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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 23, Issue 1 (Mar 2015)


Matrix of ℤ-module1

Yuichi Futa / Hiroyuki Okazaki / Yasunari Shidama
Published Online: 2015-03-31 | DOI: https://doi.org/10.2478/forma-2015-0003


In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

MSC: 11E39; 13C10; 03B35

Keywords: matrix of Z-module; matrix of linear transformation; bilinear form

MML: identifier: ZMATRLIN; version: 8.1.04 5.31.1231


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About the article

Received: 2015-02-18

Published Online: 2015-03-31

Published in Print: 2015-03-01

1This work was supported by JSPS KAKENHI 21240001 and 22300285.

Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2015-0003.

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© Yuichi Futa et al.. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

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