Cryptography and Communications

, Volume 10, Issue 6, pp 1183–1202 | Cite as

The cycle structure of LFSR with arbitrary characteristic polynomial over finite fields

  • Zuling Chang
  • Martianus Frederic EzermanEmail author
  • San Ling
  • Huaxiong Wang
Part of the following topical collections:
  1. Special Issue on Sequences and Their Applications


We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed.


Cycle structure Cyclotomic number de Bruijn sequence Decimation LFSR Periodic sequence 

Mathematics Subject Classification (2010)

11B50 94A55 94A60 



The work of Z. Chang is supported by the Joint Fund of the National Natural Science Foundation of China under Grants 61772476 and U1304604. Research Grants TL-9014101684-01 and MOE2013-T2-1-041 support the research carried out by M. F. Ezerman, S. Ling, and H. Wang.


  1. 1.
    Aguirre, G. K., Mattar, M. G., Magis-Weinberg, L.: De Bruijn cycles for neural decoding. NeuroImage 56(3), 1293–1300 (2011)CrossRefGoogle Scholar
  2. 2.
    Berger, B., Orenstein, Y.: Efficient design of compact unstructured RNA libraries covering all k-mers. In: Proceedings of WABI 2015, pp. 308–325, Atlanta, USA, Sept. 10–12, 2015. Springer, BerlinGoogle Scholar
  3. 3.
    Compeau, P., Pevzner, P., Tesler, G.: How to apply de Bruijn graphs to genome assembly. Nat. Biotechnol. 29(11), 987–991 (2011)CrossRefGoogle Scholar
  4. 4.
    Ding, C., Pei, D., Salomaa, A.: Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography. World Scientific Publishing Co. Inc., River Edge (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Fredricksen, H.: A survey of full length nonlinear shift register cycle algorithms. SIAM Rev. 24(2), 195–221 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Golomb, S. W.: Shift Register Sequences. Aegean Park Press, Laguna Hills (1981)zbMATHGoogle Scholar
  7. 7.
    Golomb, S. W., Gong, G.: Signal Design for Good Correlation: for Wireless Communication, Cryptography, and Radar. Cambridge University Press, New York (2004)zbMATHGoogle Scholar
  8. 8.
    Hauge, E. R., Helleseth, T.: De Bruijn sequences, irreducible codes and cyclotomy. Discrete Math. 159(1–3), 143–154 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lempel, A: On a homomorphism of the de Bruijn graph and its applications to the design of feedback shift registers. IEEE Trans. Comput. C-19(12), 1204–1209 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lidl, R., Niederreiter, H.: Finite Fields, ser. Encyclopaedia of Mathematics and Its Applications, vol. 20. Cambridge University Press, New York (1997)Google Scholar
  11. 11.
    Storer, T.: Cyclotomy and Difference Sets, ser. Lectures in Advanced Mathematics. Markham Pub. Co., Chicago (1967)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouChina
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

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