Families of Elliptic Curves with Rational 3-torsion

<!DOCTYPE html> <html lang="en-us" xml:lang="en-us"> <head><script type="text/javascript" src="/_static/js/bundle-playback.js?v=HxkREWBo" charset="utf-8"></script> <script type="text/javascript" src="/_static/js/wombat.js?v=txqj7nKC" charset="utf-8"></script> <script>window.RufflePlayer=window.RufflePlayer||{};window.RufflePlayer.config={"autoplay":"on","unmuteOverlay":"hidden"};</script> <script type="text/javascript" src="/_static/js/ruffle/ruffle.js"></script> <script type="text/javascript"> __wm.init("https://web.archive.org/web"); __wm.wombat("https://csrc.nist.gov/publications/detail/journal-article/2012/families-of-elliptic-curves-with-rational-3-torsion","20230203033640","https://web.archive.org/","web","/_static/", "1675395400"); </script> <link rel="stylesheet" type="text/css" href="/_static/css/banner-styles.css?v=S1zqJCYt" /> <link rel="stylesheet" type="text/css" href="/_static/css/iconochive.css?v=3PDvdIFv" />  <meta charset="utf-8"/> <title>Families of Elliptic Curves with Rational 3-torsion | CSRC</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8"/> <meta http-equiv="content-style-type" content="text/css"/> <meta http-equiv="content-script-type" content="text/javascript"/> <meta name="viewport" content="width=device-width, initial-scale=1.0"/> <meta name="msapplication-config" content="/CSRC/Media/images/favicons/browserconfig.xml"/> <meta name="theme-color" content="#000000"/> <meta name="google-site-verification" content="xbrnrVYDgLD-Bd64xHLCt4XsPXzUhQ-4lGMj4TdUUTA"/> <meta description="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 Fq-isogeny classes of each family, as well as the number of Fq-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 j-invariant 0."/>  <meta name="dcterms.title" content="Families of Elliptic Curves with Rational 3-torsion"/> <meta name="dcterms.description" content="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 Fq-isogeny classes of each family, as well as the number of Fq-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 j-invariant 0."/>  <meta name="dcterms.creator" content="Author: Dustin Moody"/> <meta name="dcterms.creator" content="Author: H. Wu"/>  <meta name="dcterms.date.created" schema="ISO8601" content="2012-01-30"/> <meta name="dcterms.identifier" content="https://csrc.nist.gov/publications/detail/journal-article/2012/families-of-elliptic-curves-with-rational-3-torsion"/> <meta name="dcterms.language" scheme="DCTERMS.RFC1766" content="EN-US"/>  <meta name="citation_title" content="Families of Elliptic Curves with Rational 3-torsion"/> <meta name="citation_publication_date" content="2012/01/30"/> <meta name="citation_doi" content="https://doi.org/10.1515/jmc-2011-0013"/> <meta name="citation_volume" content="5"/> <meta name="citation_issue" content="42433"/> <meta name="citation_firstpage" content="225"/> <meta name="citation_lastpage" content="246"/> <meta name="citation_keywords" content="elliptic curves,Hessian curves,cryptography"/> <meta name="citation_language" content="en"/> <meta name="citation_pdf_url" content="https://doi.org/10.1515/jmc-2011-0013"/> <meta name="citation_abstract_html_url" content="https://csrc.nist.gov/publications/detail/journal-article/2012/families-of-elliptic-curves-with-rational-3-torsion"/>  <meta name="citation_author" content="Moody, Dustin"/> <meta name="citation_author" content="Wu, H."/>  <meta name="og:site_name" content="CSRC | NIST"/> <meta name="og:type" content="article"/> <meta name="og:url" content="https://web.archive.org/web/20230203033640im_/https://csrc.nist.gov/publications/detail/journal-article/2012/families-of-elliptic-curves-with-rational-3-torsion"/> <meta name="og:title" content="Families of Elliptic Curves with Rational 3-torsion"/> <meta name="og:description" content="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 Fq-isogeny classes of each family, as well as the number of Fq-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 j-invariant 0."