Skip to main content

Linear Filtering of Nonlinear Shift-Register Sequences

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNSC,volume 3969)

Abstract

Nonlinear n-stage feedback shift-register sequences over the finite field \(\mathbb{F}_q\) of period q n–1 are investigated under linear operations on sequences. We prove that all members of an easily described class of linear combinations of shifted versions of these sequences possess useful properties for cryptographic applications: large periods, large linear complexities and good distribution properties. They typically also have good maximum order complexity values as has been observed experimentally. A running key generator is introduced based on certain nonlinear feedback shift registers with modifiable linear feedforward output functions.

Keywords

  • Linear Complexity
  • Stream Cipher
  • Periodic Sequence
  • Irreducible Polynomial
  • Linear Filter

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. van Aardenne-Ehrenfest, T., de Bruijn, N.G.: Circuits and trees in oriented linear graphs. Simon Steven 28, 203–217 (1951)

    MathSciNet  MATH  Google Scholar 

  2. Zong-Duo, D., Jun-Hui, Y.: Linear complexity of periodically repeated random sequences. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 168–175. Springer, Heidelberg (1991)

    CrossRef  Google Scholar 

  3. Fúster-Sabater, A., Caballero-Gil, P.: On the linear complexity on nonlinearly filtered PN-sequences. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 80–90. Springer, Heidelberg (1995)

    Google Scholar 

  4. Gammel, B.M., Göttfert, R., Kniffler, O.: Status of Achterbahn and tweaks. In: SASC 2006—Stream Ciphers Revisited, Leuven, Belgium, February 2-3, 2006. Workshop Record, pp. 302–315 (2006)

    Google Scholar 

  5. Gammel, B.M., Göttfert, R., Kniffler, O.: An NLFSR-based stream cipher. In: IEEE International Symposium on Circuits and Systems — ISCAS 2006, Island of Kos, Greece, May 21-24 (2006)

    Google Scholar 

  6. Golomb, S.W.: Shift Register Sequences. Aegean Park Press, Laguna Hills (1982)

    MATH  Google Scholar 

  7. Groth, E.J.: Generation of binary sequences with controllable complexity. IEEE Trans. Inform. Theory IT-17, 288–296 (1971)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Jansen, C.J.A.: Investigations On Nonlinear Streamcipher Systems: Construction and Evaluation Methods, Ph.D. Thesis, Technical University of Delft, Delft (1989)

    Google Scholar 

  9. Key, E.: An analysis of the structure and complexity of nonlinear binary sequence generators. IEEE Trans. Inform. Theory IT-22, 732–736 (1976)

    CrossRef  MATH  Google Scholar 

  10. Laksov, D.: Linear recurring sequences over finite fields. Math. Scand. 16, 181–196 (1965)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Lam, C.C.Y., Gong, G.: A lower bound for the linear span of filtering sequences. In: A lower bound for the linear span of filtering sequences, Workshop Record of The State of the Art of Stream Ciphers, Brugge, October 2004, pp. 220–233 (2004)

    Google Scholar 

  12. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley, Reading (1983) (Now Cambridge Univ. Press.)

    MATH  Google Scholar 

  13. Massey, J.L., Serconek, S.: A Fourier Transform Approach to the Linear Complexity of Nonlinearly Filtered Sequences. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 332–340. Springer, Heidelberg (1994)

    Google Scholar 

  14. Meidl, W., Niederreiter, H.: On the expected value of the linear complexity and the k-error linear complexity of periodic sequences. IEEE Trans. Inform. Theory 48, 2817–2825 (2002)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Mykkeltveit, J.: Nonlinear recurrences and arithmetic codes. Information and Control 33, 193–209 (1977)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Mykkeltveit, J., Siu, M.-K., Tong, P.: On the cycle structure of some nonlinear shift register sequences. Information and Control 43, 202–215 (1979)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Niederreiter, H.: Cryptology—The mathematical theory of data security. In: Mitsui, T., Nagasaka, K., Kano, T. (eds.) Prospects of Mathematical Science, pp. 189–209. World Sci. Pub., Singapore (1988)

    Google Scholar 

  18. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NFS Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992)

    CrossRef  MATH  Google Scholar 

  19. Niederreiter, H.: Sequences with almost perfect linear complexity profile. In: Price, W.L., Chaum, D. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 37–51. Springer, Heidelberg (1988)

    CrossRef  Google Scholar 

  20. Paterson, K.G.: Root counting, the DFT and the linear complexity of nonlinear filtering. Designs, Codes and Cryptography 14, 247–259 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Rueppel, R.A.: Analysis and Design of Stream Ciphers. Springer, Heidelberg (1986)

    CrossRef  MATH  Google Scholar 

  22. Selmer, E.S.: Linear Recurrence Relations over Finite Fields. Univ. of Bergen (1966)

    Google Scholar 

  23. Siegenthaler, T., Kleiner, A.W., Forré, R.: Generation of binary sequences with controllable complexity and ideal r-tuple distribution. In: Price, W.L., Chaum, D. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 15–23. Springer, Heidelberg (1988)

    CrossRef  Google Scholar 

  24. Willett, M.: The minimum polynomial for a given solution of a linear recursion. Duke Math. J. 39, 101–104 (1972)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gammel, B.M., Göttfert, R. (2006). Linear Filtering of Nonlinear Shift-Register Sequences. In: Ytrehus, Ø. (eds) Coding and Cryptography. WCC 2005. Lecture Notes in Computer Science, vol 3969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779360_28

Download citation

  • DOI: https://doi.org/10.1007/11779360_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35481-9

  • Online ISBN: 978-3-540-35482-6

  • eBook Packages: Computer ScienceComputer Science (R0)