Cryptography and Communications

, Volume 9, Issue 6, pp 665–682 | Cite as

Character values of the Sidelnikov-Lempel-Cohn-Eastman sequences

Article
  • 291 Downloads

Abstract

Binary sequences with good autocorrelation properties and large linear complexity are useful in stream cipher cryptography. The Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences have nearly optimal autocorrelation. However, the problem of determining the linear complexity of the SLCE sequences is still open. It is well known that one can gain insight into the linear complexity of a sequence if one can say something about the divisors of the gcd of a certain pair of polynomials associated with the sequence. Helleseth and Yang (IEEE Trans. Inf. Theory 49(6), 1548–1552 2002), Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) and Meidl and Winterhof (Des. Codes Crypt. 8, 159–178 2006) were able to obtain some results of this type for the SLCE sequences. Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) mention that it would be nice to obtain more such results. We derive new divisibility results for the SLCE sequences in this paper. Our approach is to exploit the fact that character values associated with the SLCE sequences can be expressed in terms of a certain type of Jacobi sum. By making use of known evaluations of Gauss and Jacobi sums in the “pure” and “small index” cases, we are able to obtain new insight into the linear complexity of the SLCE sequences.

Keywords

Linear complexity Feedback shift registers Autocorrelation Stream cipher cryptography Difference sets Almost difference sets Jacobi sums Gauss sums 

Mathematics Subject Classification (2010)

05B10 94A55 11T23 11T71 11B50 

Notes

Acknowledgments

The authors wish to thank the anonymous referees for their valuable comments. The research of Şaban Alaca is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2015-05208). Goldwyn Millar’s studies are supported by an Ontario Graduate Scholarship.

