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href="#Elliptic_curves_over_ternary_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Elliptic curves over ternary fields</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Elliptic_curves_over_ternary_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pictures_and_intros" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pictures_and_intros"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Pictures and intros</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">5 comments</span> <ul id="toc-Pictures_and_intros-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resolved_issues" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Resolved_issues"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Resolved issues</span> </div> </a> <ul id="toc-Resolved_issues-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Factoring_link" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Factoring_link"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Factoring link</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Factoring_link-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cryptographic_schemes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cryptographic_schemes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Cryptographic schemes</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Cryptographic_schemes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_section_referencing_RSA_is_wrong." class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_section_referencing_RSA_is_wrong."> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>The section referencing RSA is wrong.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">16 comments</span> <ul id="toc-The_section_referencing_RSA_is_wrong.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curve25519" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Curve25519"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Curve25519</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Curve25519-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_Set_forms_a_Group?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_Set_forms_a_Group?"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>A Set forms a Group?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">3 comments</span> <ul id="toc-A_Set_forms_a_Group?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Design_choices_and_ECC" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Design_choices_and_ECC"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Design choices and ECC</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Design_choices_and_ECC-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Broken_link_in_reference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Broken_link_in_reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Broken link in reference</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Broken_link_in_reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generator" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generator"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Generator</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Generator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_link_one" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_link_one"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>External link one</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-External_link_one-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_Computing_Attack_Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantum_Computing_Attack_Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>Quantum Computing Attack Citations</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">4 comments</span> <ul id="toc-Quantum_Computing_Attack_Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Possible_NSA_backdoor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Possible_NSA_backdoor"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>Possible NSA backdoor</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">8 comments</span> <ul id="toc-Possible_NSA_backdoor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_Order" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Group_Order"> <div class="vector-toc-text"> <span class="vector-toc-numb">18</span> <span>Group Order</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul id="toc-Group_Order-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links_modified" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links_modified"> <div class="vector-toc-text"> <span class="vector-toc-numb">19</span> <span>External links modified</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-External_links_modified-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links_modified_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links_modified_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">20</span> <span>External links modified</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-External_links_modified_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links_modified_3" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links_modified_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">21</span> <span>External links modified</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-External_links_modified_3-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithm_needed" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithm_needed"> <div class="vector-toc-text"> <span class="vector-toc-numb">22</span> <span>Algorithm needed</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Algorithm_needed-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_first_paragraph_seems_self-contradictory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_first_paragraph_seems_self-contradictory"> <div class="vector-toc-text"> <span class="vector-toc-numb">23</span> <span>The first paragraph seems self-contradictory</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-The_first_paragraph_seems_self-contradictory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Not_a_typo._521,_not_512." class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Not_a_typo._521,_not_512."> <div class="vector-toc-text"> <span class="vector-toc-numb">24</span> <span>Not a typo. 521, not 512.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Not_a_typo._521,_not_512.-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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.class-unassessed{background-color:#DCDCDC}.mw-parser-output .conflict{border:0.2em solid red}.mw-parser-output .inactive-wikiproject .wpb-header-bubbles{color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .wpb-category-box{background-color:#0a0a0a}html.skin-theme-clientpref-night .mw-parser-output .wpbs .wpb .wpb-main,html.skin-theme-clientpref-night .mw-parser-output .wpbs .wpb-header{background-color:#302f2d}html.skin-theme-clientpref-night .mw-parser-output .wpbs .banner-shell-inner{background-color:#2e2505}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .wpb-category-box{background-color:#0a0a0a}html.skin-theme-clientpref-os .mw-parser-output .wpbs .wpb .wpb-main,html.skin-theme-clientpref-os .mw-parser-output .wpbs .wpb-header{background-color:#302f2d}html.skin-theme-clientpref-os .mw-parser-output .wpbs .banner-shell-inner{background-color:#2e2505}}</style><table class="tmbox tmbox-notice mw-collapsible innercollapse wpb wpb-table"><tbody><tr class="wpb-header"><td class="wpb-header-icon"><span typeof="mw:File"><a href="/wiki/File:Crypto_key.svg" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Crypto_key.svg/35px-Crypto_key.svg.png" decoding="async" width="35" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Crypto_key.svg/53px-Crypto_key.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Crypto_key.svg/70px-Crypto_key.svg.png 2x" data-file-width="651" data-file-height="271" /></a></span></td><td class="wpb-header-combined"><a href="/wiki/Wikipedia:WikiProject_Cryptography" title="Wikipedia:WikiProject Cryptography">Cryptography</a><span class="wpb-nested-task-force">: <a href="/wiki/Wikipedia:WikiProject_Computer_science" title="Wikipedia:WikiProject Computer science">Computer science</a></span> <span class="wpb-header-bubbles import-high">High‑importance</span></td></tr><tr><td class="mbox-text wpb-main" colspan="2"><table><tbody><tr><td class="mbox-image wpb-image"><span typeof="mw:File"><a href="/wiki/File:Crypto_key.svg" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Crypto_key.svg/64px-Crypto_key.svg.png" decoding="async" width="64" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Crypto_key.svg/96px-Crypto_key.