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Segment LLL Reduction of Lattice Bases Using Modular Arithmetic
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
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
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MDPI and ACS Style
Mehrotra, S.; Li, Z. Segment LLL Reduction of Lattice Bases Using Modular Arithmetic. Algorithms 2010, 3, 224-243.
Mehrotra S, Li Z. Segment LLL Reduction of Lattice Bases Using Modular Arithmetic. Algorithms. 2010; 3(3):224-243.
Mehrotra, Sanjay; Li, Zhifeng. 2010. "Segment LLL Reduction of Lattice Bases Using Modular Arithmetic." Algorithms 3, no. 3: 224-243.