References

  1. 1.
    Akiyama, S.: On the pure Jacobi sums. Acta Arith. 75(2), 97–104 (1996)MathSciNetMATHGoogle Scholar
  2. 2.
    Alaca, S., Williams, K.: Introductory Algebraic Number Theory. Cambridge (2004)Google Scholar
  3. 3.
    Aly, H., Meidl, W.: On the linear complexity and k-error linear complexity over 𝔽p of the d-ary Sidelnikov sequence. IEEE Trans. Inform. Theory 53(12), 4755–4761 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Aly, H.,Winterhof, A.: On the k-error linear complexity over 𝔽p of Legendre and Sidelnikov sequences. Des. Codes Crypt. 40(3), 369–374 (2006)Google Scholar
  5. 5.
    Arasu, K.T., Ding, C., Helleseth, T., Kumar, V., Martinsen, H.M.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inform. Theory 47(7), 2934–2943 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Berndt, B.C., Evans, R.J.: Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer. Ill. J. Math. 23(3), 374–437 (1979)MathSciNetMATHGoogle Scholar
  7. 7.
    Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. A Wiley-Interscience Publication (1998)Google Scholar
  8. 8.
    Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn, vol. 1. Cambridge (1999)Google Scholar
  9. 9.
    Brandstätter, N., Meidl, W.: On the Linear Complexity of Sidelnikov Sequences over 𝔽d, Sequences and their Applications - SETA 2006, 47 - 60, Lecture Notes in Comput. Sci., vol. 4086. Springer, Berlin (2006)Google Scholar
  10. 10.
    Brandstätter, N., Meidl, W.: On the linear complexity of Sidelnikov sequences over nonprime fields. J. Complexity 24(5–6), 648–659 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Brandstätter, N., Meidl, W., Winterhof, A.: Addendum to Sidel’nikov sequences over nonprime fields. Inf. Process. Lett. 113, 332–336 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Brandstätter, N., Winterhof, A.: k-error linear complexity over 𝔽p of subsequences of Sidelnikov sequences of period (p r − 1)/3. J. Math. Cryptol. 3(3), 215–225 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chung, J.H., Yang, K.: Bounds on the Linear Complexity and the 1-Error Linear Complexity over 𝔽p of M-ary Sidelnikov Sequences, Sequences and their Applications - SETA 2006, 74 - 87, Lecture Notes in Comput. Sci., vol. 4086. Springer, Berlin (2006)Google Scholar
  14. 14.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin (1993)CrossRefMATHGoogle Scholar
  15. 15.
    Evans, R., Hollmann, H.D.L., Krattenthaler, C., Xiang, Q.: Gauss sums, Jacobi Sums, and p-ranks of cyclic difference sets. J. Comb. Theory Ser. A 87, 74–119 (1999)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Feng, T., Xiang, Q.: Cyclotomic constructions of skew Hadamard difference sets. J. Comb. Theory Ser. A 119, 245–256 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Garaev, M.Z., Luca, F., Shparlinski, I.E., Winterhof, A.: On the lower bound of the linear complexity over 𝔽p of Sidelnikov sequences. IEEE Trans. Inform. Theory 52(7), 3299–3304 (2006)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Golomb, S., Gong, G.: Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge (2005)Google Scholar
  19. 19.
    Helleseth, T., Kim, S.H., No, J.S.: Linear complexity over 𝔽p and trace representation of Lempel-Cohn-Eastman sequences. IEEE Trans. Inform. Theory 49(6), 1548–1552 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Helleseth, T., Maas, M., Mathiassen, J.E., Segers, T.: Linear complexity over 𝔽p of Sidel’nikov sequences. IEEE Trans. Inform. Theory 50(10), 2468–2472 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Helleseth, T., Yang, K.: On binary sequences of period n = p m − 1 with optimal autocorrelation. In: Helleseth, T., Kumar, P., Yang, K. (eds.) Proceedings of SETA01, pp 209–217 (2002)Google Scholar
  22. 22.
    Hardy, K., Muskat, J.B., Williams, K.S.: A deterministic algorithm for solving n = f u 2 + g v 2 in coprime integers u and v. Math. Comp. 91(55), 327–343 (1990)MathSciNetMATHGoogle Scholar
  23. 23.
    Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Springer-Verlag (1990)Google Scholar
  24. 24.
    Kim, Y.S., Chung, J.S., No, J.S., Chung, H.: Linear complexity over 𝔽p of ternary Sidelnikov sequences. Sequences and their applications - SETA 2006, 61–73, Lecture Notes in Comput. Sci., vol. 4086. Springer, Berlin (2006)Google Scholar
  25. 25.
    Kyureghyan, G., Pott, A.: On the linear complexity of the Sidelnikov-Lempel-Cohn-Eastman sequences. Des. Codes Crypt. 29, 149–164 (2003)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Langevin, P.: Calculs de Certaines Sommes de Gauss. J. Number Theory 63, 59–64 (1997)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lempel, A., Cohn, M., Eastman, W.L.: A class of binary sequences with optimal autocorrelation properties. IEEE Trans. Inform. Theory IT-23, 38–42 (1977)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Leung, K.H., Schmidt, B.: The field descent method. Des. Codes Crypt. 171–188 (2005)Google Scholar
  29. 29.
    Ma, S.L.: A survey of partial difference sets. Des. Codes Crypt. 221–261 (1994)Google Scholar
  30. 30.
    MacWilliams, J., Mann, H.B.: On the p-rank of the design matrix of a difference set. Inform. Control 12, 474–488 (1968)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mann, H.B.: Introduction to Algebraic Number Theory. Ohio State Press, Columbus (1955)MATHGoogle Scholar
  32. 32.
    Meidl, W., Winterhof, A.: Some notes on the linear complexity of Sidel’nikov-Lempel-Cohn-Eastman sequences. Des. Codes Crypt. 8, 159–178 (2006)CrossRefMATHGoogle Scholar
  33. 33.
    Shiratani, K., Yamada, M.: On Rationality of Jacobi Sums. Colloq. Math. 73 (2), 251–260 (1997)MathSciNetMATHGoogle Scholar
  34. 34.
    Sidelnikov, V.M.: Some k-valued pseudo-random sequences and nearly equidistant codes. Probl. Inform. Trans. 5(1), 12–16 (1969)MathSciNetGoogle Scholar
  35. 35.
    Xia, L., Yang, J.: Complete solving of explicit evaluation of Gauss sums in the index 2 case. Sci. China Math. 53(9), 2525–2542 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

Personalised recommendations