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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H. Cohen (1995): A Course in Computational Algebraic Number Theory (2nd edition) Springer-Verlag, Berlin Heidelberg New York.
A. K. Lenstra, H.W. Lenstra jr., and L. Lovász (1982): Factoring polynomials with rational coefficients, Math. Ann. 261, 515–534.
N. Tzanakis and B. M. M. de Weger (1989): On the practical solution of the Thue equation, J. Number Theory 31, 99–132.
B. M. M. de Weger (1987): Solving exponential diophantine equations using lattice basis reduction algorithms, J. Number Theory 26, 325–367.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
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