Cryptography and Communications

, Volume 4, Issue 3–4, pp 233–243 | Cite as

On the nonlinearity of maximum-length NFSR feedbacks

  • Meltem Sönmez Turan


Linear Feedback Shift Registers (LFSRs) are the main building block of many classical stream ciphers; however due to their inherent linearity, most of the LFSR-based designs do not offer the desired security levels. In the last decade, using Nonlinear Feedback Shift Registers (NFSRs) in stream ciphers became very popular. However, the theory of NFSRs is not well-understood, and there is no efficient method that constructs a cryptographically strong feedback function and also, given a feedback function it is hard to predict the period. In this paper, we study the maximum-length NFSRs, focusing on the nonlinearity of their feedback functions. First, we provide some upper bounds on the nonlinearity of the maximum-length feedback functions, and then we study the feedback functions having nonlinearity 2 in detail. We also show some techniques to improve the nonlinearity of a given feedback function using cross-joining.


Nonlinearity NFSR Stream ciphers Cryptography 

Mathematics Subject Classifications (2010)

94A55 94A60 68P25 



The author would like to thank Çağdaş Çalık for his valuable comments. The author would also like to thank the anonymous reviewers for their suggestions to improve the quality of the paper.


  1. 1.
    Golomb, S.W.: Shift Register Sequences. Holden-Day, Inc., Laguna Hills (1967)zbMATHGoogle Scholar
  2. 2.
    Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)CrossRefGoogle Scholar
  3. 3.
    Braeken, A., Lano, J.: On the (im)possibility of practical and secure nonlinear filters and combiners. In: Selected Areas in Cryptography, pp. 159–174 (2005)Google Scholar
  4. 4.
    Hell, M., Johansson, T., Meier, W.: Grain-A Stream Cipher for Constrained Environments. eSTREAM, ECRYPT Stream Cipher Project, Report 2005/010 (2005)Google Scholar
  5. 5.
    Babbage, S., Dodd, M.: The Stream Cipher MICKEY (version 1). eSTREAM, ECRYPT Stream Cipher Project, Report 2005/015 (2005)Google Scholar
  6. 6.
    De Cannière, C., Preneel, B.: Trivium Specifications. eSTREAM, ECRYPT Stream Cipher Project, Report 2005/030 (2005)Google Scholar
  7. 7.
    Fredricksen, H.: A Survey of Full Length Nonlinear Shift Register Cycle Algorithms. SIAM Rev. 24(2), 195–221 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Tsuneda, A., Kudo, K., Yoshioka, D., Inoue, T.: Maximal-period sequences generated by feedback-limited nonlinear shift registers. IEICE Trans. 90(10), 2079–2084 (2007)CrossRefGoogle Scholar
  9. 9.
    Çalık, Ç., Sönmez Turan, M., Özbudak, F.: On feedback functions of maximum length nonlinear feedback shift registers. IEICE Trans. 93(6), 1226–1231 (2010)CrossRefGoogle Scholar
  10. 10.
    de Bruijn, N.G.: A combinatorial problem. Proc. K. Ned. Acad. Wet. Ser. A 49(7), 758–764 (1946)zbMATHGoogle Scholar
  11. 11.
    Gonzalo, R., Ferrero, D., Soriano, M.: Some properties of nonlinear feedback shift registers with maximum period. In: Proc. 6th Int. Conf. Telecommunications Systems (1998)Google Scholar
  12. 12.
    Etzion, T., Lempel, A.: On the distribution of de Bruijn sequences of given complexity. IEEE Trans. Inf. Theory 30(4), 611–614 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Wu, C.K.: Distribution of Boolean functions with nonlinearity 2(n − 2). In: Proceedings of ChinaCrypt’94, pp. 10–14. Springer, China (1994)Google Scholar
  14. 14.
    Helleseth, T., Kløve, T.: The number of cross-join pairs in maximum length linear sequences. IEEE Trans. Inf. Theory 37(6), 1731–1733 (1991)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC (outside the USA) 2012

Authors and Affiliations

  1. 1.National Institute of Standards and TechnologyGaithersburgUSA

Personalised recommendations