The Shortest Feedback Shift Register That Can Generate A Given Sequence

  • Cees J. A. Jansen
  • Dick E. Boekee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 435)


In this paper the problem of finding the absolutely shortest (possibly nonlin- ear) feedback shift register, which can generate a given sequence with characters from some arbitrary finite alphabet, is considered. To this end, a new complex- ity measure is defined, called the maximum order complexity. A new theory of the nonlinear feedback shift register is developed, concerning elementary complexity properties of transposed and reciprocal sequences, and feedback functions of the maximum order feedback shift register equivalent. Moreover, Blumer’s algorithm is identified as a powerful tool for determining the maxi- mum order complexity profile of sequences, as well as their period, in linear time and memory. The typical behaviour of the maximum order complexity profile is shown and the consequences for the analysis of given sequences and the synthesis of feedback shift registers are discussed.


  1. [1]
    R. Arratia, L. Gordon and M. S. Waterman. “An Extreme Value Theory for Sequence Matching”, The Annals of Statistics, vol. 14, no. 3, pp. 971–993, 1986.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Blumer, J. Blumer, A. Ehrenfeucht, D. Haussler and R. McConnell. “Linear Size Finite Automata for the Set of all Subwords of a Word: An Outline of Results”, Bul. Eur. Assoc. Theor. Comp. Sci., no. 21, pp. 12–20, 1983.Google Scholar
  3. [3]
    D. W. Davies and W. L. Price. Security for Computer Networks, John Wiley & Sons, Inc., Chichester, 1984.Google Scholar
  4. [4]
    H. Fredricksen. “A survey of full-length nonlinear shift register cycle algorithms”, SIAM Rev., vol. 24, pp. 195–221, April 1982.CrossRefMathSciNetGoogle Scholar
  5. [5]
    S. W. Golomb. Shift Register Sequences, Holden-Day Inc., San Francisco, 1967.MATHGoogle Scholar
  6. [6]
    C. J. A. Jansen. Investigations On Nonlinear Streamcipher Systems: Construction and Evaluation Methods, PhD. Thesis, Technical University of Delft, Delft, 1989.Google Scholar
  7. [7]
    S. Karlin, G. Ghandour, F. Ost, S. Tavare and L. J. Korn. “New Approaches for Computer Analysis of Nucleic Acid Sequences”, Proc. Natl. Acad. Sci. USA, vol. 80, pp. 5660–5664, September 1983.MATHCrossRefGoogle Scholar
  8. [8]
    A. G. Konheim. Cryptography, John Wiley & Sons, Inc., New York, 1981.MATHGoogle Scholar
  9. [9]
    A. Lempel and J. Ziv. “On the Complexity of Finite Sequences”, IEEE Trans. on Info. Theory, vol. IT-22, no. 1, pp. 75–81, January 1976.CrossRefMathSciNetGoogle Scholar
  10. [10]
    F. J. MacWilliams and N. J. A. Sloane. The Theory of Error Correcting Codes, Amsterdam, North-Holland, 1978.Google Scholar
  11. [11]
    J. L. Massey. “Shift-Register Synthesis and BCH Decoding”, IEEE Trans. on Info. Theory, vol. IT-15, January 1969.Google Scholar
  12. [12]
    C. H. Meyer and S. M. Matyas. Cryptography: A New Dimension in Computer Data Security, John Wiley & Sons, New York, 1982.MATHGoogle Scholar
  13. [13]
    E. M. McCreight. “A space-economical suffix tree construction algorithm”, JACM, vol. 23, no. 2, pp. 262–272, April 1976.CrossRefMathSciNetGoogle Scholar
  14. [14]
    R. A. Rueppel. New Approaches to Stream Ciphers, PhD. Thesis, Swiss Federal Institute of Technology, Zurich, 1984.Google Scholar
  15. [15]
    C. E. Shannon. “Communication Theory of Secrecy Systems”, Bell Systems Technical Journal, vol. 28, pp. 656–715, October 1949.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Cees J. A. Jansen
    • 1
  • Dick E. Boekee
    • 2
  1. 1.Philips USFA B.V.EindhovenThe Netherlands
  2. 2.Technical University of DelftDelftThe Netherlands

Personalised recommendations