{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T15:02:03Z","timestamp":1777474923366,"version":"3.51.4"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2015,3,1]],"date-time":"2015-03-01T00:00:00Z","timestamp":1425168000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/ http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article, we formalize a matrix of \u2124-module and its properties. Specially, we formalize a matrix of a linear transformation of \u2124-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free \u2124-module\n                    <jats:italic>V<\/jats:italic>\n                    , determinant of its Gramian matrix is constant regardless of selection of its basis. \u2124-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\u00e1sz) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of \u2124-module.\n                  <\/jats:p>","DOI":"10.2478\/forma-2015-0003","type":"journal-article","created":{"date-parts":[[2015,7,14]],"date-time":"2015-07-14T07:49:39Z","timestamp":1436860179000},"page":"29-49","source":"Crossref","is-referenced-by-count":1,"title":["Matrix of \u2124-module\n                    <sup>1<\/sup>"],"prefix":"10.2478","volume":"23","author":[{"given":"Yuichi","family":"Futa","sequence":"first","affiliation":[{"name":"Japan Advanced Institute of Science and Technology, Ishikawa, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hiroyuki","family":"Okazaki","sequence":"additional","affiliation":[{"name":"Shinshu University, Nagano, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Shinshu University, Nagano, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2015,3,31]]},"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/23\/1\/article-p29.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2015-0003","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:50:49Z","timestamp":1777384249000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2015-0003"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,3,31]]},"published-print":{"date-parts":[[2015,3,1]]}},"alternative-id":["10.2478\/forma-2015-0003"],"URL":"https:\/\/doi.org\/10.2478\/forma-2015-0003","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,3,1]]}}}