Characterization of 2 n -periodic binary sequences with fixed 2-error or 3-error linear complexity | Designs, Codes and Cryptography Skip to main content
Springer Nature Link
Log in

Characterization of 2n-periodic binary sequences with fixed 2-error or 3-error linear complexity

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

The linear complexity of sequences is an important measure of the cryptographic strength of key streams used in stream ciphers. The instability of linear complexity caused by changing a few symbols of sequences can be measured using k-error linear complexity. In their SETA 2006 paper, Fu et al. (SETA, pp. 88–103, 2006) studied the linear complexity and the 1-error linear complexity of 2n-periodic binary sequences to characterize such sequences with fixed 1-error linear complexity. In this paper we study the linear complexity and the k-error linear complexity of 2n-periodic binary sequences in a more general setting using a combination of algebraic, combinatorial, and algorithmic methods. This approach allows us to characterize 2n-periodic binary sequences with fixed 2- or 3-error linear complexity. Using this characterization we obtain the counting function for the number of 2n-periodic binary sequences with fixed k-error linear complexity for k = 2 and 3.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cusick T.W., Ding C., Renvall A.: Stream Ciphers and Number Theory. North-Holland (1998).

  2. Ding C., Xiao G., Shan W.: The Stability Theory of Stream Ciphers. Springer (1991).

  3. Fu F.-W., Niederreiter H., Su M.: The characterization of 2n-periodic binary sequences with fixed 1-error linear complexity. In: Gong G., Helleseth T., Song H.-Y., Yang K. (eds.) SETA 2006. LNCS, vol. 4086, pp. 88–103. Springer (2006).

  4. Games R.A., Chan A.H.: A fast algorithm for determining the complexity of a pseudo-random sequence with period 2n. IEEE Trans. Inform. Theory 29(1), 144–146 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kavuluru R.: 2n-periodic binary sequences with fixed 2-error or 3-error linear complexity. In: Golomb S., Parker M., Pott A., Winterhof A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 252–265. Springer (2008).

  6. Kurosawa K., Sato F., Sakata T., Kishimoto W.: A relationship between linear complexity and k-error linear complexity. IEEE Trans. Inform. Theory 46(2), 694–698 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Massey J.L.: Shift register synthesis and BCH decoding. IEEE Trans. Inform. Theory 15(1), 122–127 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  8. Meidl W.: On the stability of 2n-periodic binary sequences. IEEE Trans. Inform. Theory 51(3), 1151–1155 (2005)

    Article  MathSciNet  Google Scholar 

  9. Meidl W., Niederreiter H.: Counting functions and expected values for the k-error linear complexity. Finite Fields Appl. 8, 142–154 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Meidl W., Niederreiter H.: Linear complexity, k-error linear complexity, and the discrete fourier transform. J. Complexity 18(1), 87–103 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Meidl W., Niederreiter H.: On the expected value of linear complexity and the k-error linear complexity of periodic sequences. IEEE Trans. Inform. Theory 48(11), 2817–2825 (2002)

    Article  MathSciNet  Google Scholar 

  12. Meidl W., Venkateswarlu A.: Remarks on the k-error linear complexity of p n-periodic sequences. Des. Codes Cryptogr. 42(2), 181–193 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Rueppel R.A.: Analysis and Design of Stream Ciphers. Springer (1986).

  14. Stamp M., Martin C.F.: An algorithm for the k-error linear complexity of binary sequences with period 2n. IEEE Trans. Inform. Theory 39(4), 1398–1401 (1993)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Kentucky, Lexington, KY, 40506, USA

    Ramakanth Kavuluru

Authors
  1. Ramakanth Kavuluru
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ramakanth Kavuluru.

Additional information

Communicated by P. Wild.

A portion of this paper has appeared in the proceedings of the 5th international conference on Sequences and their Applications (SETA 2008). This material is based upon work supported by the National Science Foundation under Grant No. CCF-0514660. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kavuluru, R. Characterization of 2n-periodic binary sequences with fixed 2-error or 3-error linear complexity. Des. Codes Cryptogr. 53, 75–97 (2009). https://doi.org/10.1007/s10623-009-9295-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-009-9295-x

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation