Nonlinear Complexity of Binary Sequences and Connections with Lempel-Ziv Compression

  • Konstantinos Limniotis
  • Nicholas Kolokotronis
  • Nicholas Kalouptsidis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4086)

Abstract

The nonlinear complexity of binary sequences is studied in this paper. A new recursive algorithm is presented, which produces the minimal nonlinear feedback shift register of a given sequence. Further, a connection between the nonlinear complexity and the compression capability of a sequence is established. A lower bound for the Lempel-Ziv compression ratio that a given sequence can achieve is proved, which depends on its nonlinear complexity.

Keywords

Cryptography Lempel-Ziv compression nonlinear complexity nonlinear feedback shift registers sequences 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berlekamp, E.R.: Algebraic coding theory. McGraw-Hill, New York (1968)MATHGoogle Scholar
  2. 2.
    Erdmann, D., Murphy, S.: An approximate distribution for the maximum order complexity. Des. Codes and Cryptography 10, 325–339 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Golomb, S.W.: Shift Register Sequences. Holden-Day, San Francisco (1967)MATHGoogle Scholar
  4. 4.
    Boekee, D.E., Jansen, C.J.A.: The shortest feedback shift register that can generate a given sequence. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 90–99. Springer, Heidelberg (1990)Google Scholar
  5. 5.
    Kalouptsidis, N.: Signal Processing Systems. Telecommunications and Signal Processing Series. John Wiley & Sons, Chichester (1996)Google Scholar
  6. 6.
    Key, E.L.: An analysis of the structure and complexity of nonlinear binary sequence generators. IEEE Trans. Inform. Theory 22, 732–736 (1976)MATHCrossRefGoogle Scholar
  7. 7.
    Kohavi, Z.: Switching and finite automata theory. McGraw-Hill Book Company, New York (1978)MATHGoogle Scholar
  8. 8.
    Kolokotronis, N., Kalouptsidis, N.: On the linear complexity of nonlinearly filtered PN-sequences. IEEE Trans. Inform. Theory 49, 3047–3059 (2003)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Inform. Theory 22, 75–81 (1976)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1996)MATHCrossRefGoogle Scholar
  11. 11.
    Massey, J.L.: Shift register synthesis and BCH decoding. IEEE Trans. Inform. Theory 15, 122–127 (1969)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Massey, J.L., Serconek, S.: A Fourier transform approach to the linear complexity of nonlinearly filtered sequences. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 332–340. Springer, Heidelberg (1994)Google Scholar
  13. 13.
    Massey, J.L., Serconek, S.: Linear complexity of periodic sequences: a general theory. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 358–371. Springer, Heidelberg (1996)Google Scholar
  14. 14.
    Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)CrossRefGoogle Scholar
  15. 15.
    Niederreiter, H.: Some computable complexity measures for binary sequences. In: Ding, C., Helleseth, T., Niederreiter, H. (eds.) Sequences and Their Applications, Discrete Mathematics and Theoretical Computer Science, pp. 67–78. Springer, Heidelberg (1999)Google Scholar
  16. 16.
    Rizomiliotis, P., Kalouptsidis, N.: Results on the nonlinear span of binary sequences. IEEE Trans. Inform. Theory 51, 1555–1563 (2005)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Rizomiliotis, P., Kolokotronis, N., Kalouptsidis, N.: On the quadratic span of binary sequences. IEEE Trans. Inform. Theory 51, 1840–1848 (2005)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Rueppel, R.A.: Analysis and design of stream ciphers. Springer, Berlin (1986)MATHGoogle Scholar
  19. 19.
    Stergiou, S., Voudouris, D., Papakonstantinou, G.: Multiple-value exclusive-or sum-of-products minimization algorithms. IEICE Transactions on Fundamentals E.87-A(5), 1226–1234 (2004)Google Scholar
  20. 20.
    Wyner, A.D., Ziv, J.: The sliding-window Lempel-Ziv algorithm is asymptotically optimal. Proceedings of the IEEE 82, 872–877 (1994)CrossRefGoogle Scholar
  21. 21.
    Ziv, J., Lempel, A.: A universal algorithm for sequential data compression. IEEE Trans. Inform. Theory 23, 337–343 (1977)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ziv, J., Lempel, A.: Compression of individual sequences via variable-rate coding. IEEE Trans. Inform. Theory 24, 530–536 (1978)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Konstantinos Limniotis
    • 1
  • Nicholas Kolokotronis
    • 1
  • Nicholas Kalouptsidis
    • 1
  1. 1.Department of Informatics and TelecommunicationsNational and Kapodistrian University of AthensAthensGreece

Personalised recommendations