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Storage efficient algorithm for Hermite Normal Form using LLL
Title
Storage efficient algorithm for Hermite Normal Form using LLL
Authors
Cho G.H.
;
Lee H.-S.
;
Lim S.
;
Kim Y.
Ewha Authors
이향숙
;
임선간
;
조국화
SCOPUS Author ID
이향숙
; 임선간
; 조국화
Issue Date
2021
Journal Title
Linear Algebra and Its Applications
ISSN
0024-3795
Citation
Linear Algebra and Its Applications vol. 613, pp. 183 - 200
Keywords
Hermite Normal Form
;
Lattice
;
LLL algorithm
Publisher
Elsevier Inc.
Indexed
SCIE; SCOPUS
Document Type
Article
Abstract
Computing HNF has a long history, but designing a storage efficient algorithm is a challenging issue for matrices of large sizes. One of the main challenges in the design of storage efficient HNF algorithm is to control the rank and the size of the intermediate input. In our proposed algorithm, we use a multiple extended gcd algorithm and show that the rank of the intermediate input matrix decreases as the number of iteration increases. The determinant of the intermediate input matrix of our algorithm is a factor of the determinant d of the input matrix and thus size reduction modulo d can be done in the computations of our algorithm. By using a lattice reduction algorithm and a proven quality of LLL reduced basis, we prove that the storage of the intermediate input matrix Bk of our algorithm is less than [Formula presented] in bits. Therefore, it is expected that a smaller storage for kth iteration is required as k closes to n. We compare the intermediate computations of our algorithm to MW-type algorithm which has an optimal asymptotic space requirement. Our experimental example and results on intermediate size suggest that the intermediate storage of our HNF algorithm is comparable to MW-type algorithm and well controlled by the intermediate input size. © 2020 Elsevier Inc.
DOI
10.1016/j.laa.2020.12.022
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