Constructions of Sequences from Algebraic Curves over Finite Fields

  • Chaoping Xing
Part of the Discrete Mathematics and Theoretical Computer Science book series (DISCMATH)


We survey some recent constructions of various sequences based on algebraic curves over finite fields. The sequences constructed in this paper include sequences with good linear complexity profiles and sequence families with both large linear complexities and low correlation.


Elliptic Curve Finite Field Function Field Binary Sequence Linear Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    M. Eichler, Introduction to the Theory of Algebraic Numbers and Functions Academic Press, New York, 1951.Google Scholar
  2. 2.
    T. Heileseth and PV Kumar Sequences with Low Correlation a chapter in: Handbook of Coding Theory edited by V Pless and C Huffman Elsevier Science Publishers, 1998.Google Scholar
  3. 3.
    D. Kohel, S Ling, and CP Xing Explicit sequence expansions In: Sequences and Their Applications (C Ding, T Heileseth, and H Niederreiter, eds.), pp. 308–317, Springer London 1999.CrossRefGoogle Scholar
  4. 4.
    H. Niederreiter Continued fractions for formal power series, numbers7 and linear complexity of sequences In: Contributions to General Algebra 5 (Proc. Salzburg Conf 1986), pp. 221–233, Teubner, Stuttgart 1987.Google Scholar
  5. 5.
    H. Niederreiter Sequences with almost perfect linear complexity profile In: Advances in Cryptology - EUROCRYPT’87(D Chaum and WL Price, eds.), Lecture Notes in Computer Science Vol. 304 pp. 37–51 Springer Berlin 1988.Google Scholar
  6. 6.
    H. Niederreiter The probabilistic theory of linear complexity In: Advances in Cryptology — EUROCRYPT ’88 (CG Günther ed.), Lecture Notes in Computer Science Vol. 330 pp. 191–209 Springer Berlin 1988.Google Scholar
  7. 7.
    H. Niederreiter Keystream sequences with a good linear complexity profile for every starting point In: Advances in Cryptology — EUROCRYPT ’89 (JJ Quisquater and J Vandewalle, eds.), Lecture Notes in Computer Science Vol. 434 pp. 523–532 Springer Berlin 1990.Google Scholar
  8. 8.
    H Niederreiter Finite fields and cryptology In: Finite Fields, Coding Theory, and Advances in Communications and Computing (GL Mullen and PJS Shiue, eds.), pp. 359–373 Dekker New York 1993.Google Scholar
  9. 9.
    H. Niederreiter Some computable complexity measures for binary sequences In: Sequences and Their Applications (C Ding, T Helleseth, and H Niederreiter, eds.), pp. 67–78 Springer London 1999.CrossRefGoogle Scholar
  10. 10.
    H. Niederreiter and M. Vielhaber, Linear complexity profles: Hausdorff dimensions for almost perfect profiles and measures for general profiles J. Complexity 13(1997), 353–383.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    H.-G. Rück A Note on Elliptic Curves over Finite Fields Math. of Comp 49(1987), 301–304.MATHCrossRefGoogle Scholar
  12. 12.
    R.A. Rueppel, Analysis and Design of Stream Ciphers Springer, Berlin, 1986.MATHGoogle Scholar
  13. 13.
    R.A. Rueppel Stream ciphers, Contemporary Cryptology: The Science of Information Integrity (GJ Simmons, ed.), pp. 65–134, IEEE Press New York 1992.Google Scholar
  14. 14.
    J. H. Silverman, The Arithmetic of Elliptic Curves Springer-Verlag, New York, 1986.MATHGoogle Scholar
  15. 15.
    H. Stichtenoth, Algebraic Function Fields and Codes Springer, Berlin, 1993.MATHGoogle Scholar
  16. 16.
    M.A. Tsfasman and S.G. Vlädut, Algebraic-Geometric Codes Kluwer, Dordrecht, 1991.MATHGoogle Scholar
  17. 17.
    C. P. Xing, Multi-sequences with almost perfect linear complexity profile and function fields over finite fields, Journal of Complexity, to appear.Google Scholar
  18. 18.
    C.P. Xing, VJ Kumar and CS Ding Low correlation, large linear span sequences from function fields Preprint, 2000.Google Scholar
  19. 19.
    C.P. Xing and K.Y. Lam, Sequences with almost perfect linear complexity profiles and curves over finite fields IEEE Trans. Inform. Theory, 45(1999), 1267–1270.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    C. P. Xing and N. Niederreiter, Applications of algebraic curves to constructions of codes and almost perfect sequences, to appear in Proceedings of the 5th International Conference on Finite Fields and Applications.Google Scholar
  21. 21.
    C.P. Xing, H. Niederreiter, K.Y. Lam, and C.S. Ding, Constructions of sequences with almost perfect linear complexity profile from curves over finite fields Finite Fields Appl. 5 301–313 (1999).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    C.P. Xing, KY Lam and ZH Wei A class of explicit perfect multi-sequences In: Advanced in Cryptology-Asiacrypt’99 (KY Lam, E Okamoto, CP Xing, eds.), Lecture Notes in Computer Science Vol. 1716 pp. 299–305 Springer-Verlag Berlin 1999.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2002

Authors and Affiliations

  • Chaoping Xing
    • 1
    • 2
  1. 1.Department of MathematicsNational University of SingaporeSingapore
  2. 2.Department of MathematicsUniversity of Science and Technology of ChinaAnhuiP. R. China

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