The Linear Complexity and 2-Error Linear Complexity Distribution of \(2^n\)-Periodic Binary Sequences with Fixed Hamming Weight

  • Wenlun Pan
  • Zhenzhen BaoEmail author
  • Dongdai Lin
  • Feng Liu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9977)


The linear complexity and k-error linear complexity of sequences are important measures of the strength of key-streams generated by stream ciphers. Based on the characters of the set of sequences with given linear complexity, people get the characterization of \(2^n\)-binary sequences with given k-error linear complexity for small k recently. In this paper, we put forward this study to get the distribution of linear complexity and k-error linear complexity of \(2^n\)-periodic binary sequences with fixed Hamming weight. First, we give the counting function of the number of \(2^n\)-periodic binary sequences with given linear complexity and fixed Hamming weight. Provide an asymptotic evaluation of this counting function when n gets large. Then we take a step further to study the distribution of \(2^n\)-periodic binary sequences with given 2-error linear complexity and fixed Hamming weight. Through an asymptotic analysis, we provide an estimate on the number of \(2^n\)-periodic binary sequences with given 2-error linear complexity and fixed Hamming weight.


Sequence Linear complexity k-error linear complexity Counting function Hamming weight Asymptotic analysis 



Many thanks go to the anonymous reviewers for their detailed comments and suggestions. This work was supported by the National Key R&D Program of China with No. 2016YFB0800100, CAS Strategic Priority Research Program with No. XDA06010701, National Key Basic Research Project of China with No. 2011CB302400 and National Natural Science Foundation of China with No. 61671448, No. 61379139.


  1. 1.
    Ding, C., Xiao, G., Shan, W.: The Stability Theory of Stream Ciphers. LNCS, vol. 561. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  2. 2.
    Fu, F.-W., 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. LNCS, vol. 4086, pp. 88–103. Springer, Heidelberg (2006). doi: 10.1007/11863854_8 CrossRefGoogle Scholar
  3. 3.
    Games, R., Chan, A.: A fast algorithm for determining the complexity of a binary sequence with period \(2^n\) (corresp.). IEEE Trans. Inf. Theory 29(1), 144–146 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kavuluru, R.: \(2^{n}\)-periodic binary sequences with fixed k-error linear complexity for \(k\) = 2 or 3. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 252–265. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-85912-3_23 CrossRefGoogle 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(2), 75–97 (2009)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15(1), 122–127 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Meidl, W.: On the stability of \(2^n\)-periodic binary sequences. IEEE Trans. Inf. Theory 51(3), 1151–1155 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ming, S.: Decomposing approach for error vectors of k-error linear complexity of certain periodic sequences. IEICE Trans. Fundam. Electr. Commun. Comput. Sci. E97-A(7), 1542–1555 (2014)Google Scholar
  10. 10.
    Rueppel, A.R.: Analysis and Design of Stream Ciphers. Communications and Control Engineering Series. Springer, Heidelberg (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    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(4), 1398–1401 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zhou, J.: A counterexample concerning the 3-error linear complexity of \(2^n\)-periodic binary sequences. Des. Codes Cryptogr. 64(3), 285–286 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhou, J., Liu, W.: The \(k\)-error linear complexity distribution for \(2^n\)-periodic binary sequences. Des. Codes Cryptogr. 73(1), 55–75 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Wenlun Pan
    • 1
    • 2
  • Zhenzhen Bao
    • 3
    Email author
  • Dongdai Lin
    • 1
  • Feng Liu
    • 1
    • 2
  1. 1.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  2. 2.University of Chinese Academy of SciencesBeijingChina
  3. 3.Shanghai Jiao Tong UniversityShanghaiChina

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