean function of a low algebraic degree. We obtain lower bounds on the algebraic immunity for a class of functions expressed through the inversion operation in the field GF(2n), as well as for larger classes of functions defined by their trace forms. In particular, for n ≥ 5, the algebraic immunity of the function Tr n (x −1) has a lower bound ⌊2√n + 4⌋ − 4, which is close enough to the previously obtained upper bound ⌊√n⌋ + ⌈n/⌊√n⌋⌉ − 2. We obtain a polynomial algorithm which, give a trace form of a Boolean function f, computes generating sets of functions of degree ≤ d for the following pair of spaces. Each function of the first (linear) space bounds f from below, and each function of the second (affine) space bounds f from above. Moreover, at the output of the algorithm, each function of a generating set is represented both as its trace form and as a polynomial of Boolean variables.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Meier, W., Pasalic, E., and Carlet, C., Algebraic Attacks and Decomposition of Boolean Functions, Advances in CryptologyEUROCRYPT 2004. Proc. Int. Conf. on the Theory and Applications of Cryptographic Techniques, Interlaken, Switzerland, Cachin, C. and Camenisch, J., Eds., Lect. Notes Comp. Sci., vol. 3027, Berlin: Springer, 2004, pp. 474–491.

    Google Scholar 

  2. Bayev, V.V., On Some Algorithms for Constructing Low-Degree Annihilators for Boolean Functions, Diskret. Mat., 2006, vol. 18, no. 3, pp. 138–151 [Discrete Math. Appl. (Engl. Transl.), 2006, vol. 16, no. 5, pp. 439–452].

    Google Scholar 

  3. Bayev, V.V., On the Complexity of Finding Low-Degree Annihilators for Boolean Functions, Proc. Int. Scientific Conf. on Security and Counter-Terrorism Problems, Moscow State Univ., 2005, Moscow: MCCME, 2006, pp. 198–204.

    Google Scholar 

  4. Braeken, A. and Preneel, B., On the Algebraic Immunity of Symmetric Boolean Functions, Progress in CryptologyINDOCRYPT 2005. Proc. 6th Int. Conf. on Cryptology in India, Bangalore, India, Maitra, S., Veni Madhavan, C.E., and Venkatesan, R., Eds., Lect. Notes Comp. Sci., vol. 3797, Berlin: Springer, 2005, pp. 35–48.

    Google Scholar 

  5. Dalai, D.K., Gupta, K.C., and Maitra, S., Cryptographically Significant Boolean Functions: Construction and Analysis in Terms of Algebraic Immunity, Proc. 12th Int. Workshop on Fast Software Encryption (FSE 2005), Paris, France, Gilbert, H. and Handschuh, H., Eds., Lect. Notes Comp. Sci., vol. 3557, Berlin: Springer, 2005, pp. 98–111.

    Google Scholar 

  6. Dalai, D.K., Maitra, S., and Sarkar, S., Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity, Des. Codes Cryptogr., 2006, vol. 40, no. 1, pp. 41–58.

    Article  MATH  MathSciNet  Google Scholar 

  7. Dalai, D.K. and Maitra, S., Balanced Boolean Functions with (More than) Maximum Algebraic Immunity, Proc. Int. Workshop on Coding and Cryptography (WCC 2007), Versailles, France, 2007, Augot, D., Sendrier, N., and Tillich, J.-P., Eds., pp. 99–106.

  8. Carlet, C., A Method of Construction of Balanced Functions with Optimum Algebraic Immunity, Cryptology ePrint Archive, 2006, Report 2006/149. Available at http://eprint.iacr.org/2006/149.

  9. Armknecht, F. and Krause, M., Constructing Single-and Multi-output Boolean Functions with Maximal Algebraic Immunity, Proc. 33rd Int. Colloq. on Automata, Languages, and Programming (ICALP 2006), Venice, Italy, Bugliesi, M., Preneel, B., Sassone, V., and Wegener, I., Eds., Lect. Notes Comp. Sci., vol. 4052, Berlin: Springer, 2006, Part II, pp. 180–191.

    Google Scholar 

  10. Qu, L., Feng, G., and Li, C., On the Boolean Functions with Maximum Possible Algebraic Immunity: Construction and a Lower Bound of the Count, Cryptology ePrint Archive, 2005, Report 2005/449. Available at http://eprint.iacr.org/2005/449.

  11. Li, N. and Qi, W.-F., Construction and Analysis of Boolean Functions of 2t+1 Variables with Maximum Algebraic Immunity, Advances in CryptologyASIACRYPT 2006. Proc. 12th Int. Conf. on the Theory and Application of Cryptology and Information Security, Shanghai, China, Lai, X. and Chen, K., Eds., Lect. Notes Comp. Sci., vol. 4284, Berlin: Springer, 2006, pp. 84–98.

    Google Scholar 

  12. Botev, A.A., On Properties of Correlation-Immune Functions with High Nonlinearity, Cand. Sci. (Phys.-Math.) Dissertation, Moscow: Moscow State Univ., 2005.

    Google Scholar 

  13. Tarannikov, Yu.V., On Correlation-Immune and Stable Boolean Functions, Mat. Vopr. Kibern., vol. 11, Moscow: Fizmatlit, 2002, pp. 91–148.

    Google Scholar 

  14. Nawaz, Y., Gong, G., and Gupta, K.C., Upper Bounds on Algebraic Immunity of Boolean Power Functions, Proc. 13th Int. Workshop on Fast Software Encryption (FSE 2006), Graz, Austria, Robshaw, M.J.B., Ed., Lect. Notes Comp. Sci., vol. 4047, Berlin: Springer, 2006, pp. 375–389.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia

    V. V. Bayev

Authors
  1. V. V. Bayev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Bayev.

Additional information

Original Russian Text © V.V. Bayev, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 3, pp. 81–104.

Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00154.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bayev, V.V. Some lower bounds on the algebraic immunity of functions given by their trace forms. Probl Inf Transm 44, 243–265 (2008). https://doi.org/10.1134/S0032946008030071

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946008030071

Keywords

Search

Navigation