# Matrix of ℤ-module1

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2015)

- Volume: 23, Issue: 1, page 29-49
- ISSN: 1426-2630

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topYuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. "Matrix of ℤ-module1." Formalized Mathematics 23.1 (2015): 29-49. <http://eudml.org/doc/270886>.

@article{YuichiFuta2015,

abstract = {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.},

author = {Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama},

journal = {Formalized Mathematics},

keywords = {matrix of Z-module; matrix of linear transformation; bilinear form; matrix of -module},

language = {eng},

number = {1},

pages = {29-49},

title = {Matrix of ℤ-module1},

url = {http://eudml.org/doc/270886},

volume = {23},

year = {2015},

}

TY - JOUR

AU - Yuichi Futa

AU - Hiroyuki Okazaki

AU - Yasunari Shidama

TI - Matrix of ℤ-module1

JO - Formalized Mathematics

PY - 2015

VL - 23

IS - 1

SP - 29

EP - 49

AB - 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.

LA - eng

KW - matrix of Z-module; matrix of linear transformation; bilinear form; matrix of -module

UR - http://eudml.org/doc/270886

ER -

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