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International Algorithmic Number Theory Symposium

ANTS 2002: Algorithmic Number Theory pp 148–162Cite as

An Accelerated Buchmann Algorithm for Regulator Computation in Real Quadratic Fields

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

Abstract

We present a probabilistic algorithm for computing the regulator R of a real quadratic order of discriminant Δ running in time L(1/2, 3/√8 + o(1)).

research supported by the DFG

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References

  1. Christine Abel. Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ordnungen. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany, 1994. German.

    Google Scholar 

  2. Eric Bach. Explicit bounds for primality testing and related problems. Math. Comp., 55(191):355–380, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  3. Johannes Buchmann, Christoph Thiel, and Hugh C. Williams. Short representation of quadratic integers. In Wieb Bosma and Alf J. van der Poorten, editors, Computational Algebra and Number Theory, Sydney 1992, volume 325 of Mathematics and its Applications, pages 159–185. Kluwer Academic Publishers, 1995.

    Google Scholar 

  4. Johannes Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In Catherine Goldstein, editor, Séminaire de Théorie des Nombres, Paris 1988–1989, volume 91 of Progress in Mathematics, pages 27–41. Birkhäuser, 1990.

    Google Scholar 

  5. J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In Séminaire de Théorie des Nombres, pages 27–41, Paris, 1988–89.

    Google Scholar 

  6. H. Cohen. A course in computational algebraic number theory. Springer, Heidelberg, 1993.

    Google Scholar 

  7. James L. Hafner and Kevin S. McCurley. A rigorous subexponential algorithm for computation of class groups. J. Am. Math. Soc., 2(4):837–850, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  8. Michael J. Jacobson, Jr. Applying sieving to the computation of quadratic class groups. Mathematics of Computation, 68(226):859–867, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  9. Hendrik W. Lenstra, Jr. On the calculation of regulators and class numbers of quadratic fields. In J. V. Armitage, editor, Journees Arithmetiques, Exeter 1980, volume 56 of London Mathematical Society Lecture Notes Series, pages 123–150. Cambridge University Press, 1982.

    Google Scholar 

  10. H.W. Lenstra Jr. and C. Pomerance. A rigorous time bound for factoring integers. J. Amer. Math. Soc., 5:483–516, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  11. Markus Maurer. Regulator approximation and fundamental unit computation for real quadratic orders. PhD thesis, Technische Universität Darmstadt, Fachbereich Informatik, Darmstadt, Germany, 2000.

    Google Scholar 

  12. Thom Mulders and Arne Storjohann. Diophantine linear system solving. ACM Press, 1999.

    Google Scholar 

  13. Martin Seysen. A probablistic factorization algorithm with quadratic forms of negative discriminant. Mathematics of Computation, 48:757–780, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  14. Ulrich Vollmer. Asymptotically fast discrete logarithms in quadratic number fields. In Wieb Bosma, editor, Algorithmic Number Theory Symposium IV, volume 1838 of Lecture Notes in Computer Science, pages 581–594. Springer-Verlag, 2000.

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Fachbereich Informatik Fachgebiet Kryptographie und Computeralgebra, Technische Universität Darmstadt, Alexanderstr. 10, 64283, Darmstadt

    Ulrich Vollmer

Authors
  1. Ulrich Vollmer
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Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

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Vollmer, U. (2002). An Accelerated Buchmann Algorithm for Regulator Computation in Real Quadratic Fields. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_12

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  • DOI: https://doi.org/10.1007/3-540-45455-1_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43863-2

  • Online ISBN: 978-3-540-45455-7

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