/> <meta name="article:author" content="Moody, Dustin"/> <meta name="article:author" content="Wu, H."/> <meta name="article:tag" content="elliptic curves,Hessian curves,cryptography"/> <meta name="article:published_time" content="2012-01-30"/> <meta name="og:image" content="https://web.archive.org/web/20230203033640im_/https://csrc.nist.gov/CSRC/media/images/CSRC-logo-open-graph.png"/> <link rel="apple-touch-icon" sizes="180x180" href="/web/20230203033640im_/https://csrc.nist.gov/images/icons/apple-touch-icon.png"/> <link rel="icon" type="image/png" href="/web/20230203033640im_/https://csrc.nist.gov/images/icons/favicon-32x32.png" sizes="32x32"/> <link rel="icon" type="image/png" href="/web/20230203033640im_/https://csrc.nist.gov/images/icons/favicon-16x16.png" sizes="16x16"/> <link rel="manifest" href="/web/20230203033640/https://csrc.nist.gov/images/icons/manifest.json"/> <link rel="mask-icon" href="/web/20230203033640im_/https://csrc.nist.gov/images/icons/safari-pinned-tab.svg" color="#000000"/> <link href="/web/20230203033640im_/https://csrc.nist.gov/CSRC/Media/images/favicons/favicon.ico" type="image/x-icon" rel="shortcut icon"/> <link href="/web/20230203033640im_/https://csrc.nist.gov/CSRC/Media/images/favicons/favicon.ico" type="image/x-icon" rel="icon"/>  <link href="/web/20230203033640cs_/https://csrc.nist.gov/dist/app.css" rel="stylesheet"/>  <style> .grecaptcha-badge { visibility: hidden; } </style> <script async type="text/javascript" id="_fed_an_ua_tag" src="https://web.archive.org/web/20230203033640js_/https://dap.digitalgov.gov/Universal-Federated-Analytics-Min.js?agency=nist&subagency=csrc&pua=UA-66610693-15&yt=true&exts=xsd,xml,wav,mpg,mpeg,avi,rtf,webm,ogg,ogv,oga,map,otf,eot,svg,ttf,woff"></script> <style id="antiClickjack"> body > * { display: none !important; } #antiClickjack { display: block !important; } </style> <noscript> <style id="antiClickjackNoScript"> body > * { display: block !important; } #antiClickjack { display: none !important; } </style> </noscript> <script type="text/javascript" id="antiClickjackScript"> if (self === top) { // no clickjacking var antiClickjack = document.getElementById("antiClickjack"); antiClickjack.parentNode.removeChild(antiClickjack); } else { setTimeout(tryForward(), 5000); } function tryForward() { top.location = self.location; } </script>  <script async src="https://web.archive.org/web/20230203033640js_/https://www.googletagmanager.com/gtag/js?id=G-TSQ0PLGJZP"></script> <script>window.dataLayer = window.dataLayer || []; function gtag() { dataLayer.push(arguments); } gtag('js', new Date()); gtag('config', 'G-TSQ0PLGJZP');</script> </head> <body> <div id="antiClickjack" style="display: none;"> <strong style="font-size: 1.6rem;">You are viewing this page in an unauthorized frame window.</strong> <p>This is a potential security issue, you are being redirected to <a href="https://web.archive.org/web/20230203033640/https://csrc.nist.gov/">https://csrc.nist.gov</a>.</p> </div> <section class="usa-banner" aria-label="Official government website"> <div class="usa-accordion container"> <header class="usa-banner__header"> <noscript> <p style="font-size: 0.85rem; font-weight: bold;">You have JavaScript disabled. 