ANALYSIS OF THE UPPER BOUND ON THE COMPLEXITY OF LLL ALGORITHM

- Journal title : Journal of the Korean Society for Industrial and Applied Mathematics
- Volume 20, Issue 2, 2016, pp.107-121
- Publisher : The Korean Society for Industrial and Applied Mathematics
- DOI : 10.12941/jksiam.2016.20.107

Title & Authors

ANALYSIS OF THE UPPER BOUND ON THE COMPLEXITY OF LLL ALGORITHM

PARK, YUNJU; PARK, JAEHYUN;

PARK, YUNJU; PARK, JAEHYUN;

Abstract

We analyze the complexity of the LLL algorithm, invented by Lenstra, Lenstra, and as a a well-known lattice reduction (LR) algorithm which is previously known as having the complexity of multiplications (or, bit operations) for a lattice basis matrix where B is the maximum value among the squared norm of columns of H. This implies that the complexity of the lattice reduction algorithm depends only on the matrix size and the lattice basis norm. However, the matrix structures (i.e., the correlation among the columns) of a given lattice matrix, which is usually measured by its condition number or determinant, can affect the computational complexity of the LR algorithm. In this paper, to see how the matrix structures can affect the LLL algorithm`s complexity, we derive a more tight upper bound on the complexity of LLL algorithm in terms of the condition number and determinant of a given lattice matrix. We also analyze the complexities of the LLL updating/downdating schemes using the proposed upper bound.

Keywords

Lattice Reduction;LLL algorithm;Complexity analysis;Matrix Updating/Downdating;

Language

English

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