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Crypto_key.svg/128px-Crypto_key.svg.png 2x" data-file-width="651" data-file-height="271" /></a></span></td><td class="mbox-text">This article is within the scope of <b><a href="/wiki/Wikipedia:WikiProject_Cryptography" title="Wikipedia:WikiProject Cryptography">WikiProject Cryptography</a></b>, a collaborative effort to improve the coverage of <a href="/wiki/Cryptography" title="Cryptography">Cryptography</a> on Wikipedia. If you would like to participate, please visit the project page, where you can join <a href="/wiki/Wikipedia_talk:WikiProject_Cryptography" title="Wikipedia talk:WikiProject Cryptography">the discussion</a> and see a list of open tasks.<span class="metadata wpb-metadata"><span class="wpb-project">Cryptography</span><span class="wpb-project_link">Wikipedia:WikiProject Cryptography</span><span class="wpb-banner_name">Template:WikiProject Cryptography</span><span class="wpb-assessment_cat">Cryptography</span></span></td><td class="mbox-empty-cell"></td></tr><tr><td class="assess import-high"><a href="/wiki/Category:High-importance_Cryptography_articles" title="Category:High-importance Cryptography articles">High</a></td><td class="mbox-text" colspan="2">This article has been rated as <b>High-importance</b> on the <a href="/wiki/Wikipedia:Version_1.0_Editorial_Team/Release_Version_Criteria#Importance_of_topic" title="Wikipedia:Version 1.0 Editorial Team/Release Version Criteria">importance scale</a>.</td></tr><tr><td><figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:LampFlowchart.svg" class="mw-file-description"><img alt="Taskforce icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/LampFlowchart.svg/18px-LampFlowchart.svg.png" decoding="async" width="18" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/LampFlowchart.svg/28px-LampFlowchart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/LampFlowchart.svg/37px-LampFlowchart.svg.png 2x" data-file-width="324" data-file-height="442" /></a><figcaption></figcaption></figure></td><td class="mbox-text" colspan="2">This article is supported by <b><a href="/wiki/Wikipedia:WikiProject_Computer_science" title="Wikipedia:WikiProject Computer science">WikiProject Computer science</a></b> (assessed as <a href="/wiki/Category:High-importance_Computer_science_articles" title="Category:High-importance Computer science articles">High-importance</a>).</td></tr></tbody></table></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1259570071" /><table class="tmbox tmbox-notice mw-collapsible innercollapse wpb wpb-table"><tbody><tr class="wpb-header"><td class="wpb-header-icon"><span typeof="mw:File"><a href="/wiki/File:AIGA_Currency_Exchange_-_Euro.svg" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/AIGA_Currency_Exchange_-_Euro.svg/27px-AIGA_Currency_Exchange_-_Euro.svg.png" decoding="async" width="27" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/AIGA_Currency_Exchange_-_Euro.svg/41px-AIGA_Currency_Exchange_-_Euro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/AIGA_Currency_Exchange_-_Euro.svg/54px-AIGA_Currency_Exchange_-_Euro.svg.png 2x" data-file-width="446" data-file-height="416" /></a></span></td><td class="wpb-header-combined"><a href="/wiki/Wikipedia:WikiProject_Numismatics" title="Wikipedia:WikiProject Numismatics">Numismatics</a> <span class="wpb-header-bubbles import-low">Low‑importance</span></td></tr><tr><td class="mbox-text wpb-main" colspan="2"><table><tbody><tr><td class="mbox-image wpb-image"><span typeof="mw:File"><a href="/wiki/File:AIGA_Currency_Exchange_-_Euro.svg" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/AIGA_Currency_Exchange_-_Euro.svg/60px-AIGA_Currency_Exchange_-_Euro.svg.png" decoding="async" width="60" height="56" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/AIGA_Currency_Exchange_-_Euro.svg/90px-AIGA_Currency_Exchange_-_Euro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/AIGA_Currency_Exchange_-_Euro.svg/120px-AIGA_Currency_Exchange_-_Euro.svg.png 2x" data-file-width="446" data-file-height="416" /></a></span></td><td class="mbox-text"><style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/United_States_penny%2C_obverse%2C_2002.png/28px-United_States_penny%2C_obverse%2C_2002.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/United_States_penny%2C_obverse%2C_2002.png/42px-United_States_penny%2C_obverse%2C_2002.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/United_States_penny%2C_obverse%2C_2002.png/56px-United_States_penny%2C_obverse%2C_2002.png 2x" data-file-width="903" data-file-height="901" /></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Numismatics" title="Portal:Numismatics">Numismatics portal</a></span></li></ul>This article is within the scope of <b><a href="/wiki/Wikipedia:WikiProject_Numismatics" title="Wikipedia:WikiProject Numismatics">WikiProject Numismatics</a></b>, a collaborative effort to improve the coverage of <a href="/wiki/Numismatics" title="Numismatics">numismatics</a> and <a href="/wiki/Currencies" class="mw-redirect" title="Currencies">currencies</a> on Wikipedia. If you would like to participate, please visit the project page, where you can join <a href="/wiki/Wikipedia_talk:WikiProject_Numismatics" title="Wikipedia talk:WikiProject Numismatics">the discussion</a> and see a list of open tasks.<span class="metadata wpb-metadata"><span class="wpb-project">Numismatics</span><span class="wpb-project_link">Wikipedia:WikiProject Numismatics</span><span class="wpb-banner_name">Template:WikiProject Numismatics</span><span class="wpb-assessment_cat">numismatic</span></span></td><td class="mbox-empty-cell"></td></tr><tr><td class="assess import-low"><a href="/wiki/Category:Low-importance_numismatic_articles" title="Category:Low-importance numismatic articles">Low</a></td><td class="mbox-text" colspan="2">This article has been rated as <b>Low-importance</b> on the <a href="/wiki/Wikipedia:WikiProject_Numismatics/Assessment#Importance_scale" title="Wikipedia:WikiProject Numismatics/Assessment">project's importance scale</a>.</td></tr></tbody></table></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1259570071" /><table class="tmbox tmbox-notice mw-collapsible innercollapse wpb wpb-table"><tbody><tr class="wpb-header"><td class="wpb-header-icon"><span typeof="mw:File"><a href="/wiki/File:Cryptocurrency_Gold.png" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Cryptocurrency_Gold.png/26px-Cryptocurrency_Gold.png" decoding="async" width="26" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Cryptocurrency_Gold.png/40px-Cryptocurrency_Gold.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Cryptocurrency_Gold.png/53px-Cryptocurrency_Gold.png 2x" data-file-width="633" data-file-height="600" /></a></span></td><td class="wpb-header-combined"><a href="/wiki/Wikipedia:WikiProject_Cryptocurrency" title="Wikipedia:WikiProject Cryptocurrency">Cryptocurrency</a> <span class="wpb-header-bubbles import-mid">Mid‑importance</span></td></tr><tr><td class="mbox-text wpb-main" colspan="2"><table><tbody><tr><td class="mbox-image wpb-image"><span typeof="mw:File"><a href="/wiki/File:Cryptocurrency_Gold.png" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Cryptocurrency_Gold.png/80px-Cryptocurrency_Gold.png" decoding="async" width="80" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Cryptocurrency_Gold.png/120px-Cryptocurrency_Gold.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Cryptocurrency_Gold.png/160px-Cryptocurrency_Gold.png 2x" data-file-width="633" data-file-height="600" /></a></span></td><td class="mbox-text">This article is within the scope of <b><a href="/wiki/Wikipedia:WikiProject_Cryptocurrency" title="Wikipedia:WikiProject Cryptocurrency">WikiProject Cryptocurrency</a></b>, a collaborative effort to improve the coverage of <a href="/wiki/Cryptocurrency" title="Cryptocurrency">cryptocurrency</a> on Wikipedia. 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If you would like to participate, please visit the project page, where you can join <a href="/wiki/Wikipedia_talk:WikiProject_Mathematics" title="Wikipedia talk:WikiProject Mathematics">the discussion</a> and see a list of open tasks.<span class="metadata wpb-metadata"><span class="wpb-project">Mathematics</span><span class="wpb-project_link">Wikipedia:WikiProject Mathematics</span><span class="wpb-banner_name">Template:WikiProject Mathematics</span><span class="wpb-assessment_cat">mathematics</span></span></td><td class="mbox-empty-cell"></td></tr><tr><td class="assess import-mid"><a href="/wiki/Category:Mid-priority_mathematics_articles" title="Category:Mid-priority mathematics articles">Mid</a></td><td class="mbox-text" colspan="2">This article has been rated as <b>Mid-priority</b> on the <a href="/wiki/Wikipedia:WikiProject_Mathematics/Wikipedia_1.0/Assessment#Priority_scale" class="mw-redirect" title="Wikipedia:WikiProject Mathematics/Wikipedia 1.0/Assessment">project's priority scale</a>.