Advertisement

Designs, Codes and Cryptography

, Volume 73, Issue 1, pp 55–75 | Cite as

The \(k\)-error linear complexity distribution for \(2^n\)-periodic binary sequences

  • Jianqin Zhou
  • Wanquan Liu
Article

Abstract

The linear complexity and the \(k\)-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, we investigate the \(k\)-error linear complexity distribution of \(2^n\)-periodic binary sequences in this paper based on Games–Chan algorithm. First, for \(k=2,3\), the complete counting functions for the \(k\)-error linear complexity of \(2^n\)-periodic binary sequences (with linear complexity less than \(2^n\)) are characterized. Second, for \(k=3,4\), the complete counting functions for the \(k\)-error linear complexity of \(2^n\)-periodic binary sequences with linear complexity \(2^n\) are presented. Third, as a consequence of these results, the counting functions for the number of \(2^n\)-periodic binary sequences with the \(k\)-error linear complexity for \(k = 2\) and \(3\) are obtained.

Keywords

Periodic sequence Linear complexity \(k\)-error linear complexity \(k\)-error linear complexity distribution 

Mathematics Subject Classification (2000)

94A55 94A60 11B50 

Notes

Acknowledgments

The research was supported by Anhui Natural Science Foundation (No.1208085MF106).

References

  1. 1.
    Ding C.S., Xiao G.Z., Shan W.J.: The Stability Theory of Stream Ciphers[M]. Lecture Notes in Computer Science, vol. 561, pp. 85–88. Springer, Berlin (1991).Google Scholar
  2. 2.
    Games R.A., Chan A.H.: A fast algorithm for determining the complexity of a binary sequence with period \(2^n\). IEEE Trans. Inf. Theory 29(1), 144–146 (1983).Google Scholar
  3. 3.
    Fu F., Niederreiter H., Su, M.: The characterization of \(2^n\)-periodic binary sequences with fixed 1-error linear complexity. In: Gong G., Helleseth T., Song H.-Y., Yang K. (eds.) SETA 2006, Lecture Notes in Computer Science, vol. 4086, pp. 88–103. Springer, Berlin (2006).Google Scholar
  4. 4.
    Kavuluru R.: \(2^n\)-periodic binary sequences with fixed 2-error or 3-error linear complexity. In: Golomb S., Parker M., Pott A., Winterhof A. (eds.) SETA 2008. Lecture Notes in Computer Science, vol. 5203, pp. 252–265. Springer, Berlin (2008).Google Scholar
  5. 5.
    Kavuluru R.: Characterization of \(2^n\)-periodic binary sequences with fixed 2-error or 3-error linear complexity. Des. Codes Cryptogr. 53, 75–97 (2009).Google Scholar
  6. 6.
    Kurosawa K., Sato F., Sakata T., Kishimoto W.: A relationship between linear complexity and \(k\)-error linear complexity. IEEE Trans. Inf. Theory 46(2), 694–698 (2000).Google Scholar
  7. 7.
    Meidl W.: On the stablity of \(2^{n}\)-periodic binary sequences. IEEE Trans. Inf. Theory 51(3), 1151–1155 (2005).Google Scholar
  8. 8.
    Rueppel, R.A.: Analysis and Design of Stream Ciphers (chap. 4). Springer, Berlin (1986).Google Scholar
  9. 9.
    Stamp M., Martin C.F.: An algorithm for the \(k\)-error linear complexity of binary sequences with period \(2^{n}\). IEEE Trans. Inf. Theory 39, 1398–1401 (1993).Google Scholar
  10. 10.
    Zhou J.Q.: A counterexample concerning the 3-error linear complexity of \(2^n\)-periodic binary sequences. Des. Codes Cryptogr. 64(3), 285–286 (2012).Google Scholar
  11. 11.
    Zhu F.X., Qi W.F.: The 2-error linear complexity of \(2^n\)-periodic binary sequences with linear complexity \(2^n\)-1. J. Electron. (China) 24(3), 390–395 (2007).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Computer Science SchoolAnhui University of TechnologyMa’anshanChina
  2. 2.Telecommunication SchoolHangzhou Dianzi UniversityHangzhouChina
  3. 3.Department of ComputingCurtin UniversityPerthAustralia

Personalised recommendations