{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T20:50:39Z","timestamp":1649105439663},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"4","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 free \u2124-module and its rank. We formally prove that for a free finite rank \u2124-module <jats:italic>V <\/jats:italic>, the number of elements in its basis, that is a rank of the \u2124-module, is constant regardless of the selection of its basis. \u2124-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lov\u00e1sz) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.<\/jats:p>","DOI":"10.2478\/v10037-012-0033-x","type":"journal-article","created":{"date-parts":[[2013,2,4]],"date-time":"2013-02-04T17:15:08Z","timestamp":1359998108000},"page":"275-280","source":"Crossref","is-referenced-by-count":0,"title":["Free \u2124-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\/4\/article-p275.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/forma.2012.20.issue-4\/v10037-012-0033-x\/v10037-012-0033-x.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,30]],"date-time":"2020-05-30T15:28:43Z","timestamp":1590852523000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10037-012-0033-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12,1]]},"references-count":0,"journal-issue":{"issue":"4"},"URL":"https:\/\/doi.org\/10.2478\/v10037-012-0033-x","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]]}}}