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Lattice Reduction

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Some Tapas of Computer Algebra

Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 4))

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Abstract

We shall give an introduction to the LLL-algorithm over ℤ. The algorithm is due to L. Lovász, H.W. Lenstra and A.K. Lenstra. It is concerned with the problem of finding a shortest nonzero vector in a lattice. In Section 2, we begin by introducing the relevant background material on lattices. Then we proceed to describe lattice reduction and finding shortest nonzero vectors in dimension 2. Section 4 presents the core result of this chapter: LLL-lattice reduction in any dimension. Section 5 deals with the implementation of the LLL-algorithm, and the last section discusses an application of the algorithm to the problem of finding Z-linear combinations of a given set of real numbers with small values.

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References

  1. H. Cohen (1995): A Course in Computational Algebraic Number Theory (2nd edition) Springer-Verlag, Berlin Heidelberg New York.

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  2. A. K. Lenstra, H.W. Lenstra jr., and L. Lovász (1982): Factoring polynomials with rational coefficients, Math. Ann. 261, 515–534.

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  3. N. Tzanakis and B. M. M. de Weger (1989): On the practical solution of the Thue equation, J. Number Theory 31, 99–132.

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  4. B. M. M. de Weger (1987): Solving exponential diophantine equations using lattice basis reduction algorithms, J. Number Theory 26, 325–367.

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Authors
  1. Frits Beukers
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Editors and Affiliations

  1. Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB, Eindhoven, The Netherlands

    Arjeh M. Cohen , Hans Cuypers  & Hans Sterk ,  & 

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© 1999 Springer-Verlag Berlin Heidelberg

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Beukers, F. (1999). Lattice Reduction. In: Cohen, A.M., Cuypers, H., Sterk, H. (eds) Some Tapas of Computer Algebra. Algorithms and Computation in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03891-8_3

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  • DOI: https://doi.org/10.1007/978-3-662-03891-8_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08335-8

  • Online ISBN: 978-3-662-03891-8

  • eBook Packages: Springer Book Archive

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