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

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Sequences and Their Applications – SETA 2006 (SETA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4086))

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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.

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References

  1. Berlekamp, E.R.: Algebraic coding theory. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  2. Erdmann, D., Murphy, S.: An approximate distribution for the maximum order complexity. Des. Codes and Cryptography 10, 325–339 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Golomb, S.W.: Shift Register Sequences. Holden-Day, San Francisco (1967)

    MATH  Google Scholar 

  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. Kalouptsidis, N.: Signal Processing Systems. Telecommunications and Signal Processing Series. John Wiley & Sons, Chichester (1996)

    Google Scholar 

  6. Key, E.L.: An analysis of the structure and complexity of nonlinear binary sequence generators. IEEE Trans. Inform. Theory 22, 732–736 (1976)

    Article  MATH  Google Scholar 

  7. Kohavi, Z.: Switching and finite automata theory. McGraw-Hill Book Company, New York (1978)

    MATH  Google Scholar 

  8. Kolokotronis, N., Kalouptsidis, N.: On the linear complexity of nonlinearly filtered PN-sequences. IEEE Trans. Inform. Theory 49, 3047–3059 (2003)

    Article  MathSciNet  Google Scholar 

  9. Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Inform. Theory 22, 75–81 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  11. Massey, J.L.: Shift register synthesis and BCH decoding. IEEE Trans. Inform. Theory 15, 122–127 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)

    Book  Google Scholar 

  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. Rizomiliotis, P., Kalouptsidis, N.: Results on the nonlinear span of binary sequences. IEEE Trans. Inform. Theory 51, 1555–1563 (2005)

    Article  MathSciNet  Google Scholar 

  17. Rizomiliotis, P., Kolokotronis, N., Kalouptsidis, N.: On the quadratic span of binary sequences. IEEE Trans. Inform. Theory 51, 1840–1848 (2005)

    Article  MathSciNet  Google Scholar 

  18. Rueppel, R.A.: Analysis and design of stream ciphers. Springer, Berlin (1986)

    MATH  Google Scholar 

  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. Wyner, A.D., Ziv, J.: The sliding-window Lempel-Ziv algorithm is asymptotically optimal. Proceedings of the IEEE 82, 872–877 (1994)

    Article  Google Scholar 

  21. Ziv, J., Lempel, A.: A universal algorithm for sequential data compression. IEEE Trans. Inform. Theory 23, 337–343 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ziv, J., Lempel, A.: Compression of individual sequences via variable-rate coding. IEEE Trans. Inform. Theory 24, 530–536 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, TYPA Buildings, University Campus, 15784, Athens, Greece

    Konstantinos Limniotis, Nicholas Kolokotronis & Nicholas Kalouptsidis

Authors
  1. Konstantinos Limniotis
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  2. Nicholas Kolokotronis
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  3. Nicholas Kalouptsidis
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Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    Guang Gong

  2. Department of Informatics, University of Bergen, PB 7803, 5020, Bergen, Norway

    Tor Helleseth

  3. Department of Electrical and Electronic Engineering, Yonsei University, 121-749, Seoul, Korea

    Hong-Yeop Song

  4. Dept. of Electronics and Electrical Engineering, Pohang University of Science and Technology (POSTECH), 790-784, Pohang, Kyungbuk, Korea

    Kyeongcheol Yang

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© 2006 Springer-Verlag Berlin Heidelberg

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Limniotis, K., Kolokotronis, N., Kalouptsidis, N. (2006). Nonlinear Complexity of Binary Sequences and Connections with Lempel-Ziv Compression. In: Gong, G., Helleseth, T., Song, HY., Yang, K. (eds) Sequences and Their Applications – SETA 2006. SETA 2006. Lecture Notes in Computer Science, vol 4086. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11863854_14

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  • DOI: https://doi.org/10.1007/11863854_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44523-4

  • Online ISBN: 978-3-540-44524-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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