</td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table> <p><br /> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Security_analysis_of_ECC" data-mw-thread-id="h-Security_analysis_of_ECC-20230804141800"><span data-mw-comment-start="" id="h-Security_analysis_of_ECC-20230804141800"></span>Security analysis of ECC<span data-mw-comment-end="h-Security_analysis_of_ECC-20230804141800"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=1" title="Edit section: Security analysis of ECC"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-153.46.253.213-20230804141800-Security_analysis_of_ECC"></span>Interesting research and summary of security for different ECC: <a rel="nofollow" class="external free" href="https://safecurves.cr.yp.to/">https://safecurves.cr.yp.to/</a> <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/153.46.253.213" title="Special:Contributions/153.46.253.213">153.46.253.213</a> (<a href="/wiki/User_talk:153.46.253.213#top" title="User talk:153.46.253.213">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-153.46.253.213-20230804141800-Security_analysis_of_ECC" class="ext-discussiontools-init-timestamplink">14:18, 4 August 2023 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-153.46.253.213-20230804141800-Security_analysis_of_ECC"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-153.46.253.213-20230804141800-Security_analysis_of_ECC"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Cite_required" data-mw-thread-id="h-Cite_required-2018-08-13T17:23:00.000Z"><span data-mw-comment-start="" id="h-Cite_required-2018-08-13T17:23:00.000Z"></span>Cite required<span data-mw-comment-end="h-Cite_required-2018-08-13T17:23:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=2" title="Edit section: Cite required"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bdamm-2018-08-13T17:23:00.000Z-Cite_required"></span>From an earlier revision of the article: </p><p><i>For comparison, in 2001 some experts are suggesting these sizes for various public key systems for a security level appropriate to major business transactions that require secrecy:</i> </p><p>RSA (based on difficulty of factorisation) 1024 bits. </p><p>DSA (based on difficulty of discrete log for integers modulo a prime) 1024 bits. </p><p>ECC (based on difficulty of discrete log for discrete ECC system) 200 bits.<i></i> </p><p>I have removed this until it can be backed up firmly by a cite - instead, I have added external links to research papers in this field. -- <a href="/wiki/User:The_Anome" title="User:The Anome">The Anome</a> </p><p>I refer you to <a href="/wiki/Wikipedia:What_Wikipedia_is_not" title="Wikipedia:What Wikipedia is not">What Wikipedia is not</a>, item 9, and <a href="/wiki/Wikipedia:Most_common_Wikipedia_faux_pas" class="mw-redirect" title="Wikipedia:Most common Wikipedia faux pas">Most common Wikipedia faux pas</a> "Deleting useful content". You have deleted some useful inline information and replaced it with external links. Bad idea. If you actually know anything about this subject and don't like my numbers, then change them, they are fairly fuzzy and there is no recognized reliable method for generating them. But don't delete them. You didn't even give a reason for deleting them. It is NOT necessary to give a cite for every single factlet on the whole of Wikipedia, and lack of a cite is NOT a good reason to delete content. I'll be back in a few days to revert the edit and maybe add some more discussion. -- Geronimo Jones </p><p>See www.nist.gov/encryption for a list of recommended elliptic curves. ANSI X9 requires a minimum of 80 bits of *symmetric key equivalent* security. THis means use of SHA-1 with 160 bit output, use of RSA/DSA with 1024 bit keys and use of ECC with 160 bit keys. Don Johnson </p><p>The references of 256 bit ECC keys providing 128-bit security need citation. <a href="/wiki/User:Bdamm" title="User:Bdamm">Bdamm</a> (<a href="/w/index.php?title=User_talk:Bdamm&action=edit&redlink=1" class="new" title="User talk:Bdamm (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Bdamm-2018-08-13T17:23:00.000Z-Cite_required" class="ext-discussiontools-init-timestamplink">17:23, 13 August 2018 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bdamm-2018-08-13T17:23:00.000Z-Cite_required"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Bdamm-2018-08-13T17:23:00.000Z-Cite_required"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Non-mathematical_description_needed" data-mw-thread-id="h-Non-mathematical_description_needed-2006-04-16T12:10:00.000Z"><span data-mw-comment-start="" id="h-Non-mathematical_description_needed-2006-04-16T12:10:00.000Z"></span>Non-mathematical description needed<span data-mw-comment-end="h-Non-mathematical_description_needed-2006-04-16T12:10:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=3" title="Edit section: Non-mathematical description needed"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-ArnoldReinhold-2006-04-16T12:10:00.000Z-Non-mathematical_description_needed"></span>I'm sure as ECC becomes more common, lay-people will be looking for information about it. A lot of these people (like me) are rather put off by seeing mathematical functions in the introductory section. Could someone write a lay description of ECC that doesn't use mathematical symbols? </p> <dl><dd>I tried to do this. --<a href="/wiki/User:ArnoldReinhold" title="User:ArnoldReinhold">agr</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-ArnoldReinhold-2006-04-16T12:10:00.000Z-Non-mathematical_description_needed" class="ext-discussiontools-init-timestamplink">12:10, 16 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-ArnoldReinhold-2006-04-16T12:10:00.000Z-Non-mathematical_description_needed"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-ArnoldReinhold-2006-04-16T12:10:00.000Z-Non-mathematical_description_needed"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"></span>A completely non-mathematical description of ECC is no more than that of PK crypto, so I doubt that it is actually possible to have it. <a href="/w/index.php?title=User:GBL&action=edit&redlink=1" class="new" title="User:GBL (page does not exist)">GBL</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z" class="ext-discussiontools-init-timestamplink">16:48, 20 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"></span></dd></dl></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-82.43.137.103-2007-02-18T00:08:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"></span>I agree with the original poster - there needs to be an intermediate paragraph describing what's happening in simplified, analogous text before it launches into all the TeX stuff. <a href="/w/index.php?title=User:82.43.137.103&action=edit&redlink=1" class="new" title="User:82.43.137.103 (page does not exist)">82.43.137.103</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-82.43.137.103-2007-02-18T00:08:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z" class="ext-discussiontools-init-timestamplink">00:08, 18 February 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-82.43.137.103-2007-02-18T00:08:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-82.43.137.103-2007-02-18T00:08:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-93.97.48.217-2010-08-01T11:50:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"></span>I agree, this is endemic on wikipedia maths articles. <a href="/wiki/Special:Contributions/93.97.48.217" title="Special:Contributions/93.97.48.217">93.97.48.217</a> (<a href="/wiki/User_talk:93.97.48.217" title="User talk:93.97.48.217">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-93.97.48.217-2010-08-01T11:50:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z" class="ext-discussiontools-init-timestamplink">11:50, 1 August 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-93.97.48.217-2010-08-01T11:50:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-93.97.48.217-2010-08-01T11:50:00.000Z-ArnoldReinhold-2006-04-16T12:10:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-74.93.34.241-2010-04-19T13:38:00.000Z-Non-mathematical_description_needed"></span>I think a simplified mathematical explanation is desperately needed. It should be possible to make an example that uses numbers small enough to fit on an 8 digit calculator and is more easily understood. How is key generation done? How do I use the keys to en/de-crypt something. I understand math, but I'm not a genius or have a PhD in it. How about explaining this in a way that a common person can understand it? Note that they did a fairly good job explaining RSA on that page, but have dropped the ball here. <span style="font-size: smaller;" class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/74.93.34.241" title="Special:Contributions/74.93.34.241">74.93.34.241</a> (<a href="/w/index.php?title=User_talk:74.93.34.241&action=edit&redlink=1" class="new" title="User talk:74.93.34.241 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-74.93.34.241-2010-04-19T13:38:00.000Z-Non-mathematical_description_needed" class="ext-discussiontools-init-timestamplink">13:38, 19 April 2010 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-74.93.34.241-2010-04-19T13:38:00.000Z-Non-mathematical_description_needed"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-74.93.34.241-2010-04-19T13:38:00.000Z-Non-mathematical_description_needed"></span> </p><p><span data-mw-comment-start="" id="c-72.177.115.161-2012-01-07T06:58:00.000Z-Non-mathematical_description_needed"></span>+1 <a