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Algorithms 2010, 3(3), 224-243; doi:10.3390/a3030224
Article

Segment LLL Reduction of Lattice Bases Using Modular Arithmetic

*  and
Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208, USA
* Author to whom correspondence should be addressed.
Received: 28 May 2010 / Accepted: 29 June 2010 / Published: 12 July 2010
(This article belongs to the Special Issue Algorithms for Applied Mathematics)
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Abstract

The algorithm of Lenstra, Lenstra, and Lovász (LLL) transforms a given integer lattice basis into a reduced basis. Storjohann improved the worst case complexity of LLL algorithms by a factor of O(n) using modular arithmetic. Koy and Schnorr developed a segment-LLL basis reduction algorithm that generates lattice basis satisfying a weaker condition than the LLL reduced basis with O(n) improvement than the LLL algorithm. In this paper we combine Storjohann’s modular arithmetic approach with the segment-LLL approach to further improve the worst case complexity of the segment-LLL algorithms by a factor of n0.5.
Keywords: Lattice; LLL basis reduction; reduced basis; successive minima; segments; modular arithmetic; fast matrix multiplication Lattice; LLL basis reduction; reduced basis; successive minima; segments; modular arithmetic; fast matrix multiplication
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Mehrotra, S.; Li, Z. Segment LLL Reduction of Lattice Bases Using Modular Arithmetic. Algorithms 2010, 3, 224-243.

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