Word-Oriented Transformation Shift Registers and Their Linear Complexity

  • Sartaj Ul Hasan
  • Daniel Panario
  • Qiang Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)

Abstract

We discuss the problem of counting the number of primitive transformation shift registers and its equivalent formulation in terms of Singer cycles in a corresponding general linear group. We also introduce the notion of word-oriented nonlinearly filtered primitive transformation shift registers based on a Langford arrangement and study their linear complexity.

Keywords

Linear feedback shift register Singer cycle transformation shift register linear complexity Langford arrangement nonlinearly filtered primitive transformation shift register 

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References

  1. 1.
    Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs, Discrete Mathematics and its Applications, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007)Google Scholar
  2. 2.
    Dewar, M., Panario, D.: Linear transformation shift registers. IEEE Trans. Inform. Theory 49(8), 2047–2052 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dewar, M., Panario, D.: Mutual Irreducibility of Certain Polynomials. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) Fq7 2003. LNCS, vol. 2948, pp. 59–68. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Dewar, M., Panario, D.: Tables for linear transformation shift registers, http://www.math.carleton.ca/~daniel/research/tsr/
  5. 5.
    Gao, F., Yang, Y., Tan, G.: Some results on word-oriented nonlinear feedback shift registers. In: The Proceedings of International Conference on Electronics and Optoelectronics (ICEOE 2011), pp. V4-357–V4-359 (2011)Google Scholar
  6. 6.
    Ghorpade, S.R., Hasan, S.U., Kumari, M.: Primitive polynomials, Singer cycles, and word oriented linear feedback shift registers. Des. Codes Cryptogr. 58(2), 123–134 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ghorpade, S.R., Ram, S.: Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields. Finite Fields Appl. 17(5), 461–472 (2011)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ghorpade, S.R., Ram, S.: Enumeration of splitting subspaces over finite fields. To Appear in: Aubry, Y., Ritzenthaler, C., Zykin, A. (eds.) Arithmetic, Geometry, Cryptography and Coding Theory, Luminy, France. Contemp. Math. Amer. Math. Society (2011)Google Scholar
  9. 9.
    Golomb, S.W.: Shift Register Sequences. Holden-Day, San Francisco (1967)Google Scholar
  10. 10.
    Golomb, S.W., Gong, G.: Signal Design for Good Correlation. Cambridge University Press (2005)Google Scholar
  11. 11.
    Groth, E.J.: Generation of binary sequences with controllable complexity. IEEE Trans. Inform. Theory 17, 288–296 (1971)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Key, E.L.: An analysis of the structure and complexity of nonlinear binary sequence generators. IEEE Trans. Inform. Theory 6, 732–736 (1976)CrossRefGoogle Scholar
  13. 13.
    Langford, C.D.: Problem. Math. Gaz. 42, 228 (1958)Google Scholar
  14. 14.
    Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1983)MATHGoogle Scholar
  15. 15.
    Niederreiter, H.: Factorization of polynomials and some linear-algebra problems over finite fields. Linear Algebra Appl. 192, 301–328 (1993)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Niederreiter, H.: The multiple-recursive matrix method for pseudorandom number generation. Finite Fields Appl. 1, 3–30 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Niederreiter, H.: Pseudorandom vector generation by the multiple-recursive matrix method. Math. Comp. 64, 279–294 (1995)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Niederreiter, H.: Improved bound in the multiple-recursive matrix method for pseudorandom number and vector generation. Finite Fields Appl. 2, 225–240 (1996)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Preneel, B.: Introduction to the Proceedings of the Second Workshop on Fast Software Encryption. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 1–5. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  20. 20.
    Tsaban, B., Vishne, U.: Efficient feedback shift registers with maximal period. Finite Fields Appl. 8, 256–267 (2002)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Zeng, G., Han, W., He, K.: Word-oriented feedback shift register: σ-LFSR (Cryptology ePrint Archive: Report 2007/114), http://eprint.iacr.org/2007/114
  22. 22.
    Zeng, G., Yang, Y., Han, W., Fan, S.: Word Oriented Cascade Jump σ−LFSR. In: Bras-Amorós, M., Høholdt, T. (eds.) AAECC 2009. LNCS, vol. 5527, pp. 127–136. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sartaj Ul Hasan
    • 1
  • Daniel Panario
    • 1
  • Qiang Wang
    • 1
  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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