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class="sr-only">Share to Facebook</span></a> <a href="https://web.archive.org/web/20230203033640/https://twitter.com/share?url=https%3A%2F%2Fcontent.csrc.e1a.nist.gov%2Fpublications%2Fdetail%2Fjournal-article%2F2012%2Ffamilies-of-elliptic-curves-with-rational-3-torsion" class="social-twitter"><i class="fa fa-twitter fa-fw" aria-hidden="true"></i><span class="sr-only">Share to Twitter</span></a> </div> <p class="hidden-lg hidden-md">     <a href="#pubs-documentation" class="btn btn-lg btn-info" id="pub-topics-anchor-sm">Documentation</a> </p> <div class="row"> <div class="col-md-8 col-sm-12 publication-panel"> <p> <strong>Published:</strong> <span id="pub-release-date">January 30, 2012</span><br/> <strong>Citation:</strong> <span id="pub-citation"><i>Journal of Mathematical Cryptology</i> vol. 5, no. 42433, (January 2012) pp. 225-246 </span><br/> </p> <h4>Author(s)</h4> <p> <strong id="pub-authors-container" data-total="2"> <span id="pub-author-0">Dustin Moody</span>, <span id="pub-author-1">H. Wu</span> </strong> </p> <h4>Announcement</h4> <p id="pub-announcement"></p> <div class="bs-callout bs-callout-success pub-abstract-callout"> <h4 id="pubs-abstract-header">Abstract</h4> <div class="hidden-sm hidden-xs hidden-xxs" id="pub-detail-abstract-info">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 Fq-isogeny classes of each family, as well as the number of Fq-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 j-invariant 0.</div> <div class="hidden-lg hidden-md"> <div id="pub-detail-abstract-min"> 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 Fq-isogeny classes of each family, as well as the number of... <a href="#pubs-abstract-header" id="pub-detail-abs-show">See full abstract</a> </div> <div id="pub-detail-abstract-all" style="display: none;"> 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 Fq-isogeny classes of each family, as well as the number of Fq-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 j-invariant 0.<br/> <a href="#pubs-abstract-header" id="pub-detail-abs-hide">Hide full abstract</a> </div> </div> <h4>Keywords</h4> <span id="pub-keywords-container" data-total="3"> <span id="pub-keyword-0">elliptic curves</span>; <span id="pub-keyword-1">Hessian curves</span>; <span id="pub-keyword-2">cryptography</span> </span> </div> <h5>Control Families</h5> <p> <span id="pub-control-fam-container" data-total="0">None selected</span> </p> </div> <div class="col-md-4 col-sm-12"> <div class="bs-callout bs-callout-success" id="pubs-documentation"> <h4>Documentation</h4> <p> <strong>Publication:</strong><br/> <a href="https://web.archive.org/web/20230203033640/https://doi.org/10.1515/jmc-2011-0013" data-identity="first-link-display" id="pub-doc-0"><i class="fa fa-external-link" aria-hidden="true"></i> Journal Article (DOI)</a><br/> </p> <p> <strong>Supplemental Material:</strong><br/> <span id="pub-supp-container" data-total="0">None available</span><br/> </p> <p> <strong>Document History:</strong><br/> <span id="pub-history-container" data-total="1"> 01/30/12: <span id="pub-history-link-0" data-current-document="true">Journal Article (Final)</span><br/> </span> </p> </div> </div> </div> </div> <div id="footer-pusher"></div> </div> <footer id="footer"> <div class="container"> <div class="row"> <div class="col-sm-6"> <span class="hidden-xs"> <a href="https://web.archive.org/web/20230203033640/https://www.nist.gov/" title="National Institute of Standards and Technology" rel="home" target="_blank" class="footer-nist-logo" id="footer-nist-logo-link"> <img src="/web/20230203033640im_/https://csrc.nist.gov/CSRC/Media/images/nist-logo-brand-white.svg" alt="National Institute of Standards and Technology logo" id="footer-nist-logo"/> </a> </span> <div class="row footer-contact-container"> <div 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Families of Elliptic Curves with Rational 3-torsion | CSRC