On the Linear Complexity of Sidel’nikov Sequences over \({\mathbb {F}}_d\)

  • Nina Brandstätter
  • Wilfried Meidl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4086)

Abstract

We study the linear complexity of sequences over the prime field \({\mathbb{F}_d}\) introduced by Sidel’nikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences. The obtained results extend known results on the binary case. Finally we present an upper bound on the aperiodic autocorrelation.

Keywords

Sidel’nikov sequence Linear complexity Linear complexity profile Aperiodic autocorrelation 

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References

  1. 1.
    Baumert, L.D., Fredricksen, H.: The cyclotomic numbers of order eighteen with applications to difference sets. Math. Comp. 21, 204–219 (1967)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. In: A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1998)Google Scholar
  3. 3.
    Cusick, T.W., Ding, C., Renvall, A.: Stream Ciphers and Number Theory. North-Holland Publishing Co., Amsterdam (1998)MATHGoogle Scholar
  4. 4.
    Granville, A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. In: Organic mathematics, Burnaby, BC, 1995, CMS Conf. Proc. 20, Amer. Math. Soc., Providence, RI, pp. 253–276 (1997)Google Scholar
  5. 5.
    Hasse, H.: Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik. J. Reine Angew. Math. 175, 50–54 (1936)CrossRefGoogle Scholar
  6. 6.
    Helleseth, T., Yang, K.: On binary sequences with period n = p m − 1 with optimal autocorrelation. In: Helleseth, T., Kumar, P., Yang, K. (eds.) Proceedings of SETA 2001, pp. 209–217 (2002)Google Scholar
  7. 7.
    Kim, Y.-S., Chung, J.-S., No, J.-S., Chung, H.: On the autocorrelation distributions of Sidel’nikov sequences. IEEE Trans. Inf. Th. 51, 3303–3307 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Kyureghyan, G.M., Pott, A.: On the linear complexity of the Sidelnikov-Lempel-Cohn-Eastman sequences. Designs, Codes, and Cryptography 29, 149–164 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lempel, A., Cohn, M., Eastman, W.L.: A class of balanced binary sequences with optimal autocorrelation properties. IEEE Trans. Inf. Th. 23, 38–42 (1977)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lidl, R., Niederreiter, H.: Finite Fields. Addison-Wesley, Reading (1983)MATHGoogle Scholar
  11. 11.
    Lucas, M.E.: Sur les congruences des nombres euleriennes et des coefficients differentiels des fuctions trigonometriques. suivant un-module premier, Bull. Soc. Math. France 6, 122–127 (1878)Google Scholar
  12. 12.
    Meidl, W., Winterhof, A.: Some notes on the linear complexity of Sidel’nikov-Lempel-Cohn-Eastman sequences. Designs, Codes, and Cryptography 38, 159–178 (2006)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Shparlinski, I.: Cryptographic Applications of Analytic Number Theory. Complexity Lower Bounds and Pseudorandomness. In: Progress in Computer Science and Applied Logic, Birkhäuser, Basel, vol. 22 (2003)Google Scholar
  14. 14.
    Sidel’nikov, V.M.: Some k-valued pseudo-random sequences and nearly equidistant codes. Problems of Information Transmission 5, 12–16 (1969); translated from Problemy Peredači Informacii (Russian) 5, 16–22 (1969)Google Scholar
  15. 15.
    Storer, T.: Cyclotomy and Difference Sets, vol. III. Markham Publishing Co., Chicago (1967)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nina Brandstätter
    • 1
  • Wilfried Meidl
    • 2
  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.Sabanci UniversityOrhanli, Tuzla, IstanbulTurkey

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