An improved method of computing the regulator of a real quadratic function field

  • Andreas Stein
  • Hugh C. Williams
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1423)

Abstract

There exists an effective algorithm for computing the regulator of a real quadratic congruence function field K=k(X)(√D) of genus g=deg(D)/2−1 in O(q2/5g) polynomial operations. In those cases where the regulator exceeds 108, this algorithm tends to be far better than the Baby step-Giant step algorithm which performs O(q2/5) polynomial operations. We show how we increased the speed of the O(q2/5g)-algorithm such that we are able to large values of regulators of real quadratic congruence function fields of small genus.

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References

  1. 1.
    Artin, E.: Quadratische Körper im Gebiete der höheren Kongruenzen I, II. Mathematische Zeitschrift 19 (1924) 153–246MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Buchmann, J., Williams, H.C.: On the Computation of the Class Number of an Algebraic Number Field. Math.Comp. 53 (1989) 679–688MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Lenstra, H.W., Jr.: On the Calculation of Regulators and Class Numbers of Quadratic Fields. London Math.Soc.Lec.Note Ser. 56 (1982) 123–150MATHMathSciNetGoogle Scholar
  4. 4.
    Perron, O.: Die Lehre von den Kettenbrüchen. Teubner, Leipzig (1913)Google Scholar
  5. 5.
    Scheidler, R., Stein, A., Williams, H.C.: Key-exchange in Real Quadratic Congruence Function Fields. Designs, Codes and Cryptography 7, Nr.1/2 (1996) 153–174MATHMathSciNetGoogle Scholar
  6. 6.
    Schmidt, F.K.: Analytische Zahlentheorie in Körpern der Charakteristik p. Mathematische Zeitschrift 33 (1931) 1–32MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Schoof, R.J.: Quadratic Fields and Factorization. Computational Methods in Number Theory (H.W.Lenstra and R.Tijdemans, eds.). Math.Centrum Tracts 155 II, Amsterdam (1983) 235–286Google Scholar
  8. 8.
    Shanks, D.: The Infrastructure of a Real Quadratic Field and its Applications. Proc.1972 Number Th.Conf., Boulder, Colorado (1972) 217–224Google Scholar
  9. 9.
    Shanks, D.: Class Number, A Theory of Factorization and Genera. Proc.Symp.Pure Math. 20 (1971) 415–440MATHMathSciNetGoogle Scholar
  10. 10.
    SIMATH Manual Chair of Prof.Dr.H.G.Zimmer, University of Saarland (1997)Google Scholar
  11. 11.
    Stein, A., Zimmer, H.G.: An Algorithm for Determining the Regulator and the Fundamental Unit of a Hyperelliptic Congruence Function Field. Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation, ISSAC, Bonn, July 15–17, ACM Press (1991) 183–184Google Scholar
  12. 12.
    Stein, A.: Algorithmen in reell-quadratischen Kongruenzfunktionenkörpern PhD Thesis, UniversitÄt des Saarlandes, Saarbrücken (1996)Google Scholar
  13. 13.
    Stein, A.: Equivalences between Elliptic Curves and Real Quadratic Congruence Function Fields. Journal de Theorie des Nombres de Bordeaux 9 (1997) 75–95MATHMathSciNetGoogle Scholar
  14. 14.
    Stein, A., Williams, H.C.: Some Methods for Evaluating the Regulator of a Real Quadratic Function Field. Experimental Mathematics (to appear)Google Scholar
  15. 15.
    Stephens, A.J., Williams, H.C.: Some Computational Results on a Problem Concerning Powerful Numbers. Mathematics of Computation 50 (1988) 619–632MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Stephens, A.J., Williams, H.C.: Computation of Real Quadratic Fields with Class Number One. Mathematics of Computation 51 (1988) 809–824MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Weis, B., Zimmer, H.G.: Artin's Theorie der quadratischen Kongruenzfunktionenkörper und ihre Anwendung auf die Berechnung der Einheiten-und Klassengruppen. Mitt.Math.Ges.Hamburg Sond., XII, No. 2 (1991)Google Scholar
  18. 18.
    Williams, H.C., Wunderlich, M.C.: On the Parallel Generation of the Residues for the Continued Fraction Algorithm. Mathematics of Computation 48 (1987) 405–423MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Andreas Stein
    • 1
  • Hugh C. Williams
    • 1
  1. 1.University of ManitobaWinnipegCanada

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