On the Linearity of Cryptographic Sequence Generators

  • Amparo Fuster-Sabater
  • Oscar Delgado-Mohatar
  • Ljiljana Brankovic
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6017)


In this paper we show that the output sequences of the generalized self-shrinking generator are particular solutions of a binary homogeneous linear difference equation. In fact, all these sequences are just linear combinations of primary sequences weighted by binary coefficients. We show that in addition to the output sequences of the generalized self-shrinking generator, the complete class of solutions of the corresponding binary homogeneous linear difference equation also includes other balanced sequences that are very suitable for cryptographic applications, as they have the same period and even greater linear complexity than the generalized self-shrinking sequences. Cryptographic parameters of all the above mentioned sequences can be analyzed in terms of linear equation solutions.


binary sequence linear difference equation generalized self-shrinking generator cryptography 


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  1. 1.
    Coppersmith, D., Krawczyk, H., Mansour, Y.: The Shrinking Generator. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 22–39. Springer, Heidelberg (1994)Google Scholar
  2. 2.
    Dickson, L.E.: Linear Groups with an Exposition of the Galois Field Theory, pp. 3–71. Dover, New York (1958); An updated reprint can be found, zbMATHGoogle Scholar
  3. 3.
    Fúster-Sabater, A., Caballero-Gil, P.: Strategic Attack on the Shrinking Generator. Theoretical Computer Science 409(3), 530–536 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fúster-Sabater, A., Caballero-Gil, P.: Cryptanalytic Attack on Cryptographic Sequence Generators: The Class of Clock-Controlled Shrinking Generators. In: Gervasi, O., Murgante, B., Laganà, A., Taniar, D., Mun, Y., Gavrilova, M.L. (eds.) ICCSA 2008, Part II. LNCS, vol. 5073, pp. 668–679. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Golomb, S.W.: Shift Register-Sequences. Aegean Park Press, Laguna Hill (1982)Google Scholar
  6. 6.
    Gong, G.: Theory and Applications of q-ary Interleaved Sequences. IEEE Trans. Information Theory 41(2), 400–411 (1995)zbMATHCrossRefGoogle Scholar
  7. 7.
    Gomulkiewicz, M., Kutylowski, M., Wlaz, P.: Fault Jumping Attacks against Shrinking Generator.In: Dagstuhl Seminar, Proceedings 06111, Complexity of Boolean Functions (2006)
  8. 8.
    Hu, Y., Xiao, G.: Generalized Self-Shrinking Generator. IEEE Trans. Inform. Theory 50, 714–719 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Jennings, S.M.: Multiplexed Sequences: Some Properties. In: Beth, T. (ed.) EUROCRYPT 1982. LNCS, vol. 149, Springer, Heidelberg (1983)CrossRefGoogle Scholar
  10. 10.
    Key, E.L.: An Analysis of the Structure and Complexity of Nonlinear Binary Sequence Generators. IEEE Trans. Informat. Theory 22(6), 732–736 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1986)zbMATHGoogle Scholar
  12. 12.
    Meier, W., Staffelbach, O.: The Self-Shrinking Generator. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 205–214. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  13. 13.
    Menezes, A.J., et al.: Handbook of Applied Cryptography. CRC Press, New York (1997)zbMATHGoogle Scholar
  14. 14.
    Mihaljevic, M.J.: A Faster Cryptanalysis of the Self-Shrinking Generator. In: Pieprzyk, J.P., Seberry, J. (eds.) ACISP 1996. LNCS, vol. 1172, Springer, Heidelberg (1996)Google Scholar
  15. 15.
    Zenner, E., Krause, M., Lucks, S.: Improved cryptanalysis of the self-shrinking generator. In: Varadharajan, V., Mu, Y. (eds.) ACISP 2001. LNCS, vol. 2119, pp. 21–35. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Zhang, B., Feng, D.: New Guess-and-Determine Attack on the Self-Shrinking Generator. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 54–68. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Amparo Fuster-Sabater
    • 1
  • Oscar Delgado-Mohatar
    • 1
  • Ljiljana Brankovic
    • 2
  1. 1.Institute of Applied PhysicsC.S.I.C.MadridSpain
  2. 2.School of Electrical Engineering and Computer ScienceUniversity of NewcastleCallaghanAustralia

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