Skip to main content
SpringerLink
Log in

Constructing sequences with high nonlinear complexity using the Weierstrass semigroup of a pair of distinct points of a Hermitian curve

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

Using the Weierstrass semigroup of a pair of distinct points of a Hermitian curve over a finite field, we construct sequences with improved high nonlinear complexity. In particular we improve the bound obtained in Niederreiter and Xing (IEEE Trans Inf Theory 60(10):6696–6701, 2014, Theorem 3) considerably and the bound in Niederreiter and Xing (2014, Theorem 4) for some parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Carvallo, C., Kato, T.: On Weierstrass semigroups and sets: review of new results. Geom. Dedic. 239, 195–210 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carvallo, C., Torres, F.: On Goppa codes and Weierstrass points at several points. Des. Codes. Cryptogr. 35, 211–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Castellanos, C., Tizziotti, G.: On Weierstrass semigroup at m points on curves of the form f(y) = g(x). J. Pure Appl. Algebra 222, 1803–1809 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding, C., Xiao, G., Shan, W.: The Stability Theory of Stream Ciphers, Lecture Notes in Computer Science, 561. Springer, Berlin, x+187 pp. ISBN: 3-540-54973-0 (1991)

  5. Garcia, A., Kim, S.J., Lax, R.F.: Consecutive Weierstrass gaps and minimum distance of Goppa codes. J. Pure Appl. Algebra 84(2), 199–207 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Garcia, A., Stichtenoth, H., Xing, C.-P.: On subfields of the Hermitian function field. Compos. Math. 120(2), 137–170 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goppa, V.D.: Codes associated with divisors. Prob. Pereda. Inf. 13, 33–39 (1977)

    MathSciNet  MATH  Google Scholar 

  8. Goppa, V.D.: Codes on algebraic curves. Sov. Math. Dokl. 24, 75–91 (1981)

    MathSciNet  MATH  Google Scholar 

  9. Matthews, G.L.: Weierstrass semigroups and codes from a quotient of the Hermitian curve. Des. Codes Cryptogr. 37(3), 473–492 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Meidl, W., Winterhof, A.: Linear complexity of sequences and multisequences. In: Handbook of Finite Fields, Chapman and Hall/CRC, 1068 pp. ISBN: 9781439873786 (2013)

  11. Meidl, W., Niederreiter, H.: Multisequences with high joint nonlinear complexity. Des. Codes Cryptogr. 81(2), 337–346 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. In: CBMS-NSF Regional Conference Series in Applied Mathematics, 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, vi+241 pp. ISBN: 0-89871-295-5 (1992)

  13. Niederreiter, H., Xing, C.: Rational Points on Curves over Finite Fields. Cambridge University Press, Cambridge, x+245 pp. ISBN: 0-521-66543-4 (2001)

  14. Niederreiter, H., Xing, C.: Algebraic geometry in coding theory and cryptography. Cambridge University Press, Princeton University Press, Princeton, NJ, xii+260 pp. ISBN: 978-0-691-10288-7 (2009)

  15. Niederreiter, H., Xing, C.: Sequences with high nonlinear complexity. IEEE Trans. Inf. Theory 60(10), 6696–6701 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rueppel, R. A.: Stream Ciphers. Contemporary Cryptology. IEEE, New York, pp. 65–134 (1992)

  17. Stichtenoth, H.: Algebraic function fields and codes, second edition. Graduate Texts in Mathematics, 254. Springer, Berlin, xiv+355 pp. ISBN: 978-3-540-76877-7 (2009)

  18. Yang, S., Hu, C.: Pure Weierstrass gaps from a quotient of the Hermitian curve. Finite Fields Appl. 50, 251–271 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Department of Mathematical Sciences, Aalborg University for supporting a one month visiting professor position for the second listed author. This work was supported by The Danish Council for Independent Research (Grant No. DFF–4002-00367), by the Spanish MINECO/FEDER (Grant No. MTM2015-65764-C3-2-P, MTM2015-69138-REDT and RYC-2016-20208 (AEI/FSE/UE)) and by METU Coordinatorship of Scientific Research Projects via grant for projects BAP-01-01-2016-008 and BAP-07-05-2017-007.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark

    Olav Geil, Ferruh Özbudak & Diego Ruano

  2. Department of Mathematics, Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak

  3. IMUVA (Mathematics Research Institute), University of Valladolid, Valladolid, Spain

    Diego Ruano

Authors
  1. Olav Geil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ferruh Özbudak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Diego Ruano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ferruh Özbudak.

Additional information

Communicated by Fernando Torres.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Geil, O., Özbudak, F. & Ruano, D. Constructing sequences with high nonlinear complexity using the Weierstrass semigroup of a pair of distinct points of a Hermitian curve. Semigroup Forum 98, 543–555 (2019). https://doi.org/10.1007/s00233-018-9976-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-018-9976-8

Keywords

Search

Navigation