href="/wiki/Special:Contributions/72.177.115.161" title="Special:Contributions/72.177.115.161">72.177.115.161</a> (<a href="/w/index.php?title=User_talk:72.177.115.161&action=edit&redlink=1" class="new" title="User talk:72.177.115.161 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-72.177.115.161-2012-01-07T06:58:00.000Z-Non-mathematical_description_needed" class="ext-discussiontools-init-timestamplink">06:58, 7 January 2012 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-72.177.115.161-2012-01-07T06:58:00.000Z-Non-mathematical_description_needed"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-72.177.115.161-2012-01-07T06:58:00.000Z-Non-mathematical_description_needed"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Elliptic_curves_over_ternary_fields" data-mw-thread-id="h-Elliptic_curves_over_ternary_fields-2006-04-02T14:25:00.000Z"><span data-mw-comment-start="" id="h-Elliptic_curves_over_ternary_fields-2006-04-02T14:25:00.000Z"></span>Elliptic curves over ternary fields<span data-mw-comment-end="h-Elliptic_curves_over_ternary_fields-2006-04-02T14:25:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=4" title="Edit section: Elliptic curves over ternary fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Augustojd-2006-04-02T14:25:00.000Z-Elliptic_curves_over_ternary_fields"></span>Hello. In the introduction, the article states that elliptic curves used in cryptography are defined over prime or binary fields. However, mainly due to pairing-based cryptography, there has been interest in elliptic curves over ternary fields as well. <a href="/wiki/User:Augustojd" title="User:Augustojd">augustojd</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Augustojd-2006-04-02T14:25:00.000Z-Elliptic_curves_over_ternary_fields" class="ext-discussiontools-init-timestamplink">14:25, 2 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Augustojd-2006-04-02T14:25:00.000Z-Elliptic_curves_over_ternary_fields"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Augustojd-2006-04-02T14:25:00.000Z-Elliptic_curves_over_ternary_fields"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Pictures_and_intros" data-mw-thread-id="h-Pictures_and_intros-2006-04-18T08:29:00.000Z"><span data-mw-comment-start="" id="h-Pictures_and_intros-2006-04-18T08:29:00.000Z"></span>Pictures and intros<span data-mw-comment-end="h-Pictures_and_intros-2006-04-18T08:29:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=5" title="Edit section: Pictures and intros"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-GBL-2006-04-18T08:29:00.000Z-Pictures_and_intros"></span>Removed from todo: </p> <dl><dd>Please add a graph such as [a picture of EC over real numbers]</dd></dl> <p>If a picture does not communicate any information there is no reason to include it (there is already such a picture in EC—there is no need to copy it to ECC). BTW, this talk page needs major clean up <a href="/w/index.php?title=User:GBL&action=edit&redlink=1" class="new" title="User:GBL (page does not exist)">GBL</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-GBL-2006-04-18T08:29:00.000Z-Pictures_and_intros" class="ext-discussiontools-init-timestamplink">08:29, 18 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-GBL-2006-04-18T08:29:00.000Z-Pictures_and_intros"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-GBL-2006-04-18T08:29:00.000Z-Pictures_and_intros"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Matt_Crypto-2006-04-18T08:45:00.000Z-GBL-2006-04-18T08:29:00.000Z"></span>I don't think we need the mathematic intro, either. A reader can read the EC article if they need it. <a href="/wiki/User:Matt_Crypto" title="User:Matt Crypto">— Matt <small>Crypto</small></a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Matt_Crypto-2006-04-18T08:45:00.000Z-GBL-2006-04-18T08:29:00.000Z" class="ext-discussiontools-init-timestamplink">08:45, 18 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Matt_Crypto-2006-04-18T08:45:00.000Z-GBL-2006-04-18T08:29:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Matt_Crypto-2006-04-18T08:45:00.000Z-GBL-2006-04-18T08:29:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-ArnoldReinhold-2006-04-18T09:18:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z"></span>There isn't much overlap between the math inro here and the EC article. ECC is a very specialized application of EC. Thanks for the cleanup here, btw. --<a href="/wiki/User:ArnoldReinhold" title="User:ArnoldReinhold">agr</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-ArnoldReinhold-2006-04-18T09:18:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z" class="ext-discussiontools-init-timestamplink">09:18, 18 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-ArnoldReinhold-2006-04-18T09:18:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-ArnoldReinhold-2006-04-18T09:18:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-18T09:18:00.000Z"></span>IMO <i>Mathematical introduction</i> is needed since it is about EC over finite fields and it is not described elsewhere, OTOH <i>Introduction</i> is about PK crypto and general EC and thus can be safely removed here and merged into PK and EC. <a href="/w/index.php?title=User:GBL&action=edit&redlink=1" class="new" title="User:GBL (page does not exist)">GBL</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-18T09:18:00.000Z" class="ext-discussiontools-init-timestamplink">16:48, 20 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-18T09:18:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-GBL-2006-04-20T16:48:00.000Z-ArnoldReinhold-2006-04-18T09:18:00.000Z"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-ArnoldReinhold-2006-04-21T12:26:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z"></span>Sorry, I don't agree. Even if a purely non-technical intro is not feasible, a less technical intro that summarizes the subject is totally appropriate and badly needed in this case.--<a href="/wiki/User:ArnoldReinhold" title="User:ArnoldReinhold">agr</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-ArnoldReinhold-2006-04-21T12:26:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z" class="ext-discussiontools-init-timestamplink">12:26, 21 April 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-ArnoldReinhold-2006-04-21T12:26:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-ArnoldReinhold-2006-04-21T12:26:00.000Z-Matt_Crypto-2006-04-18T08:45:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Resolved_issues" data-mw-thread-id="h-Resolved_issues"><span data-mw-comment-start="" id="h-Resolved_issues"></span>Resolved issues<span data-mw-comment-end="h-Resolved_issues"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=6" title="Edit section: Resolved issues"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>mathematical description of ECC was added</li> <li>109-bit EC provides only 55 bits of security</li> <li>a sentence <i>given integers j and k ...</i> was revised (it is not in the article any more)</li> <li>MQV</li> <li><i>Victor Miller</i> link was incorrect</li></ul> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Factoring_link" data-mw-thread-id="h-Factoring_link-2006-08-10T16:21:00.000Z"><span data-mw-comment-start="" id="h-Factoring_link-2006-08-10T16:21:00.000Z"></span>Factoring link<span data-mw-comment-end="h-Factoring_link-2006-08-10T16:21:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=7" title="Edit section: Factoring link"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Lordspaz-2006-08-10T16:21:00.000Z-Factoring_link"></span>The link for factoring in "recent advances in factoring" points to the general factorization article; wouldn't the Integer factorization article be more appropriate in this case? <a href="/w/index.php?title=User:Lordspaz&action=edit&redlink=1" class="new" title="User:Lordspaz (page does not exist)">lordspaz</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Lordspaz-2006-08-10T16:21:00.000Z-Factoring_link" class="ext-discussiontools-init-timestamplink">16:21, 10 August 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Lordspaz-2006-08-10T16:21:00.000Z-Factoring_link"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Lordspaz-2006-08-10T16:21:00.000Z-Factoring_link"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Cryptographic_schemes" data-mw-thread-id="h-Cryptographic_schemes-2006-08-27T11:13:00.000Z"><span data-mw-comment-start="" id="h-Cryptographic_schemes-2006-08-27T11:13:00.000Z"></span>Cryptographic schemes<span data-mw-comment-end="h-Cryptographic_schemes-2006-08-27T11:13:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=8" title="Edit section: Cryptographic schemes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-83.64.176.129-2006-08-27T11:13:00.000Z-Cryptographic_schemes"></span>Just to note... </p><p>> (Another factor is that ElGamal scheme is vulnerable to chosen-ciphertext attacks.) That's certainly not a real factor as e.g. plain RSA is vulnerable to chosen-ciphertext attacks as well. That's what the padding schemes are for (PCKS, OAEP, SAEP...). </p><p>> ...cryptography based on integer factorization (e.g., RSA) and finite-field cryptography (e.g., DSA). Well, both RSA and standard DSA are based on finite-field cryptography. </p><p><a href="/w/index.php?title=User:83.64.176.129&action=edit&redlink=1" class="new" title="User:83.64.176.129 (page does not exist)">83.64.176.129</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-83.64.176.129-2006-08-27T11:13:00.000Z-Cryptographic_schemes" class="ext-discussiontools-init-timestamplink">11:13, 27 August 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-83.64.176.129-2006-08-27T11:13:00.000Z-Cryptographic_schemes"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-83.64.176.129-2006-08-27T11:13:00.000Z-Cryptographic_schemes"></span> </p><p><span data-mw-comment-start="" id="c-BrunoX-2006-11-29T16:24:00.000Z-Cryptographic_schemes"></span>==== Actually, RSA is based on Rings, not finite-field - <a href="/wiki/User:BrunoX" title="User:BrunoX">BrunoX</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-BrunoX-2006-11-29T16:24:00.000Z-Cryptographic_schemes" class="ext-discussiontools-init-timestamplink">16:24, 29 November 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-BrunoX-2006-11-29T16:24:00.000Z-Cryptographic_schemes"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-BrunoX-2006-11-29T16:24:00.000Z-Cryptographic_schemes"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="The_section_referencing_RSA_is_wrong." data-mw-thread-id="h-The_section_referencing_RSA_is_wrong.-2008-02-27T21:09:00.000Z"><span data-mw-comment-start="" id="h-The_section_referencing_RSA_is_wrong.-2008-02-27T21:09:00.000Z"></span>The section referencing RSA is wrong.<span data-mw-comment-end="h-The_section_referencing_RSA_is_wrong.-2008-02-27T21:09:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=9" title="Edit section: The section referencing RSA is wrong."><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Grendelkhan-2008-02-27T21:09:00.000Z-The_section_referencing_RSA_is_wrong."></span>The introductory states that "... a user picks two large random primes as his private key, and publishes their product as his public key. The difficulty of factoring ensures that no one else can derive the private key (i.e., the two prime factors) from the public one within a reasonable amount of time." This is wrong. Consider the article <a href="/wiki/RSA#Operation" class="mw-disambig" title="RSA">RSA</a>; in short, RSA generates two primes, <i>p</i> and <i>q</i>, but these are <i>not</i> the private key. The user then creates two exponents <i>d</i> and <i>e</i>, such that <i>de</i> = <i>k</i>(<i>p</i>-1)(<i>q</i>-1) for some <i>k</i>. (There are other restrictions on <i>e</i>, and I'm unsure if the two are really interchangeable.) Unless certain shortcuts are taken, both <i>p</i> and <i>q</i> are deleted at the end of the key generation process (though <i>n</i> = <i>pq</i> is retained). </p><p>In any case, this is a rather crucial distinction: in the system described in the article currently, the public key doesn't contain any information that the holder of the private key (assuming they somehow don't have the public key) doesn't already have, and so it doesn't make sense that it could be used to encrypt data that only they could decrypt. </p><p>I've rewritten it, but I'm not very happy with my layman's explanations of things. Future editors, please reference the actual operation of RSA before writing about it; there are a lot of misconceptions about cryptography out there. <a href="/wiki/User:Grendelkhan" title="User:Grendelkhan">grendel</a>|<a href="/wiki/User_talk:Grendelkhan" title="User talk:Grendelkhan">khan</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Grendelkhan-2008-02-27T21:09:00.000Z-The_section_referencing_RSA_is_wrong." class="ext-discussiontools-init-timestamplink">21:09, 27 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Grendelkhan-2008-02-27T21:09:00.000Z-The_section_referencing_RSA_is_wrong."><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Grendelkhan-2008-02-27T21:09:00.000Z-The_section_referencing_RSA_is_wrong."></span> </p><p><br /><span data-mw-comment-start="" id="c-DRLB-2008-02-27T21:34:00.000Z-The_section_referencing_RSA_is_wrong."></span> In <a href="/wiki/RSA" class="mw-disambig" title="RSA">RSA</a>, the private key has several equivalent forms, including (<i>n</i>,<i>d</i>) and (<i>p</i>,<i>q</i>). The previous version article was written using the latter in mind, which is fine. This emphasizes the dependence of RSA on <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a>, while ignoring other details (such as the <a href="/wiki/RSA_problem" title="RSA problem">RSA problem</a> being required to hard too). </p><p>With this new edit, the article now appears to suggest that <i>p</i>, one of the primes, is to be included in the public key. This would be wrong. Given <i>n</i> and <i>p</i>, one can recover <i>q</i>, and therefore determine the private key. From your talk page comment above, I gather that you meant <i>e</i> was the value to be made public, not <i>p</i>, but this was not made sufficiently clear in the article edit. </p><p>Either the current version should be clarified, or the article should be put back the way it was. <a href="/wiki/User:DRLB" title="User:DRLB">DRLB</a> (<a href="/wiki/User_talk:DRLB" title="User talk:DRLB">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-DRLB-2008-02-27T21:34:00.000Z-The_section_referencing_RSA_is_wrong." class="ext-discussiontools-init-timestamplink">21:34, 27 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-DRLB-2008-02-27T21:34:00.000Z-The_section_referencing_RSA_is_wrong."><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-DRLB-2008-02-27T21:34:00.000Z-The_section_referencing_RSA_is_wrong."></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Grendelkhan-2008-02-28T21:30:00.000Z-DRLB-2008-02-27T21:34:00.000Z"></span>Yeah, I see how it looks like I'm talking about <i>p</i> and <i>q</i> rather than <i>d</i> and <i>e</i> (the user does create two primes, but they use those to create two <i>different</i> primes which are actually used in the keys--I don't stress that there are two sets of primes). Is it particularly vital to make this distinction? I'm not sure it adds anything to a layman's overview. <a href="/wiki/User:Grendelkhan" title="User:Grendelkhan">grendel</a>|<a href="/wiki/User_talk:Grendelkhan" title="User talk:Grendelkhan">khan</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Grendelkhan-2008-02-28T21:30:00.000Z-DRLB-2008-02-27T21:34:00.000Z" class="ext-discussiontools-init-timestamplink">21:30, 28 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Grendelkhan-2008-02-28T21:30:00.000Z-DRLB-2008-02-27T21:34:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Grendelkhan-2008-02-28T21:30:00.000Z-DRLB-2008-02-27T21:34:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-DRLB-2008-02-28T22:47:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z"></span>Well, neither <i>d</i> nor <i>e</i> need to be prime (although in practice <i>e</i> is often a fixed, but not a random prime), so by saying one prime is included in the public key and the other in the private key is strongly suggestive that <i>p</i> is made public, especially to those already familiar with RSA, as typically, the first thing one learns about RSA about is the secret primes <i>p</i> and <i>q</i>, whose product is difficult to factor. Maybe the article's reference to RSA should be simplified, just to say that is another public key cryptography scheme, which requires integer factorization to be hard, letting <a href="/wiki/RSA" class="mw-disambig" title="RSA">RSA</a> explain the details. <small>—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:DRLB" title="User:DRLB">DRLB</a> (<a href="/wiki/User_talk:DRLB" title="User talk:DRLB">talk</a> • <a href="/wiki/Special:Contributions/DRLB" title="Special:Contributions/DRLB">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-DRLB-2008-02-28T22:47:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z" class="ext-discussiontools-init-timestamplink">22:47, 28 February 2008 (UTC)</a><span data-mw-comment-end="c-DRLB-2008-02-28T22:47:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z"></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-DRLB-2008-02-28T22:47:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span></dd></dl></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Brusegadi-2008-02-28T05:07:00.000Z-DRLB-2008-02-27T21:34:00.000Z"></span>To keep it simple the private is (p,q) and the public N=p*q. e and d can be introduced in the RSA article along with the RSA problem. <a href="/wiki/User:Brusegadi" title="User:Brusegadi">Brusegadi</a> (<a href="/wiki/User_talk:Brusegadi" title="User talk:Brusegadi">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Brusegadi-2008-02-28T05:07:00.000Z-DRLB-2008-02-27T21:34:00.000Z" class="ext-discussiontools-init-timestamplink">05:07, 28 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Brusegadi-2008-02-28T05:07:00.000Z-DRLB-2008-02-27T21:34:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Brusegadi-2008-02-28T05:07:00.000Z-DRLB-2008-02-27T21:34:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Grendelkhan-2008-02-28T21:30:00.000Z-Brusegadi-2008-02-28T05:07:00.000Z"></span>That's