{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T18:25:48Z","timestamp":1777487148497,"version":"3.51.4"},"reference-count":6,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2019,10,1]],"date-time":"2019-10-01T00:00:00Z","timestamp":1569888000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,10,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article, we formalize in Mizar [1], [2] a binary operation of points on an elliptic curve over\n                    <jats:bold>GF<\/jats:bold>\n                    (\n                    <jats:bold>p<\/jats:bold>\n                    ) in affine coordinates. We show that the operation is unital, complementable and commutative. Elliptic curve cryptography [3], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.\n                  <\/jats:p>","DOI":"10.2478\/forma-2019-0026","type":"journal-article","created":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T04:32:28Z","timestamp":1582086748000},"page":"315-320","source":"Crossref","is-referenced-by-count":0,"title":["Operations of Points on Elliptic Curve in Affine Coordinates"],"prefix":"10.2478","volume":"27","author":[{"given":"Yuichi","family":"Futa","sequence":"first","affiliation":[{"name":"Tokyo University of Technology Tokyo , 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":[[2020,2,17]]},"reference":[{"key":"2026042801430415772_j_forma-2019-0026_ref_001_w2aab3b8c11b1b7b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026042801430415772_j_forma-2019-0026_ref_002_w2aab3b8c11b1b7b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801430415772_j_forma-2019-0026_ref_003_w2aab3b8c11b1b7b1ab1ab3Aa","doi-asserted-by":"crossref","unstructured":"[3] I. Blake, G. Seroussi, and N. Smart. Elliptic Curves in Cryptography. Number 265 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.10.1017\/CBO9781107360211","DOI":"10.1017\/CBO9781107360211"},{"key":"2026042801430415772_j_forma-2019-0026_ref_004_w2aab3b8c11b1b7b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Set of points on elliptic curve in projective coordinates. Formalized Mathematics, 19(3):131\u2013138, 2011. doi:10.2478\/v10037-011-0021-6.10.2478\/v10037-011-0021-6","DOI":"10.2478\/v10037-011-0021-6"},{"key":"2026042801430415772_j_forma-2019-0026_ref_005_w2aab3b8c11b1b7b1ab1ab5Aa","doi-asserted-by":"crossref","unstructured":"[5] Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Operations of points on elliptic curve in projective coordinates. Formalized Mathematics, 20(1):87\u201395, 2012. doi:10.2478\/v10037-012-0012-2.10.2478\/v10037-012-0012-2","DOI":"10.2478\/v10037-012-0012-2"},{"key":"2026042801430415772_j_forma-2019-0026_ref_006_w2aab3b8c11b1b7b1ab1ab6Aa","unstructured":"[6] Artur Korni\u0142owicz. Recursive definitions. Part II. Formalized Mathematics, 12(2):167\u2013172, 2004."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/content.sciendo.com\/view\/journals\/forma\/27\/3\/article-p315.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2019-0026","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T14:31:15Z","timestamp":1777386675000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2019-0026"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,1]]},"references-count":6,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,2,17]]},"published-print":{"date-parts":[[2019,10,1]]}},"alternative-id":["10.2478\/forma-2019-0026"],"URL":"https:\/\/doi.org\/10.2478\/forma-2019-0026","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2019,10,1]]}}}