{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T04:55:39Z","timestamp":1764996939724,"version":"3.46.0"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"3-4","license":[{"start":{"date-parts":[[2012,1,25]],"date-time":"2012-01-25T00:00:00Z","timestamp":1327449600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,2]]},"abstract":"<jats:title>Abstract.<\/jats:title>\n                  <jats:p>\n                    In this paper we look at three families of elliptic curves with rational 3-torsion over a finite field. These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves. We find the number of\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0013_eq1.png\"\/>\n                    -isogeny classes of each family, as well as the number of\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0013_eq2.png\"\/>\n                    -isomorphism classes of the generalized DIK curves. We also include some formulas for efficient computation on these curves, improving upon known results. In particular, we find better formulas for doubling and addition on the original tripling-oriented DIK curves and also for addition and tripling on elliptic curves with\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"jmc-2011-0013_eq3.png\"\/>\n                    -invariant 0.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2011-0013","type":"journal-article","created":{"date-parts":[[2012,3,21]],"date-time":"2012-03-21T10:07:09Z","timestamp":1332324429000},"page":"225-246","source":"Crossref","is-referenced-by-count":2,"title":["Families of elliptic curves with rational 3-torsion"],"prefix":"10.1515","volume":"5","author":[{"given":"Dustin","family":"Moody","sequence":"first","affiliation":[{"name":"National Institute of Standards and Technology \u2013 Computer Security Division, 100 Bureau Drive Stop 8930, Gaithersburg, Maryland 20899-8930, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongfeng","family":"Wu","sequence":"additional","affiliation":[{"name":"North China University of Technology \u2013 College of Sciences, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2012,1,25]]},"container-title":["Journal of Mathematical Cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2011-0013\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2011-0013\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:20:13Z","timestamp":1764980413000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2011-0013\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,25]]},"references-count":0,"journal-issue":{"issue":"3-4","published-online":{"date-parts":[[2012,1,25]]},"published-print":{"date-parts":[[2012,2]]}},"alternative-id":["10.1515\/jmc-2011-0013"],"URL":"https:\/\/doi.org\/10.1515\/jmc-2011-0013","relation":{},"ISSN":["1862-2984","1862-2976"],"issn-type":[{"type":"electronic","value":"1862-2984"},{"type":"print","value":"1862-2976"}],"subject":[],"published":{"date-parts":[[2012,1,25]]}}}