still not really accurate. The public key can't just include <i>n</i>; it also has to include <i>e</i>. The essential nature of RSA is that key operations are symmetric: what can be done with one key can only be undone with the other, and vice versa. The system as described is very much a one-way scheme, where the public key can be derived solely from the private key, but not vice versa, which is not how RSA works. The rewritten explanation describes it properly, as does somehow shoehorning in a mention of <i>e</i> in the original description. We shouldn't sacrifice accuracy while we're trying to make the explanation simpler. <a href="/wiki/User:Grendelkhan" title="User:Grendelkhan">grendel</a>|<a href="/wiki/User_talk:Grendelkhan" title="User talk:Grendelkhan">khan</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Grendelkhan-2008-02-28T21:30:00.000Z-Brusegadi-2008-02-28T05:07:00.000Z" class="ext-discussiontools-init-timestamplink">21:30, 28 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Grendelkhan-2008-02-28T21:30:00.000Z-Brusegadi-2008-02-28T05:07:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Grendelkhan-2008-02-28T21:30:00.000Z-Brusegadi-2008-02-28T05:07:00.000Z"></span></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-ArnoldReinhold-2008-02-28T22:54:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z"></span>While it's true that to use a public key, one has to also know the value of e, it is often not specified explicitly, rather all users of a given encryption product typically agree on a default value for e. For example, PGP uses 17 by default and will try other popular values. So e is a technicality that does not have to be discussed here. It has no confidentiality implications in usual practice. All the public key confidentiality is in <i>n.</i> --<a href="/wiki/User:ArnoldReinhold" title="User:ArnoldReinhold">agr</a> (<a href="/wiki/User_talk:ArnoldReinhold" title="User talk:ArnoldReinhold">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-ArnoldReinhold-2008-02-28T22:54:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z" class="ext-discussiontools-init-timestamplink">22:54, 28 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-ArnoldReinhold-2008-02-28T22:54:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-ArnoldReinhold-2008-02-28T22:54:00.000Z-Grendelkhan-2008-02-28T21:30:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Brusegadi-2008-02-29T03:02:00.000Z-ArnoldReinhold-2008-02-28T22:54:00.000Z"></span>I agree. The most important technical aspect might be gcd(e,(p - 1)(q - 1))=1 , and the fact that if you choose e such that d is smaller than N^(1/4) then RSA becomes vulnerable (D. Boneh and G. Durfee. Cryptanalysis of RSA with private key d less than N^(.292) . IEEE Trans. Inform. Theory, 46(4):1339-1349, 2000. ) according to my textbook is good further reading... In summary, I see what grendel says, but I am not sure if adding the bit about e would make things too complicated. Is there a third opinion thats somewhere in between proposals? <a href="/wiki/User:Brusegadi" title="User:Brusegadi">Brusegadi</a> (<a href="/wiki/User_talk:Brusegadi" title="User talk:Brusegadi">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Brusegadi-2008-02-29T03:02:00.000Z-ArnoldReinhold-2008-02-28T22:54:00.000Z" class="ext-discussiontools-init-timestamplink">03:02, 29 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Brusegadi-2008-02-29T03:02:00.000Z-ArnoldReinhold-2008-02-28T22:54:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Brusegadi-2008-02-29T03:02:00.000Z-ArnoldReinhold-2008-02-28T22:54:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-ArnoldReinhold-2008-02-29T04:19:00.000Z-Brusegadi-2008-02-29T03:02:00.000Z"></span>Right, but that level of detail isn't needed in this article. --<a href="/wiki/User:ArnoldReinhold" title="User:ArnoldReinhold">agr</a> (<a href="/wiki/User_talk:ArnoldReinhold" title="User talk:ArnoldReinhold">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-ArnoldReinhold-2008-02-29T04:19:00.000Z-Brusegadi-2008-02-29T03:02:00.000Z" class="ext-discussiontools-init-timestamplink">04:19, 29 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-ArnoldReinhold-2008-02-29T04:19:00.000Z-Brusegadi-2008-02-29T03:02:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-ArnoldReinhold-2008-02-29T04:19:00.000Z-Brusegadi-2008-02-29T03:02:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-85.2.102.219-2008-02-29T06:35:00.000Z-ArnoldReinhold-2008-02-29T04:19:00.000Z"></span>Agreed. The article is about elliptic curve cryptography in general and not just one specific cryptosystem. Hence the introduction should compare the mathematical problems that are the basis of different classes of cryptosystems, i.e. integer factorisation vs. discrete logarithms over GF(p) vs. discrete logarithms over elliptic curves. Details about RSA do not belong here, simply because there are many other cryptosystmes that are also based on integer factorisation. <a href="/wiki/Special:Contributions/85.2.102.219" title="Special:Contributions/85.2.102.219">85.2.102.219</a> (<a href="/w/index.php?title=User_talk:85.2.102.219&action=edit&redlink=1" class="new" title="User talk:85.2.102.219 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-85.2.102.219-2008-02-29T06:35:00.000Z-ArnoldReinhold-2008-02-29T04:19:00.000Z" class="ext-discussiontools-init-timestamplink">06:35, 29 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-85.2.102.219-2008-02-29T06:35:00.000Z-ArnoldReinhold-2008-02-29T04:19:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-85.2.102.219-2008-02-29T06:35:00.000Z-ArnoldReinhold-2008-02-29T04:19:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Brusegadi-2008-02-29T06:44:00.000Z-85.2.102.219-2008-02-29T06:35:00.000Z"></span>Yet, we still should be careful about equating solving RSA to integer factorization... It is not proven that the only way to crack RSA is by factorization... Am I being too picky? <a href="/wiki/User:Brusegadi" title="User:Brusegadi">Brusegadi</a> (<a href="/wiki/User_talk:Brusegadi" title="User talk:Brusegadi">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Brusegadi-2008-02-29T06:44:00.000Z-85.2.102.219-2008-02-29T06:35:00.000Z" class="ext-discussiontools-init-timestamplink">06:44, 29 February 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Brusegadi-2008-02-29T06:44:00.000Z-85.2.102.219-2008-02-29T06:35:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Brusegadi-2008-02-29T06:44:00.000Z-85.2.102.219-2008-02-29T06:35:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-85.2.99.242-2008-03-01T19:12:00.000Z-Brusegadi-2008-02-29T06:44:00.000Z"></span>If you find such a mistake, fix it. Right now the article only claims that finding the private key is as difficult as factoring, which is correct. Still there is much room to improve the introduction. E.g., right now the introduction appears to claim that DSA is based on factorisation. <a href="/wiki/Special:Contributions/85.2.99.242" title="Special:Contributions/85.2.99.242">85.2.99.242</a> (<a href="/w/index.php?title=User_talk:85.2.99.242&action=edit&redlink=1" class="new" title="User talk:85.2.99.242 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-85.2.99.242-2008-03-01T19:12:00.000Z-Brusegadi-2008-02-29T06:44:00.000Z" class="ext-discussiontools-init-timestamplink">19:12, 1 March 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-85.2.99.242-2008-03-01T19:12:00.000Z-Brusegadi-2008-02-29T06:44:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-85.2.99.242-2008-03-01T19:12:00.000Z-Brusegadi-2008-02-29T06:44:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-ArnoldReinhold-2008-03-02T02:21:00.000Z-85.2.99.242-2008-03-01T19:12:00.000Z"></span>Brusegadi's point is that in fact we don't know if finding the private key is as difficult as factoring. That is the <a href="/wiki/RSA_problem" title="RSA problem">RSA problem</a>. I tried to clarify the article on this point, without getting too far off topic.