Quotient Module of Z-module

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2012)

  • Volume: 20, Issue: 3, page 205-214
  • ISSN: 1426-2630

Abstract

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In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattices [14]. Some theorems in this article are described by translating theorems in [20] and [19] into theorems of Z-module.

How to cite

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Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. "Quotient Module of Z-module." Formalized Mathematics 20.3 (2012): 205-214. <http://eudml.org/doc/267786>.

@article{YuichiFuta2012,
abstract = {In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattices [14]. Some theorems in this article are described by translating theorems in [20] and [19] into theorems of Z-module.},
author = {Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
language = {eng},
number = {3},
pages = {205-214},
title = {Quotient Module of Z-module},
url = {http://eudml.org/doc/267786},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Yuichi Futa
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Quotient Module of Z-module
JO - Formalized Mathematics
PY - 2012
VL - 20
IS - 3
SP - 205
EP - 214
AB - In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattices [14]. Some theorems in this article are described by translating theorems in [20] and [19] into theorems of Z-module.
LA - eng
UR - http://eudml.org/doc/267786
ER -

References

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Citations in EuDML Documents

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

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