Journal of the Korean Society for Industrial and Applied Mathematics

  • 제20권2호
  • /
  • Pages.107-121
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  • 2016
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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ANALYSIS OF THE UPPER BOUND ON THE COMPLEXITY OF LLL ALGORITHM

  • PARK, YUNJU (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, KOREA SCIENCE ACADEMY OF KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY) ;
  • PARK, JAEHYUN (DEPARTMENT OF ELECTRONIC ENGINEERING, PUKYONG NATIONAL UNIVERSITY)
  • 투고 : 2016.02.16
  • 심사 : 2016.05.25
  • 발행 : 2016.06.25
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초록

We analyze the complexity of the LLL algorithm, invented by Lenstra, Lenstra, and $Lov{\acute{a}}sz$ as a a well-known lattice reduction (LR) algorithm which is previously known as having the complexity of $O(N^4{\log}B)$ multiplications (or, $O(N^5({\log}B)^2)$ bit operations) for a lattice basis matrix $H({\in}{\mathbb{R}}^{M{\times}N})$ 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.

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참고문헌

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피인용 문헌

  1. Improved Cryptanalysis of a Fully Homomorphic Symmetric Encryption Scheme vol.2019, pp.None, 2019, https://doi.org/10.1155/2019/8319508