Results on the Immunity of Boolean Functions against Probabilistic Algebraic Attacks

  • Meicheng Liu
  • Dongdai Lin
  • Dingyi Pei
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6812)

Abstract

In this paper, we study the immunity of Boolean functions against probabilistic algebraic attacks. We first show that there are functions, using as filters in a linear feedback shift register based nonlinear filter generator, such that probabilistic algebraic attacks outperform deterministic ones. Then we introduce two notions, algebraic immunity distance and k-error algebraic immunity, to measure the ability of Boolean functions resistant to probabilistic algebraic attacks. We analyze both lower and upper bounds on algebraic immunity distance, and also present the relations among algebraic immunity distance, k-error algebraic immunity, algebraic immunity and high order nonlinearity.

Keywords

Boolean functions algebraic attacks algebraic immunity algebraic immunity distance k-error algebraic immunity high order nonlinearity 
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References

  1. 1.
    Armknecht, F.: Improving fast algebraic attacks. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 65–82. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Armknecht, F., Carlet, C., Gaborit, P., Künzli, S., Meier, W., Ruatta, O.: Efficient computation of algebraic immunity for algebraic and fast algebraic attacks. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 147–164. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Braeken, A., Preneel, B.: Probabilistic algebraic attacks. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 290–303. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Carlet, C.: On the higher order nonlinearities of algebraic immune functions. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 584–601. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Carlet, C., Feng, K.: An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 425–440. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Cohen, G., Honkala, I., Litsyn, S., Lobstein, A.: Covering codes. North-Holland, Amsterdam (1997)MATHGoogle Scholar
  7. 7.
    Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 345–359. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Courtois, N.T.: Fast algebraic attacks on stream ciphers with linear feedback. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 176–194. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Dalai, D.K., Maitra, S., Sarkar, S.: Basic theory in construction of Boolean functions with maximum possible annihilator immunity. Designs, Codes and Cryptography 40(1), 41–58 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dalai, D.K., Gupta, K.C., Maitra, S.: Results on algebraic immunity for cryptographically significant boolean functions. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 92–106. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Du, Y., Pei, D.: Construction of Boolean functions with maximum algebraic immunity and count of their annihilators at lowest degree. Sci. China Inf. Sci, 53(4), 780–787 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hawkes, P., Rose, G.: Rewriting variables: The complexity of fast algebraic attacks on stream ciphers. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 390–406. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Li, N., Qu, L., Qi, W., et al.: On the construction of Boolean Functions with optimal algebraic immunity. IEEE Trans. Inform. Theory 54(3), 1330–1334 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Li, N., Qi, W.: Boolean functions of an odd number of variables with maximum algebraic immunity. Sci. China Ser. F-Inf. Sci. 50(3), 307–317 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liu, M., Pei, D., Du, Y.: Identification and construction of Boolean functions with maximum algebraic immunity. Sci. China. Inf. Sci, 53(7), 1379–1396 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Liu, M., Du, Y., Pei, D., Lin, D.: On designated-weight Boolean functions with highest algebraic immunity. Sci. China. Math, 53(11), 2847–2854 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lobanov, M.: Tight bounds between algebraic immunity and nonlinearities of high orders., http://eprint.iacr.org/2007/444
  18. 18.
    MacWilliams, F.J., Sloane, N.J.A.: The theory of error correcting codes. North-Holland, New York (1977)MATHGoogle Scholar
  19. 19.
    Mesnager, S.: Improving the Lower Bound on the Higher Order Nonlinearity of Boolean Functions With Prescribed Algebraic Immunity. IEEE Transactions on Information Theory 54(8), 3656–3662 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Meier, W., Pasalic, E., Carlet, C.: Algebraic attacks and decomposition of boolean functions. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 474–491. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Pasalic, E.: Probabilistic versus deterministic algebraic cryptanalysis – a performance comparison. IEEE Transactions on Information Theory 55(11), 5233–5240 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pometun, S.: Study of Probabilistic Scenarios of Algebraic Attacks on Stream Ciphers. Journal of Automation and Information Sciences 41(2), 67–80 (2009), http://eprint.iacr.org/2007/448 CrossRefGoogle Scholar
  23. 23.
    Qu, L., Feng, G., Li, C.: On the Boolean functions with maximum possible algebraic immunity: construction and a lower bound of the count., http://eprint.iacr.org/2005/449
  24. 24.
    Tu, Z., Deng, Y.: Algebraic Immunity Hierarchy of Boolean Functions. ChinaCrypt (2007), http://eprint.iacr.org/2007/259
  25. 25.
    Zhang, X., Pieprzyk, J., Zheng, Y.: On algebraic immunity and annihilators. In: Rhee, M.S., Lee, B. (eds.) ICISC 2006. LNCS, vol. 4296, pp. 65–80. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Meicheng Liu
    • 1
  • Dongdai Lin
    • 1
  • Dingyi Pei
    • 2
  1. 1.The State Key Laboratory of Information Security, Institute of SoftwareChinese Academy of SciencesBeijingChina
  2. 2.College of Mathematics and Information SciencesGuangzhou UniversityGuangzhouChina

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