Submodule of free Z-module

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2013)

  • Volume: 21, Issue: 4, page 273-282
  • ISSN: 1426-2630

Abstract

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In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their proofs are different.

How to cite

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Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. "Submodule of free Z-module." Formalized Mathematics 21.4 (2013): 273-282. <http://eudml.org/doc/267363>.

@article{YuichiFuta2013,
abstract = {In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their proofs are different.},
author = {Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {free Z-module; submodule of free Z-module; free -module; submodule of free -module},
language = {eng},
number = {4},
pages = {273-282},
title = {Submodule of free Z-module},
url = {http://eudml.org/doc/267363},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Yuichi Futa
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Submodule of free Z-module
JO - Formalized Mathematics
PY - 2013
VL - 21
IS - 4
SP - 273
EP - 282
AB - In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their proofs are different.
LA - eng
KW - free Z-module; submodule of free Z-module; free -module; submodule of free -module
UR - http://eudml.org/doc/267363
ER -

References

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