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Lattice basis reduction in function fields

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Algorithmic Number Theory (ANTS 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1423))

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Abstract

We present an algorithm for lattice basis reduction in function fields. In contrast to integer lattices, there is a simple algorithm which provably computes a reduced basis in polynomial time. This algorithm works only with the coefficients of the polynomials involved, so there is no polynomial arithmetic needed. This algorithm can be generically extended to compute a reduced basis starting from a generating system for a lattice. Moreover, it can be applied to lattices over the field of puiseux expansions of a function field. In that case, this algorithm represents one major step towards an efficient arithmetic in Jacobians of curves.

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References

  1. E. Artin: Quadratische Körper im Gebiete der höheren Kongruenzen I. Mathematische Zeitschrift 19 (1924). pp. 153–206. In: S. Lang, J. Tate (eds.): The collected papers of Emil Artin. Reading, Mass.: Addison Wesley 1965.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Coates: Construction of rational functions on a curve. Proc. Camb. Phil. Soc. 68 (1970).

    Google Scholar 

  3. S. Galbraith, S. Paulus: Unique representation of divisor class groups of function fields of degree > 2. In preparation.

    Google Scholar 

  4. A. K. Lenstra, H. W. Lenstra Jr., L. Lovasz: Factoring polynomials with rational coefficients. Math. Ann. 261 (1982).

    Google Scholar 

  5. A. K. Lenstra: Factoring multivariate polynomials over finite fields. J. Computer & System Sciences 30 (1985) No. 2.

    Google Scholar 

  6. K. Mahler: An analogue of Minkowski's geometry of numbers in a field of series. Annals of Math. 42 (1941).

    Google Scholar 

  7. S. Paulus, H. G. Rück: Real and imaginary quadratic representations of hyperelliptic function fields. To appear in Mathematics of Computation.

    Google Scholar 

  8. M. E. Pohst, H. Zassenhaus: Algorithmic algebraic number thoery. Cambrigde University Press: Cambridge 1989.

    Google Scholar 

  9. M. E. Pohst, M. Schörning: On integral basis reduction in global function fields. Proceedings of ANTS II. Lecture Notes in Computer Science 1122. Springer Verlag 1996.

    Google Scholar 

  10. G. Walsh: A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function. Department of Mathematics, University of Ottawa. Unpublished.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Theoretical Computer Science, Darmstadt University of Technology, 64283, Darmstadt, Germany

    Sachar Paulus

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  1. Sachar Paulus
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Joe P. Buhler

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© 1998 Springer-Verlag Berlin Heidelberg

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Paulus, S. (1998). Lattice basis reduction in function fields. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054893

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  • DOI: https://doi.org/10.1007/BFb0054893

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64657-0

  • Online ISBN: 978-3-540-69113-6

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