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Design sequences with high linear complexity over finite fields using generalized cyclotomy

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Abstract

Based on the generalized cyclotomy theory, we design some classes of sequences with high linear complexity over the finite fields. First, we construct a new class of sequence from some generalized cyclotomic sequences of different orders with different prime powers period. Then we obtain the discrete Fourier transform, defining pairs and the linear complexity of the new sequences. Finally, we study the linear complexity of a special class of q−ary (q prime) sequences.

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References

  1. Çesmelioglu, A., Meidl, W.: A general approach to construction and determination of the linear complexity of sequences based on cosets. Sequences and Their Applications 6338, 125–138 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Cusick, T.W., Ding, C., Renvall, A.: Stream Ciphers and Number Theory. North-Holland Publishing Co., Amsterdam (1998)

    MATH  Google Scholar 

  3. Ding, C., Helleseth, T.: New generalized cyclotomy and its applications. Finite Fields Appl. 4(2), 140–166 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dai, Z., Gong, G., Song, H.-Y.: Trace representation and linear complexity of binary e-th residue sequences, International Workshop on Coding and Cryptography. Versailles, France (2003)

    Google Scholar 

  5. Du, X., Chen, Z.: A generalization of the Hall’s sextic residue sequences. Inf. Sci. 222, 784–794 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, X., Chen, Z.: Trace representations of generalized cyclotomic sequences of length pq with arbitrary order. Chin. J. Electron. 18(3), 460–464 (2009)

    Google Scholar 

  7. Edemskiy, V.: About computation of the linear complexity of generalized cyclotomic sequences with period p n+1. Des. Codes Crypt. 61, 251–260 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan, C., Ge, G.: A unified approach to Whiteman’s and Ding-Helleseth’s generalized cyclotomy over residue class rings. IEEE Trans. Inf. Theory 60, 1326–1336 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Golomb, S.W., Gong, G.: Signal Design for Good Correlation: For Wireless Communications, Cryptography and Radar Applications. Cambridge University Press (2005)

  10. Green, D.H., Smith, M.D., Martzoukos, N.: Linear complexity of polyphase power residue sequences. IEE Proc Commun. 149(4), 195–201 (2002)

    Article  Google Scholar 

  11. Hu, L., Yue, Q., Wang, M.: The linear complexity of Whiteman’s generalized cyclotomic sequences of period p m+1 q n+1. IEEE Trans. Inf. Theory 58(8), 5534–5543 (2012)

    Article  MATH  Google Scholar 

  12. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  13. Kim, Y.J., Song, H.Y.: Linear complexity of prime n-square sequences. IEEE International Symposium on Information Theory, 2405–2408 (2008)

  14. Li, D., Chang, Z., Wen, Q.: Linear complexity of generalized cyclotomic quaternary sequences of length 2p m+1 q n+1. IET Inf. Secur. 7, 1751–8717 (2015)

    Google Scholar 

  15. Lidl, R., Niederreiter, H.: Finite Fields, Addison-Wesley, MA (1983)

  16. Liu, F., Peng, D., Tang, X.: On the autocorrelation and the linear complexity of q-ary prime n-square sequences. Sequences and Their Applications 6338, 139–150 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Wu, C., Chen, Z., Du, X.: The linear complexity of q-ary generalized cyclotomic sequences of period p m. J. Wuhan Univ. 59(2), 129–136 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Yan, T., Li, S., Xiao, G.: On the linear complexity of generalized cyclotomic sequences with the period p m. Appl. Math. Lett. 21, 187–193 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors acknowledge the patient referees for their valuable and constructive comments which helped to improve this work.

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Authors and Affiliations

  1. Novgorod State University, Str. B. St. Petersburgskaya, 41, 173003, Veliky Novgorod, Russia

    Vladimir Edemskiy

  2. College of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu, 730070, People’s Republic of China

    Xiaoni Du

Authors
  1. Vladimir Edemskiy
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  2. Xiaoni Du
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Corresponding author

Correspondence to Vladimir Edemskiy.

Additional information

V. Edemskiy was supported by the Ministry of Education and Science of Russia as a part of state-sponsored project No. 1.949.2014/K. X. Du was partially supported by the National Natural Science Foundation of China (grant No. 61462077) and the Program for New Century Excellent Talents in University (NCET-12-0620).

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Edemskiy, V., Du, X. Design sequences with high linear complexity over finite fields using generalized cyclotomy. Cryptogr. Commun. 9, 683–691 (2017). https://doi.org/10.1007/s12095-016-0209-2

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  • DOI: https://doi.org/10.1007/s12095-016-0209-2

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