Advertisement

Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 703–717 | Cite as

On the linear complexity of Legendre–Sidelnikov sequences

Article

Abstract

Linear complexity is an important cryptographic index of sequences. We study the linear complexity of \(p(q-1)\)-periodic Legendre–Sidelnikov sequences, which combine the concepts of Legendre sequences and Sidelnikov sequences. We get lower and upper bounds on the linear complexity in different cases, and experiments show that the upper bounds can be attained. Remarkably, we associate the linear complexity of Legendre–Sidelnikov sequences with some famous primes including safe prime and Fermat prime. If \(2\) is a primitive root modulo \(\frac{q-1}{2}\), and \(q\) is a safe prime greater than 7, the linear complexity is the period if \(p\equiv 3 \pmod 8\); \(p(q-1)-p+1\) if \(p\equiv q \equiv 7 \pmod 8\), and \(p(q-1)-\frac{p-1}{2}\) if \(p \equiv 7 \pmod 8, q \equiv 3 \pmod 8\). If \(q\) is a Fermat prime, the linear complexity is the period if \(p \equiv 3 \pmod 8\), and \(p(q-1)-q+2\) if \(p \equiv 5 \pmod 8\). It is very interesting that the Legendre–Sidelnikov sequence has maximal linear complexity and is balanced if we choose \(p=q\) to be some safe prime.

Keywords

Linear complexity Legendre–Sidelnikov sequence Legendre sequence Sidelnikov sequence Cryptography 

Mathematics Subject Classification

94A60 94A55 65C10 68P25 

Notes

Acknowledgments

The author would like to thank anonymous referees for valuable comments, and express his sincere thanks for hospitality during his visit to RICAM, Austrian Academy of Science, and some useful discussions with Arne Winterhof that led to this work. He is supported by National Natural Science Foundation of China (No. 61003070), and partially by NSFC (No. 61070014, 61373018).

References

  1. 1.
    Brandstatter N., Pirsic G., Winterhof A.: Correlation of the two-prime Sidel’nikov sequence. Des. Codes Cryptogr. 59, 59–68 (2011).Google Scholar
  2. 2.
    Cusick T.W., Ding C., Renvall A.: Stream Ciphers and Number Theory. North-Holland, Amsterdam (1998).Google Scholar
  3. 3.
    Ding C.: Linear complexity of generalized cyclotomic binary sequences of order 2. Finite Fields Appl. 3, 159–174 (1997).Google Scholar
  4. 4.
    Ding C.: Autocorrelation values of generalized cyclotomic sequences of order two. IEEE Trans. Inf. Theory 44(4), 1699–1702 (1998).Google Scholar
  5. 5.
    Ding C., Helleseth T., Shan W.: On the linear complexity of Legendre sequences. IEEE Trans. Inf. Theory 44(3), 1276–1278 (1998).Google Scholar
  6. 6.
    Granville A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. Organic mathematics (Burnaby BC, 1995). CMS Conference Proceedings, vol. 20, pp. 253–276. American Mathematical Society, Providence (1997).Google Scholar
  7. 7.
    Hasse H.: Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik. J. Reine Angew. Math. 175, 50–54 (1936).Google Scholar
  8. 8.
    Helleseth T., Yang K.: On binary sequences with period \(n = p^m-1\) with optimal autocorrelation. In: Helleseth T., Kumar P., Yang K. (eds.) SETA 2001. Lecture Notes in Computer Science, pp. 209–217. Springer, Berlin (2002).Google Scholar
  9. 9.
    Jungnickel D.: Finite Fields. BI-Wissenschaftsverlag, Mannheim (1993).Google Scholar
  10. 10.
    Kyureghyan G.M., Pott A.: On the linear complexity of the Sidelnikov–Lempel–Cohn–Eastman sequences. Des. Codes Cryptogr. 29, 149–164 (2003).Google Scholar
  11. 11.
    Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley, Reading (1983).Google Scholar
  12. 12.
    Lucas M.E.: Sur les congruences des nombres euleriennes et des coefficients differentiels des fuctions trigonometriques, suivant un-module premier. Bull. Soc. Math. France 6, 122–127 (1878).Google Scholar
  13. 13.
    Meidl W., Winterhof A.: Some notes on the linear complexity of Sidel’nikov–Lempel–Cohn–Eastman sequences. Des. Codes Cryptogr. 38(2), 159–178 (2006).Google Scholar
  14. 14.
    Sidel’nikov V.M.: Some \(k\)-valued pseudo-random sequences and nearly equidistant codes. Probl. Inf. Transm. 5(1), 12–16 (1969).Google Scholar
  15. 15.
    Storer T.: Cyclotomy and Difference Sets. Markham Publishing, Chicago (1967).Google Scholar
  16. 16.
    Su M.: On the \(d\)-ary generalized Legendre-Sidelnikov sequence. In: Helleseth T., Jedwab J. (eds.) Proceedings of Sequences and their Applications 2012, Waterloo, ON, Canada. Lecture Notes in Computer Science, pp. 233–244 (2012).Google Scholar
  17. 17.
    Su M., Winterhof A.: Autocorrelation of Legendre–Sidelnikov sequences. IEEE Trans. Inf. Theory 56, 1714–1718 (2010).Google Scholar
  18. 18.
    Su M., Winterhof A.: Correlation measure of order \(k\) and linear complexity profile of Legendre-Sidelnikov sequences. IEICE Trans. 95–A(11), 1851–1854 (2012).Google Scholar
  19. 19.
    Topuzoğlu A., Winterhof A.: Pseudorandom Sequences. Topics in Geometry, Coding Theory and Cryptography, vol. 6, pp. 135–166. Springer, Dordrecht (2007).Google Scholar
  20. 20.
    Winterhof A.: A note on the linear complexity profile of the discrete logarithm in finite fields. In: Feng K.Q., Niederreiter H., Xing C.P. (eds.) Coding, Cryptography and Combinatorics, pp. 359–368. Birkhäuser, Basel (2004).Google Scholar
  21. 21.
  22. 22.

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceNankai UniversityTianjinChina

Personalised recommendations