{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T05:56:09Z","timestamp":1648619769857},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n\t\t\t\t<jats:p>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 <jats:italic>V <\/jats:italic>and a prime number <jats:italic>p<\/jats:italic>, a quotient module <jats:italic>V\/pV <\/jats:italic>has the structure of a vector space over F<jats:italic>p<\/jats:italic>. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\u00b4asz) 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.<\/jats:p>","DOI":"10.2478\/v10037-012-0024-y","type":"journal-article","created":{"date-parts":[[2013,2,4]],"date-time":"2013-02-04T17:14:25Z","timestamp":1359998065000},"page":"205-214","source":"Crossref","is-referenced-by-count":1,"title":["Quotient Module of Z-module"],"prefix":"10.2478","volume":"20","author":[{"given":"Yuichi","family":"Futa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hiroyuki","family":"Okazaki","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","container-title":["Formalized Mathematics"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/20\/3\/article-p205.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/forma.2012.20.issue-3\/v10037-012-0024-y\/v10037-012-0024-y.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,30]],"date-time":"2020-05-30T15:28:39Z","timestamp":1590852519000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10037-012-0024-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12,1]]},"references-count":0,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.2478\/v10037-012-0024-y","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2012,12,1]]}}}