{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T01:26:01Z","timestamp":1777425961005,"version":"3.51.4"},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,1,1]]},"abstract":"Operations of Points on Elliptic Curve in Projective Coordinates<\/jats:title>\n In this article, we formalize operations of points on an elliptic curve over GF<\/jats:bold>(p<\/jats:bold>). Elliptic curve cryptography [7], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security. We prove that the two operations of points: compellProjCo<\/jats:sub> and addellProjCo<\/jats:sub> are unary and binary operations of a point over the elliptic curve.<\/jats:p>","DOI":"10.2478\/v10037-012-0012-2","type":"journal-article","created":{"date-parts":[[2012,9,13]],"date-time":"2012-09-13T01:22:44Z","timestamp":1347499364000},"page":"87-95","source":"Crossref","is-referenced-by-count":1,"title":["Operations of Points on Elliptic Curve in Projective Coordinates"],"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":"Daichi","family":"Mizushima","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","reference":[{"issue":"1","key":"1","first-page":"91","article-title":"The ordinal numbers","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"2","first-page":"175","article-title":"Binary operations","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"3","first-page":"55","article-title":"Functions and their basic properties","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"4","first-page":"153","article-title":"Functions from a set to a set","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"5","first-page":"47","article-title":"Some basic properties of sets","volume":"1","author":"Czes\u0142aw Byli\u0144ski","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"131","DOI":"10.2478\/v10037-011-0021-6","article-title":"Set of points on elliptic curve in projective coordinates","volume":"19","author":"Yuichi Futa","year":"2011","journal-title":"Formalized Mathematics"},{"key":"7","volume-title":"Elliptic Curves in Cryptography","author":"G. 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