Quotient Module of Z-module
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...
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| Format: | Article |
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Sciendo
2012-12-01
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| Series: | Formalized Mathematics |
| Online Access: | https://doi.org/10.2478/v10037-012-0024-y |
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doaj-a878d0d55e9245bb8d7e6da4b99277462021-09-05T18:16:48ZengSciendoFormalized Mathematics1426-26301898-99342012-12-0120320521410.2478/v10037-012-0024-yQuotient Module of Z-moduleFuta Yuichi0Okazaki Hiroyuki1Shidama Yasunari2Shinshu University, Nagano, JapanShinshu University, Nagano, JapanShinshu University, Nagano, JapanIn 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.https://doi.org/10.2478/v10037-012-0024-y |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Futa Yuichi Okazaki Hiroyuki Shidama Yasunari |
| spellingShingle |
Futa Yuichi Okazaki Hiroyuki Shidama Yasunari Quotient Module of Z-module Formalized Mathematics |
| author_facet |
Futa Yuichi Okazaki Hiroyuki Shidama Yasunari |
| author_sort |
Futa Yuichi |
| title |
Quotient Module of Z-module |
| title_short |
Quotient Module of Z-module |
| title_full |
Quotient Module of Z-module |
| title_fullStr |
Quotient Module of Z-module |
| title_full_unstemmed |
Quotient Module of Z-module |
| title_sort |
quotient module of z-module |
| publisher |
Sciendo |
| series |
Formalized Mathematics |
| issn |
1426-2630 1898-9934 |
| publishDate |
2012-12-01 |
| description |
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. |
| url |
https://doi.org/10.2478/v10037-012-0024-y |
| work_keys_str_mv |
AT futayuichi quotientmoduleofzmodule AT okazakihiroyuki quotientmoduleofzmodule AT shidamayasunari quotientmoduleofzmodule |
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1717786072509841408 |