--<a href="/wiki/User:ArnoldReinhold" title="User:ArnoldReinhold">agr</a> (<a href="/wiki/User_talk:ArnoldReinhold" title="User talk:ArnoldReinhold">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-ArnoldReinhold-2008-03-02T02:21:00.000Z-85.2.99.242-2008-03-01T19:12:00.000Z" class="ext-discussiontools-init-timestamplink">02:21, 2 March 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-ArnoldReinhold-2008-03-02T02:21:00.000Z-85.2.99.242-2008-03-01T19:12:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-ArnoldReinhold-2008-03-02T02:21:00.000Z-85.2.99.242-2008-03-01T19:12:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-85.2.53.248-2008-03-02T07:58:00.000Z-ArnoldReinhold-2008-03-02T02:21:00.000Z"></span>That is wrong. Rivest, Shamir and Adleman already showed in their original RSA paper that N can be factored in propabilistic polynimial time given N,e and d. But it is still unknown whether there exists another method for decrypting RSA that does not require the private key. <a href="/wiki/Special:Contributions/85.2.53.248" title="Special:Contributions/85.2.53.248">85.2.53.248</a> (<a href="/w/index.php?title=User_talk:85.2.53.248&action=edit&redlink=1" class="new" title="User talk:85.2.53.248 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-85.2.53.248-2008-03-02T07:58:00.000Z-ArnoldReinhold-2008-03-02T02:21:00.000Z" class="ext-discussiontools-init-timestamplink">07:58, 2 March 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-85.2.53.248-2008-03-02T07:58:00.000Z-ArnoldReinhold-2008-03-02T02:21:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-85.2.53.248-2008-03-02T07:58:00.000Z-ArnoldReinhold-2008-03-02T02:21:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Brusegadi-2008-03-02T08:24:00.000Z-85.2.53.248-2008-03-02T07:58:00.000Z"></span>Yeah, the RSA problem is to decipher without the private key, with e, N and Cipher text. We know we can do it if we factor, the question is, can we do it without factoring? Thus, factoring is not equivalent to solving RSA although the first implies the second... (I am pretty sure of this, I was once penalized for saying they were equivalent...) <a href="/wiki/User:Brusegadi" title="User:Brusegadi">Brusegadi</a> (<a href="/wiki/User_talk:Brusegadi" title="User talk:Brusegadi">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Brusegadi-2008-03-02T08:24:00.000Z-85.2.53.248-2008-03-02T07:58:00.000Z" class="ext-discussiontools-init-timestamplink">08:24, 2 March 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Brusegadi-2008-03-02T08:24:00.000Z-85.2.53.248-2008-03-02T07:58:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Brusegadi-2008-03-02T08:24:00.000Z-85.2.53.248-2008-03-02T07:58:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-ArnoldReinhold-2008-03-02T12:41:00.000Z-Brusegadi-2008-03-02T08:24:00.000Z"></span>Sorry, you're right, I missed that distinction, but I think the current article text still covers this point.--<a href="/wiki/User:ArnoldReinhold" title="User:ArnoldReinhold">agr</a> (<a href="/wiki/User_talk:ArnoldReinhold" title="User talk:ArnoldReinhold">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-ArnoldReinhold-2008-03-02T12:41:00.000Z-Brusegadi-2008-03-02T08:24:00.000Z" class="ext-discussiontools-init-timestamplink">12:41, 2 March 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-ArnoldReinhold-2008-03-02T12:41:00.000Z-Brusegadi-2008-03-02T08:24:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-ArnoldReinhold-2008-03-02T12:41:00.000Z-Brusegadi-2008-03-02T08:24:00.000Z"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Curve25519" data-mw-thread-id="h-Curve25519-2008-06-14T06:51:00.000Z"><span data-mw-comment-start="" id="h-Curve25519-2008-06-14T06:51:00.000Z"></span>Curve25519<span data-mw-comment-end="h-Curve25519-2008-06-14T06:51:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=10" title="Edit section: Curve25519"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-NoDepositNoReturn-2008-06-14T06:51:00.000Z-Curve25519"></span>I believe that <a href="/wiki/Curve25519" title="Curve25519">Curve25519</a> can be considered a cipher in its own right, and have added a page for it; however, I lack the time to write a full article for it, so I have redirected it here for the time being (rather than provide a meaningless stub.) I am not sure whether a Curve25519 section in the ECC page makes more sense than its own page; I suspect that it is best handled in a dedicated page. But at least now there's something for it. <a href="/wiki/User:NoDepositNoReturn" title="User:NoDepositNoReturn">NoDepositNoReturn</a> (<a href="/wiki/User_talk:NoDepositNoReturn" title="User talk:NoDepositNoReturn">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-NoDepositNoReturn-2008-06-14T06:51:00.000Z-Curve25519" class="ext-discussiontools-init-timestamplink">06:51, 14 June 2008 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-NoDepositNoReturn-2008-06-14T06:51:00.000Z-Curve25519"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-NoDepositNoReturn-2008-06-14T06:51:00.000Z-Curve25519"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-CesarB-2013-10-12T15:40:00.000Z-NoDepositNoReturn-2008-06-14T06:51:00.000Z"></span>Looking at Bernstein's article "Curve25519: new Diffie-Hellman speed records", is see that the domain he use is y<sup>2</sup> = x<sup>3</sup> + <b>a</b>x<sup>2</sup> + x which is different from the one presented on this page. Being a non expert in elliptic curve cryptography I would like to know if this makes a significant difference ? If I take this page by the word, curve25519 is not performing elliptic cryptography. <small><span class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:134.158.16.169&action=edit&redlink=1" class="new" title="User:134.158.16.169 (page does not exist)">134.158.16.169</a> (<a href="/w/index.php?title=User_talk:134.158.16.169&action=edit&redlink=1" class="new" title="User talk:134.158.16.169 (page does not exist)">talk</a> • <a href="/wiki/Special:Contributions/134.158.16.169" title="Special:Contributions/134.158.16.169">contribs</a>) 15:36, 25 March 2010‎</span></small></dd></dl> <dl><dd><dl><dd>That is a <a href="/wiki/Montgomery_curve" title="Montgomery curve">Montgomery curve</a>. If you look at that article, there is a section on how to convert it into the Weierstrass form used by this article. --<a href="/wiki/User:CesarB" title="User:CesarB">cesarb</a> (<a href="/wiki/User_talk:CesarB" title="User talk:CesarB">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-CesarB-2013-10-12T15:40:00.000Z-NoDepositNoReturn-2008-06-14T06:51:00.000Z" class="ext-discussiontools-init-timestamplink">15:40, 12 October 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-CesarB-2013-10-12T15:40:00.000Z-NoDepositNoReturn-2008-06-14T06:51:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-CesarB-2013-10-12T15:40:00.000Z-NoDepositNoReturn-2008-06-14T06:51:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="A_Set_forms_a_Group?" data-mw-thread-id="h-A_Set_forms_a_Group?-2009-10-04T19:12:00.000Z"><span id="A_Set_forms_a_Group.3F"></span><span data-mw-comment-start="" id="h-A_Set_forms_a_Group?-2009-10-04T19:12:00.000Z"></span>A Set forms a Group?<span data-mw-comment-end="h-A_Set_forms_a_Group?-2009-10-04T19:12:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=11" title="Edit section: A Set forms a Group?"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Skippydo-2009-10-04T19:12:00.000Z-A_Set_forms_a_Group?"></span>It's unhelpful to say that a set of points (x,y) forms a group, without giving some hint as to what the group operator is. How do (x1,y1) and (x2,y2) combine to form (x3,y3), also a solution? And why is the "point at infinity" (which point at infinity?) the identity element for this combination? <a href="/w/index.php?title=User:213.123.226.227&action=edit&redlink=1" class="new" title="User:213.123.226.227 (page does not exist)">213.123.226.227</a> </p> <dl><dd>Good point, I've added a line about the source of the group. <a href="/wiki/User:Skippydo" title="User:Skippydo">Skippydo</a> (<a href="/wiki/User_talk:Skippydo" title="User talk:Skippydo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Skippydo-2009-10-04T19:12:00.000Z-A_Set_forms_a_Group?" class="ext-discussiontools-init-timestamplink">19:12, 4 October 2009 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Skippydo-2009-10-04T19:12:00.000Z-A_Set_forms_a_Group?"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Skippydo-2009-10-04T19:12:00.000Z-A_Set_forms_a_Group?"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-212.159.69.4-2010-02-03T20:15:00.000Z-Skippydo-2009-10-04T19:12:00.000Z"></span>Speaking as an informed layman (i.e. I have some post-secondary mathematics study under my belt, but am neither a mathematician nor a cryptographer), I have no idea what the sentence you added there means, and the articles linked in it are basically incomprehensible to me.</dd> <dd>What I do know is that the article still states that a set of points forms a group, which to my (admittedly basic) understanding of discrete mathematics simply cannot be true: a group consists of a set of items and an operator on those items.