Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1023–1038 | Cite as

Solving the FCSR synthesis problem for multi-sequences by lattice basis reduction

  • Weihua Liu
  • Andrew Klapper
  • Zhixiong Chen


Register synthesis for multi-sequences has significance for the security of word-oriented stream ciphers. Feedback with carry shift registers (FCSRs) are promising alternatives to linear feedback shift registers for the design of stream ciphers. In this paper, we solve the FCSR synthesis problem for multi-sequences by two rational approximation algorithms using lattice theory. One is based on the lattice reduction greedy algorithm proposed by Nguyen and Stehlé (ACM Trans Algorithms (TALG) 5(4):46, 2009). The other is based on the LLL algorithm which is a polynomial time lattice reduction algorithm. Both of these rational approximation algorithms can find the smallest common FCSR for a given multi-sequence but with different numbers of known terms. When the number of sequences within the multi-sequence is less than or equal to 3, the former is suggested because it has better time complexity and fewer terms are needed. Otherwise, the latter will have better time complexity.


Multi-sequences Lattice basis reduction algorithm FCSR synthesis problem 

Mathematics Subject Classification




This material is based upon work supported by the National Science Foundation under Grant No. CNS-1420227. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Zhixiong Chen was partially supported by the National Natural Science Foundation of China under Grant No. 61373140 and China Scholarship Council.


  1. 1.
    Ajtai M.: The shortest vector problem in \( L^2\) is NP-hard for randomized reductions. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing STOC 98, pp. 10–19. ACM, New York (1998)Google Scholar
  2. 2.
    Arnault F., Berger T.P., Necer A.: Feedback with carry shift registers synthesis with the Euclidean algorithm. IEEE Trans. Inf. Theory 50(5), 910–917 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dwork C.: Lattices and their application to cryptography. Stanford University, Lecture Notes (1998).Google Scholar
  4. 4.
    Feng G., Tzeng K.: A generalized Euclidean algorithm for multisequence shift-register synthesis. IEEE Trans. Inf. Theory 35(3), 584–594 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goresky M., Klapper A.: Algebraic Shift Register Sequences. Cambridge University Press, Cambridge (2012).zbMATHGoogle Scholar
  6. 6.
    Hu H., Hu L., Feng D.: On the expected value of the joint 2-adic complexity of periodic binary multisequences. In: Gong G., Helleseth T., Song H., Yang K. (eds.) Sequences and Their Applications—SETA 2006, pp. 199–208. Springer, Berlin (2006).CrossRefGoogle Scholar
  7. 7.
    Klapper A., Goresky M.: Cryptanalysis based on 2-adic rational approximation. In: Coppersmith D. (ed.) Advances in Cryptology—CRYPTO’95, pp. 262–273. Springer, Berlin (1995).Google Scholar
  8. 8.
    Klapper A., Goresky M.: Feedback shift registers, 2-adic span, and combiners with memory. J. Cryptol. 10(2), 111–147 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Klapper A., Xu J.: Register synthesis for algebraic feedback shift registers based on non-primes. Des. Codes Cryptogr. 31(3), 227–250 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lenstra A., Lenstra H., Lovász L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Massey J.L.: Shift register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15(1), 122–127 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nguyen P.Q., Stehlé D.: An LLL algorithm with quadratic complexity. SIAM J. Comput. 39(3), 874–903 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nguyen Phong Q., Stehlé D.: Low-dimensional lattice basis reduction revisited. ACM Trans. Algorithms (TALG) 5(4), 46 (2009).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Sakata S.: Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array. J. Symb. Comput. 5(3), 321–337 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schmidt G., Sidorenko V.R.: Multi-sequence linear shift-register synthesis: the varying length case. In: 2006 IEEE International Symposium on Information Theory, pp. 1738–1742 (2006).Google Scholar
  16. 16.
    Wang L., Zhu Y.: \(f [x]\)-lattice basis reduction algorithm and multisequence synthesis. Sci. China Ser. Inf. Sci. 44(5), 321–328 (2001).MathSciNetzbMATHGoogle Scholar
  17. 17.
    Yang M., Lin D., Xuan G.: Generalized Fourier transform and the joint \(N\)-adic complexity of a multisequence. IEICE Trans. Fundam. Electron. Comput. Sci. E97.A(9), 1982–1986 (2014).Google Scholar
  18. 18.
    Zhao L., Wen Q.: On the joint 2-adic complexity of binary multisequences. RAIRO Theor. Inf. Appl. 46(03), 401–412 (2012).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceThe William Paterson University of New JerseyWayneUSA
  2. 2.Department of Computer ScienceUniversity of KentuckyLexingtonUSA
  3. 3.Provincal Key Laboratory of Applied MathematicsPutian UniversityPutianChina

Personalised recommendations