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A lower bound on the 2-adic complexity of the modified Jacobi sequence

  • Yuhua Sun
  • Qiang Wang
  • Tongjiang Yan
Article
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Abstract

Let p, q be distinct primes satisfying gcd(p −  1, q −  1) = d and let D i , i =  0, 1, · · · ,d −  1, be Whiteman’s generalized cyclotomic classes with \(\mathbb {Z}_{pq}^{\ast }=\cup _{i = 0}^{d-1}D_{i}\). In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets \(D_{0}^{\ast }=\cup _{i = 0}^{\frac {d}{2}-1}D_{2i}\) and \(D_{1}^{\ast }=\cup _{i = 0}^{\frac {d}{2}-1}D_{2i + 1}\). As an application, we determine a lower bound on the 2-adic complexity of the modified Jacobi sequence. Our result shows that the 2-adic complexity of the modified Jacobi sequence is at least pqpq − 1 with period N = pq. This indicates that the 2-adic complexity of the modified Jacobi sequence is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).

Keywords

Gauss period Generalized cyclotomic class Modified Jacobi sequence 2-adic complexity 

Mathematics Subject Classification (2010)

11B50 94A55 94A60 

Notes

Acknowledgements

Parts of this work were done during a very pleasant visit of the first author to the School of Mathematics and Statistics at Carleton University. She wishes to thank the hosts for their hospitality. We also thank anonymous referees for their helpful suggestions.

References

  1. 1.
    Bai, E., Liu, X., Xiao, G.: Linear complexity of new generalized cyclotomic sequences of order two of length pq. IEEE Trans. Inform. Theory 51, 1849–1853 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cai, H., Liang, H., Tang, X.: Constructions of optimal 2-D optical orthogonal codes via generalized cyclotomic classes. IEEE Trans. Inform. Theory 61, 688–695 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Z., Du, X., Xiao, G.: Sequences related to Legendre/Jacobi sequences. Inf. Sci. 177(21), 4820–4831 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cusick, T.W., Ding, C, Renvall, A.: Stream ciphers and number theory. Elsevier, Amsterdam (2015)zbMATHGoogle Scholar
  5. 5.
    Davis, P.J.: Circulant matrices. Chelsea, New York (1994)zbMATHGoogle Scholar
  6. 6.
    Ding, C., Xing, C.: Several classes of (2m − 1,w, 2) optical orthogonal codes. Discret. Appl. Math. 128, 103–120 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ding, C., Xing, C.: Cyclotomic optical orthogonal codes of composite lengths. IEEE Trans. Inform. Theory 52, 263–268 (2004)Google Scholar
  8. 8.
    Ding, C.: Cyclotomic constructions of cyclic codes with length being the product of two primes. IEEE Trans. Inform. Theory 58, 2231–2236 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ding, C.: Cyclic codes from the two-prime sequences. IEEE Trans. Inform. Theory 58, 3881–3891 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding, C., Helleseth, T.: On the linear complexity of Legendre sequences. IEEE Trans. Inform. Theory 44, 1693–1698 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Green, D.H., Green, P.R.: Modified Jacobi sequences. IEEE Proceedings-Computers and Digital Techniques 147(4), 241–251 (2000)CrossRefGoogle Scholar
  13. 13.
    Green, D.H., Choi, J.: Linear complexity of modified Jacobi sequences. IEE Proceedings-Computers and Digital Techniques 149(3), 97–101 (2002)CrossRefGoogle Scholar
  14. 14.
    Hu, L., Yue, Q.: Gauss periods and codebooks from generalized cyclotomic sets of order four. Des. Codes Crypt. 69, 233–246 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, X., Ma, W., Yan, T., Zhao, X.: Linear complexity of a new generalized cyclotomic sequence of order two of length pq. IEICE Trans. 96-A, 1001–1005 (2013)CrossRefGoogle Scholar
  16. 16.
    Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inform. Theory 15, 122–127 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klapper, A., Goresky, M.: Feedback shift registers, 2-adic span, and combiners with memory. J. Cryptol. 10, 111–147 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tang, X., Ding, C.: New classes of balanced quaternary and almost balanced binary sequences with optimal autocorrelation Value. IEEE Trans. Inform. Theory 56, 6398–6405 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tang, X., Fan, P., Matsufuji, S.: Lower bounds on the maximum correlation of sequences with low or zero correlation zone. Electron. Lett. 36, 551–552 (2000)CrossRefGoogle Scholar
  20. 20.
    Tian, T., Qi, W.: 2-Adic complexity of binary m-sequences. IEEE Trans. Inform. Theory 56, 450–454 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Whiteman, A.L.: A family of difference sets. Ill. J. Math. 6, 107–121 (1962)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Xiong, H., Qu, L., Li, C.: A new method to compute the 2-adic complexity of binary sequences. IEEE Trans. Inform. Theory 60, 2399–2406 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xiong, H., Qu, L., Li, C.: 2-Adic complexity of binary sequences with interleaved structure. Finite Fields Appl. 33, 14–28 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yan, T.: Study on constructions and properties of pseudo-random sequence. Ph. D Thesis (2007)Google Scholar
  25. 25.
    Xiong, T., Hall, J.I.: Modifications of modified Jacobi sequences. IEEE Trans. Inform. Theory 57, 493–504 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yan, T., Du, X., Xiao, G., Huang, X.: Linear complexity of binary Whiteman generalized cyclotomic sequences of order 2k. Inf. Sci. 179, 1019–1023 (2009)CrossRefzbMATHGoogle Scholar
  27. 27.
    Zeng, X., Cai, H., Tang, X., Yang, Y.: Optimal frequency sequences of odd length. IEEE Trans. Inform. Theory 59, 3237–3248 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Xiao, Z., Zeng, X., Sun, Z.: 2-Adic complexity of two classes of generalized cyclotomic binary sequences. Int. J. Found. Comput. Sci. 27, 879–893 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhou, Z., Tang, X., Gong, G.: A new classes of sequences with zero or low correlation zone based on interleaving technique. IEEE Trans. Inform. Theory 54, 4267–4273 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of SciencesChina University of PetroleumQingdaoChina
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  3. 3.Qilu University of Technology (Shandong Academy of Sciences), Shandong Computer Science Center (National Supercomputer Center in Jinan, Shandong Provincial Key Laboratory of Computer NetworksJinanChina
  4. 4.Key Laboratory of Applied MathematicsFujian Province University (Putian University)PutianChina

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