Journal of Systems Science and Complexity

, Volume 28, Issue 5, pp 1231–1242 | Cite as

A note on determine the greatest common subfamily of two NFSRs by Gröbner basis

  • Zhongxiao WangEmail author
  • Wenfeng Qi
  • Tian Tian


For nonlinear feedback shift registers (NFSRs), their greatest common subfamily may be not unique. Given two NFSRs, the authors only consider the case that their greatest common subfamily exists and is unique. If the greatest common subfamily is exactly the set of all sequences which can be generated by both of them, the authors can determine it by Gröbner basis theory. Otherwise, the authors can determine it under some conditions and partly solve the problem.


Greatest common subfamily Gröbner basis nonlinear feedback shift register stream cipher 


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  1. [1]
    Meier W and Staffelbach O, Fast correlation attacks on certain stream ciphers, Journal of Cryptology, 1989, 1(3): 159–176.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Canteaut A and Trabbia M, Improved fast correlation attacks using parity-check equations of weight 4 and 5, Advances in Cryptology-EUROCRYPT 2000 (ed. by Preneel B), Bruges, 2000.Google Scholar
  3. [3]
    Courtois N and Meier W, Algebraic attacks on stream ciphers with linear feedback, Advances in Cryptology-EUROCRYPT 2003 (ed. by Biham E), Warsaw, 2003.Google Scholar
  4. [4]
    Courtois N, Fast algebraic attacks on stream ciphers with linear feedback, Advances in Cryptology-CRYPTO 2003 (ed. by Boneh D), California, 2003.Google Scholar
  5. [5]
    Hell M, Johansson T, and Meier W, New Stream Cipher Designs: The Grain Family of Stream Ciphers, Springer-Verlag, Berlin, 2008.Google Scholar
  6. [6]
    Babbage S and Dodd M, New Stream Cipher Designs: The MICKEY Stream Ciphers, Springer-Verlag, Berlin, 2008.Google Scholar
  7. [7]
    Cannière C D and Preneel B, New Stream Cipher Designs: Trivium, Springer-Verlag Berlin, 2008.Google Scholar
  8. [8]
    Kjeldsen K, On the cycle of structure of a set of nonlinear shift registers with symmetric feedback functions, Journal of Combinatorial Theory Series A, 1976, 1(3): 154–169.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Hu H and Gong G, Periods on two kinds of nonlinear feedback shift registers with time varying feedback functions, International Journal of Foundations of Computer Science, 2011, 22(6): 1317–1329.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Annexstein F S, Generating De Bruijn sequences: An efficient implementation, IEEE Transactions on Computers, 1997, 46(2): 198–200.CrossRefGoogle Scholar
  11. [11]
    Jansen C J, Investigations on nonlinear streamcipher systems: Construction and evaluation methods, Doctor’s degree thesis, Technical University of Delft, Netherlands, 1989.Google Scholar
  12. [12]
    Erdmann D and Murphy S, An approximate distribution for the maximum order complexity, Designs, Codes and Cryptography, 2005, 10(4): 1555–1563.Google Scholar
  13. [13]
    Dubrova E, A transformation from the Fibonacci to the Galois NLFSRs, IEEE Transactions on Information Theory, 2009, 55(11): 5263–5271.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Lidl R and Niederreiter H, Finite Fields, Cambridge University Press Oxford, 1997.Google Scholar
  15. [15]
    Cox D, Little J, and O’Shea D, Ideals, Varieties and Algorithms, Springer-Verlag, Berlin, 1996.zbMATHGoogle Scholar
  16. [16]
    Gao X S and Huang Z, Characteristic set algorithms for equation solving in finite fields, Journal of Symbolic Computation, 2012, 47(6): 655–679.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Lang S, Algebra, Springer-Verlag Berlin, 2002.CrossRefzbMATHGoogle Scholar
  18. [18]
    Becker T and Weispfenning V, Gröbner Bases, a Computationnal Approach to Commutative Algebra, Springer-Verlag, Berlin, 1993.Google Scholar
  19. [19]
    Golomb S W, Shift Register Sequences, Aegean Park Press California, 1982.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Luoyang University of Foreign LanguageLuoyangChina
  2. 2.State Key Laboratory of Mathematical Engineering and Advanced ComputingZhengzhou Information Science and Technology InstituteZhengzhouChina

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