# Quotient Module of Z-module

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2012)

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

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topYuichi 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 -

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## Citations in EuDML Documents

top- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Free ℤ-module
- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Submodule of free Z-module
- Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module
- Yuichi Futa, Yasunari Shidama, Lattice of ℤ-module
- Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama, Torsion Z-module and Torsion-free Z-module
- Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Torsion Part of ℤ-module
- Yuichi Futa, Yasunari Shidama, Divisible ℤ-modules

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