On the Stability of m-Sequences

  • Alex J. Burrage
  • Ana Sălăgean
  • Raphael C. -W. Phan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7089)


We study the stability of m-sequences in the sense of determining the number of errors needed for decreasing the period of the sequences, as well as giving lower bounds on the k-error linear complexity of the sequences. For prime periods the results are straightforward so we concentrate on composite periods. We give exact results for the case when the period is reduced by a factor which is a Mersenne number and for the case when it is reduced by a prime p such that the order of 2 modulo p equals p − 1. The general case is believed to be difficult due to its similarity to a well studied problem in coding theory. We also provide results about the relative frequencies of the different cases. We formulate a conjecture regarding the minimum number of errors needed for reducing the period at all. Finally we apply our results to the LFSR components of several well known stream ciphers.


Binary Sequence Linear Complexity Brute Force Period Length Cyclic Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berbain, C., Billet, O., Canteaut, A., Courtois, N., Debraize, B., Gilbert, H., Goubin, L., Gouget, A., Granboulan, L., Lauradoux, C., Minier, M., Pornin, T., Silbert, H.: DECIMv2. In: eStream Candidate, ECRYPT Stream Cipher Workshop SKEW (2005)Google Scholar
  2. 2.
    Dawson, E., Clark, A., Golić, J., Millan, W., Penna, L., Simpson, L.: The LILI-128 Keystream Generator. In: Proc. 1st NESSIE Workshop (2000)Google Scholar
  3. 3.
    Ding, C., Xiao, G., Shan, W.: The Stability Theory of Stream Ciphers. LNCS, vol. 561. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  4. 4.
    Ding, C.: The Weight Distribution of Some Irreducible Cyclic Codes. IEEE Transactions on Information Theory 55(3), 955–960 (2009)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Ekdahl, P., Johansson, T.: Another Attack on A5/1. IEEE Transactions on Information Theory 49(1) (2003)Google Scholar
  6. 6.
    Golomb, S.W.: Shift Register Sequences. Aegean Park Press (1982)Google Scholar
  7. 7.
    Hell, M., Johansson, T., Meier, W.: Grain - A Stream Cipher for Constrained Environments. International Journal of Wireless and Mobile Computing, Special Issue on Security of Computer Network and Mobile Systems (2006)Google Scholar
  8. 8.
    Hu, H.: Periods on Two Kinds of Nonlinear Feedback Shift Registers with Time Varying Feedback Functions Technical Reports, Center for Applied Cryptographic Research (2011)Google Scholar
  9. 9.
    Kaida, T., Uehara, S., Imamura, K.: An algorithm for the k-error linear complexity of sequences over GF(p m) with period p n, p a prime. Inform. Comput. 151, 134–147 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Lauder, A.G.B., Paterson, K.G.: Computing the error linear complexity spectrum of a binary sequence of period 2n. IEEE Transactions on Information Theory 49, 273–280 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland (1978)Google Scholar
  12. 12.
    Meidl, W., Aly, H., Winterhof, A.: On the k-error linear complexity of cyclotomic sequences. Journal Mathematical Cryptography (2007)Google Scholar
  13. 13.
    Pomerance, C.: Recent developments in primality testing. The Mathematical Intelligencer 3, 97–105 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Stamp, M., Martin, C.F.: An algorithm for the k-error linear complexity of binary sequences of period 2n. IEEE Transactions on Information Theory 39, 1398–1401 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Surböck, F., Weinrichter, H.: Interlacing Properties of Shift-Register Sequences with Generator Polynomials Irreducible over GF(p). IEEE Transactions on Information Theory 24(3), 386–389 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Zhang, M., Carroll, C., Chan, A.H.: The Software-Oriented Stream Cipher SSC2. In: Schneier, B. (ed.) FSE 2000. LNCS, vol. 1978, p. 31. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alex J. Burrage
    • 1
  • Ana Sălăgean
    • 1
  • Raphael C. -W. Phan
    • 2
  1. 1.Computer ScienceLoughborough UniversityLeicestershireUK
  2. 2.Electronic, Electrical & Systems EngineeringLoughborough UniversityLeicestershireUK

Personalised recommendations