Free ℤ-module

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2012)

  • Volume: 20, Issue: 4, page 275-280
  • ISSN: 1426-2630

Abstract

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In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.

How to cite

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Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. "Free ℤ-module." Formalized Mathematics 20.4 (2012): 275-280. <http://eudml.org/doc/267931>.

@article{YuichiFuta2012,
abstract = {In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.},
author = {Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {275-280},
title = {Free ℤ-module},
url = {http://eudml.org/doc/267931},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Yuichi Futa
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Free ℤ-module
JO - Formalized Mathematics
PY - 2012
VL - 20
IS - 4
SP - 275
EP - 280
AB - In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.
LA - eng
UR - http://eudml.org/doc/267931
ER -

References

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Citations in EuDML Documents

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  1. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Submodule of free Z-module
  2. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Matrix of ℤ-module1
  3. Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module
  4. Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama, Torsion Z-module and Torsion-free Z-module
  5. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Torsion Part of ℤ-module
  6. Yuichi Futa, Yasunari Shidama, Divisible ℤ-modules

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