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Lattice Reduction by Random Sampling and Birthday Methods

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Abstract

We present a novel practical algorithm that given a lattice basis b1, ..., bn finds in O(n 2( k/6 )k/4) average time a shorter vector than b 1 provided that b 1 is ( k/6 )n/(2k) times longer than the length of the shortest, nonzero lattice vector. We assume that the given basis b 1, ..., b n has an orthogonal basis that is typical for worst case lattice bases. The new reduction method samples short lattice vectors in high dimensional sublattices, it advances in sporadic big jumps. It decreases the approximation factor achievable in a given time by known methods to less than its fourth-th root. We further speed up the new method by the simple and the general birthday method.

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Author information

Authors and Affiliations

  1. Fachbereiche Mathematik/Biologie-Informatik, Universität Frankfurt, PSF 111932, D-60054, Frankfurt am Main, Germany

    Claus Peter Schnorr

Authors
  1. Claus Peter Schnorr
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takusstr. 9, 14195, Berlin, Germany

    Helmut Alt

  2. LIRMM, 161, Rue Ada, 34392, Montpellier Cedex 5, France

    Michel Habib

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© 2003 Springer-Verlag Berlin Heidelberg

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Schnorr, C.P. (2003). Lattice Reduction by Random Sampling and Birthday Methods. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_14

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  • DOI: https://doi.org/10.1007/3-540-36494-3_14

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00623-7

  • Online ISBN: 978-3-540-36494-8

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