A new algorithm on the minimal rational fraction representation of feedback with carry shift registers

  • Yubo Li
  • Zhichao Yang
  • Kangquan Li
  • Longjiang QuEmail author


In 1994, Klapper and Goresky (Proceedings of the 1993 Cambridge Security Workshop, Lecture Notes in Computer Science, vol 809, Cambridge, pp 174–178, 1994) proposed a new device called feedback with carry shift register to generate pseudo-random sequences instead of using the traditional device linear feedback shift register. They raised an algorithm called as rational approximation algorithm to recover the device for a given sequence (Klapper and Goresky, Advances in Cryptology, Crypto’95, Lecture Notes in Computer Science, vol 963, Springer, Berlin, pp 262–274, 1995). In this paper, we propose a new algorithm by introducing a new parameter and get the best rational approximation of the sequence much more quickly, especially when the size of the sequence increases dramatically. Unlike most of known algorithms, we can solve the minimal lattice basis instead of one shortest vector. Besides, we can prove that the solution of each step is optimal regardless of the length of the input sequence theoretically.


Sequence FCSR Lattice Rational approximation Rational fraction representation 

Mathematics Subject Classification

94A55 94A60 03G10 11P21 



  1. 1.
    Arnault F., Berger T.P., Necer A.: Feedback with carry shift registers synthesis with the Euclidean algorithm. IEEE Trans. Inf. Theor. 50(5), 910–917 (2004).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Conway J.H., Sloane N.J.A.: Sphere Packings, Lattice and Groups, 2nd edn. Springer, New York (1993). Scholar
  3. 3.
    de Weger B.M.M.: Approximation lattices of \(p\)-adic numbers. J. Num. Theory 24, 70–88 (1986).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Goresky M., Klapper A.: Feedback Registers Based on Ramified Extensions of the 2-adic Numbers, vol. 950, pp. 215–222. Advances in Cryptology-Eurocrypt’94, LNCSSpringer, Berlin (1995).zbMATHGoogle Scholar
  5. 5.
    Goresky M., Klapper A.: Large Periods Nearly de Bruijn FCSR Sequences, vol. 921, pp. 263–273. Advances in Cryptology-Eurocrypt’95, LNCSSpringer, Berlin (1995).zbMATHGoogle Scholar
  6. 6.
    Goresky M., Klapper A.: Algebraic Shift Register Sequences. Cambridge University Press, Cambridge (2009).zbMATHGoogle Scholar
  7. 7.
    Klapper A., Goresky M.: 2-adic shift registers, fast software encryption. In: Proceedings of the 1993 Cambridge Security Workshop, Lecture Notes in Computer Science, vol. 809, Cambridge, pp. 174–178 (1994).Google Scholar
  8. 8.
    Klapper, A., Goresky, M.: Cryptanalysis based on 2-adic rational approximation. In: Advances in Cryptology, Crypto’95, Lecture Notes in Computer Science, vol. 963, Springer, Berlin, pp. 262–274 (1995).CrossRefGoogle Scholar
  9. 9.
    Klapper A., Goresky M.: Feedback shift registers, 2-adic span, and combiners with memory. J. Cryptol. 10, 11–147 (1997).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Klapper A., Xu J.: Algebraic feedback shift registers. Theor. Comput. Sci. 226(1), 61–92 (1999).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Klapper A., Xu J.: Register synthesis for algebraic feedback shift registers based on non primes. Preprint (2002).Google Scholar
  12. 12.
    Liu W., Klapper A.: A lattice rational approximation algorithm for AFSRs over quadratic integer rings. In: SETA 2014, LNCS 8865, pp. 200–211 (2014).Google Scholar
  13. 13.
    Mahler K.: On a geometrical representation of \(p\)-adic numbers. Ann. Math. 41, 8–56 (1940).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Massey J.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory IT–15, 122–127 (1969).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Meidl W.: Extended Games–Chan algorithm for the 2-adic complexity of FCSR-sequences. Theor. Comput. Sci. 290, 2045–2051 (2003).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Schönhage A., Strassen V.: Schnelle multiplikation grosser zahlen. Computing 7, 281–292 (1971).MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.College of Liberal Arts and SciencesNational University of Defense TechnologyChangshaChina
  2. 2.College of ComputerNational University of Defense TechnologyChangshaChina
  3. 3.State Key Laboratory of CryptologyBeijingChina

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