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A Survey of Feedback with Carry Shift Registers

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Sequences and Their Applications - SETA 2004 (SETA 2004)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3486))

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Abstract

Feedback with carry shift registers (FCSRs) are arithmetic analogs of linear feedback shift registers (LFSRs). In this paper we survey some of the basic properties of FCSRs. For comparison, we first review some basic facts about LFSRs. We then define FCSRs and discuss their relation to the N-adic numbers. This leads to the analysis of periodicity of FCSR sequences, their exponential representation, and a description of maximal period FCSR sequences. We also discuss an arithmetic analog of cross-correlations, the FCSR register synthesis problem, and how FCSRs can be efficiently implemented in parallel architecture.

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References

  1. Arnault, F., Berger, T., Necer, A.: A new class of stream ciphers combining LFSR and FCSR architectures. In: Menezes, A., Sarkar, P. (eds.) INDOCRYPT 2002. LNCS, vol. 2551, pp. 22–33. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  2. Arnault, F., Berger, T., Necer, A.: Feedback with carry shift registers synthesis with the Euclidean algorithm. IEEE Trans. Info. Thy. 50, 910–917 (2004)

    Article  MathSciNet  Google Scholar 

  3. Chan, A., Games, R.: On the quadratic span of de Bruijn sequences. IEEE Trans. Info. Thy. 36, 822–829 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Couture, R., L’Ecuyer, P.: On the lattice structure of certain linear congruential sequences related to AWC/SWB generators. Math. Comp. 62, 799–808 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Goresky, M., Klapper, A.: Fibonacci and Galois Mode Implementation of Feedback with Carry Shift Registers. IEEE Trans. Info. Thy. 48, 2826–2836 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Goresky, M., Klapper, A., Murty, R., Shparlinski, I.: On Decimations of ℓ-sequences. SIAM J. Disc. Math 18, 130–140 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Goresky, M., Klapper, A.: Polynomial pseudo-noise sequences based on algebraic feedback shift registers, under review

    Google Scholar 

  8. Hooley, C.: On Artin’s conjecture. J. Reine Angew. Math. 22, 209–220 (1967)

    Article  MathSciNet  Google Scholar 

  9. Klapper, A.: Distributional properties of d-FCSR sequences. J. Complexity 20, 305–317 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Klapper, A., Goresky, M.: Feedback Shift Registers, Combiners with Memory, and 2-Adic Span. J. Cryptology 10, 111–147 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Klapper, A., Goresky, M.: Arithmetic Cross-Correlations of FCSR Sequences. IEEE Trans. Info. Thy. 43, 1342–1346 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Klapper, A., Xu, J.: Algebraic feedback shift registers. Theoretical Comp. Sci. 226, 61–93 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Klapper, A., Xu, J.: Register synthesis for algebraic feedback shift registers based on non-primes. Designs, Codes, and Crypt. 31, 227–255 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lidl, R., Niederreiter, H.: Finite Fields. Encycl. Math. Appl. 20 (1983)

    Google Scholar 

  15. Mahler, K.: On a geometrical representation of p-adic numbers. Ann. Math. 41, 8–56 (1940)

    Article  MathSciNet  Google Scholar 

  16. Marsaglia, G., Zaman, A.: A new class of random number generators. Annals of Appl. Prob. 1, 462–480 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Massey, J.: Shift-register synthesis and BCH decoding. IEEE Trans. Info. Thy. IT 15, 122–127 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  18. Massey, J., Rueppel, R.: Method of, and Apparatus for, Transforming a Digital Data Sequence into an Encoded Form. U.S. Patent 4, 797, 922 (1989)

    Google Scholar 

  19. Meier, W., Staffelbach, O.: Correlation properties of combiners with memory in stream ciphers. J. Cryptology 5, 67–86 (1992)

    MATH  MathSciNet  Google Scholar 

  20. Reeds, J., Sloane, N.: Shift-register synthesis (modulo m). SIAM J. Comp. 14, 505–513 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  22. de Weger, B.M.: Approximation lattices of p-adic numbers. J. Num. Th. 24, 70–88 (1986)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Kentucky, 779 A Anderson Hall, Lexington, KY, 40506-0046, USA

    Andrew Klapper

Authors
  1. Andrew Klapper
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, PB 7803, 5020, Bergen, Norway

    Tor Helleseth

  2. University of Ilinois at Urbana-Champaign, 1406 West Green Street, IL 61801, Urbana, USA

    Dilip Sarwate

  3. Department of Electrical and Electronic Engineering, Yonsei University, 121-749, Seoul, Korea

    Hong-Yeop Song

  4. Dept. of Electronics and Electrical Engineering, Pohang University of Science and Technology (POSTECH), 790-784, Pohang, Kyungbuk, Korea

    Kyeongcheol Yang

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© 2005 Springer-Verlag Berlin Heidelberg

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Klapper, A. (2005). A Survey of Feedback with Carry Shift Registers. In: Helleseth, T., Sarwate, D., Song, HY., Yang, K. (eds) Sequences and Their Applications - SETA 2004. SETA 2004. Lecture Notes in Computer Science, vol 3486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11423461_3

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  • DOI: https://doi.org/10.1007/11423461_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26084-4

  • Online ISBN: 978-3-540-32048-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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