</dd> <dd>Can somebody, please, add a simple explanation of what the operator in question actually is, and hence correct the clearly incorrect statement that "this set [of points on a curve plus a point at infinity] forms an Abelian group, with the point at infinity as identity element." <a href="/wiki/Special:Contributions/212.159.69.4" title="Special:Contributions/212.159.69.4">212.159.69.4</a> (<a href="/wiki/User_talk:212.159.69.4" title="User talk:212.159.69.4">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-212.159.69.4-2010-02-03T20:15:00.000Z-Skippydo-2009-10-04T19:12:00.000Z" class="ext-discussiontools-init-timestamplink">20:15, 3 February 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-212.159.69.4-2010-02-03T20:15:00.000Z-Skippydo-2009-10-04T19:12:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-212.159.69.4-2010-02-03T20:15:00.000Z-Skippydo-2009-10-04T19:12:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Dimawik-2010-02-03T23:06:00.000Z-212.159.69.4-2010-02-03T20:15:00.000Z"></span>I have added some wording referring to the group operation defined in the <a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a> article. The explanations there are admittedly very <a href="/wiki/Bourbaki" class="mw-disambig" title="Bourbaki">Bourbakish</a> and therefore totally incomprehensible, but the theory shall not be replicated here - but instead fixed in the <a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a> article, IMHO. <a href="/wiki/User:Dimawik" title="User:Dimawik">Dimawik</a> (<a href="/wiki/User_talk:Dimawik" title="User talk:Dimawik">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Dimawik-2010-02-03T23:06:00.000Z-212.159.69.4-2010-02-03T20:15:00.000Z" class="ext-discussiontools-init-timestamplink">23:06, 3 February 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dimawik-2010-02-03T23:06:00.000Z-212.159.69.4-2010-02-03T20:15:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Dimawik-2010-02-03T23:06:00.000Z-212.159.69.4-2010-02-03T20:15:00.000Z"></span></dd> <dd>The group operation is referred to as "addition".</dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Design_choices_and_ECC" data-mw-thread-id="h-Design_choices_and_ECC-2009-12-13T08:46:00.000Z"><span data-mw-comment-start="" id="h-Design_choices_and_ECC-2009-12-13T08:46:00.000Z"></span>Design choices and ECC<span data-mw-comment-end="h-Design_choices_and_ECC-2009-12-13T08:46:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=12" title="Edit section: Design choices and ECC"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-192.118.32.80-2009-12-13T08:46:00.000Z-Design_choices_and_ECC"></span>This article is written almost entirely from the mathematical POV. There are other important POVs which should be reflected here. </p><p>For example, what guides a design choice to incorporate ECC vs. alternatives? How does ECC compare to alternatives such as RSA? e.g. the key length is shorter, computational complexity on each side of an exchange, etc. <span style="font-size: smaller;" class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/192.118.32.80" title="Special:Contributions/192.118.32.80">192.118.32.80</a> (<a href="/wiki/User_talk:192.118.32.80" title="User talk:192.118.32.80">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-192.118.32.80-2009-12-13T08:46:00.000Z-Design_choices_and_ECC" class="ext-discussiontools-init-timestamplink">08:46, 13 December 2009 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-192.118.32.80-2009-12-13T08:46:00.000Z-Design_choices_and_ECC"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-192.118.32.80-2009-12-13T08:46:00.000Z-Design_choices_and_ECC"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Broken_link_in_reference" data-mw-thread-id="h-Broken_link_in_reference-2010-06-17T19:32:00.000Z"><span data-mw-comment-start="" id="h-Broken_link_in_reference-2010-06-17T19:32:00.000Z"></span>Broken link in reference<span data-mw-comment-end="h-Broken_link_in_reference-2010-06-17T19:32:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=13" title="Edit section: Broken link in reference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-84.157.71.42-2010-06-17T19:32:00.000Z-Broken_link_in_reference"></span>The link hxxp://www.nsa.gov/ia/programs/suiteb_cryptography/index.shtml in reference 3 is broken. –<a href="/wiki/Special:Contributions/84.157.71.42" title="Special:Contributions/84.157.71.42">84.157.71.42</a> (<a href="/w/index.php?title=User_talk:84.157.71.42&action=edit&redlink=1" class="new" title="User talk:84.157.71.42 (page does not exist)">talk</a>) <span style="font-size: smaller;" class="autosigned">—Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">undated</a> comment added <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-84.157.71.42-2010-06-17T19:32:00.000Z-Broken_link_in_reference" class="ext-discussiontools-init-timestamplink">19:32, 17 June 2010 (UTC)</a>.</span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-84.157.71.42-2010-06-17T19:32:00.000Z-Broken_link_in_reference"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-84.157.71.42-2010-06-17T19:32:00.000Z-Broken_link_in_reference"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Generator" data-mw-thread-id="h-Generator-2010-12-22T20:24:00.000Z"><span data-mw-comment-start="" id="h-Generator-2010-12-22T20:24:00.000Z"></span>Generator<span data-mw-comment-end="h-Generator-2010-12-22T20:24:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=14" title="Edit section: Generator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Doctorhook-2010-12-22T20:24:00.000Z-Generator"></span>The article currently states that "For cryptographic application the order of G [...] must be prime." This agrees with SP800-56A, but the only references I have seen that justify a restriction on the order only say that it should be large and divisible by a large prime. (E.g., <i>Algebraic aspects of cryptography</i>, Neal Koblitz.) Does anyone know of a reason why the order would <b>have</b> to be prime? <a href="/wiki/User:Doctorhook" title="User:Doctorhook">Doctorhook</a> (<a href="/wiki/User_talk:Doctorhook" title="User talk:Doctorhook">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Doctorhook-2010-12-22T20:24:00.000Z-Generator" class="ext-discussiontools-init-timestamplink">20:24, 22 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Doctorhook-2010-12-22T20:24:00.000Z-Generator"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Doctorhook-2010-12-22T20:24:00.000Z-Generator"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Schlafly-2010-12-22T22:48:00.000Z-Doctorhook-2010-12-22T20:24:00.000Z"></span>You're right, it does not have to be prime. If it is not prime, then some complexity would be added to some protocols. For example, someone might choose a public key that has order 2, and that is useless. So it is customary to choose the generator to have prime order. <a href="/wiki/User:Schlafly" title="User:Schlafly">Roger</a> (<a href="/wiki/User_talk:Schlafly" title="User talk:Schlafly">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Schlafly-2010-12-22T22:48:00.000Z-Doctorhook-2010-12-22T20:24:00.000Z" class="ext-discussiontools-init-timestamplink">22:48, 22 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Schlafly-2010-12-22T22:48:00.000Z-Doctorhook-2010-12-22T20:24:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Schlafly-2010-12-22T22:48:00.000Z-Doctorhook-2010-12-22T20:24:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="External_link_one" data-mw-thread-id="h-External_link_one-2011-11-21T03:07:00.000Z"><span data-mw-comment-start="" id="h-External_link_one-2011-11-21T03:07:00.000Z"></span>External link one<span data-mw-comment-end="h-External_link_one-2011-11-21T03:07:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=15" title="Edit section: External link one"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Gryllida-2011-11-21T03:07:00.000Z-External_link_one"></span>The first of the external links (<a rel="nofollow" class="external autonumber" href="http://sagenb.org/home/pub/1126/">[1]</a>) shows a page with no substantial or interactive content to me? <a href="/wiki/User:Gryllida" title="User:Gryllida">Gryllida</a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Gryllida-2011-11-21T03:07:00.000Z-External_link_one" class="ext-discussiontools-init-timestamplink">03:07, 21 November 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Gryllida-2011-11-21T03:07:00.000Z-External_link_one"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Gryllida-2011-11-21T03:07:00.000Z-External_link_one"></span> </p><p><span data-mw-comment-start="" id="c-BeEs1-2011-12-01T11:08:00.000Z-External_link_one"></span>This is a link to a SAGE notebook. SAGE is an important open-source computer-algebra system. A notebook has to be run in your browser to use it -- when you just open it, without starting it, you see the source-code which may appear to be similar to TeX. To start the notebook just click "Evaluate". [BE] <small><span class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:BeEs1&action=edit&redlink=1" class="new" title="User:BeEs1 (page does not exist)">BeEs1</a> (<a href="/wiki/User_talk:BeEs1" title="User talk:BeEs1">talk</a> • <a href="/wiki/Special:Contributions/BeEs1" title="Special:Contributions/BeEs1">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-BeEs1-2011-12-01T11:08:00.000Z-External_link_one" class="ext-discussiontools-init-timestamplink">11:08, 1 December 2011 (UTC)</a></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-BeEs1-2011-12-01T11:08:00.000Z-External_link_one"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-BeEs1-2011-12-01T11:08:00.000Z-External_link_one"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Quantum_Computing_Attack_Citations" data-mw-thread-id="h-Quantum_Computing_Attack_Citations-2012-06-25T22:20:00.000Z"><span data-mw-comment-start="" id="h-Quantum_Computing_Attack_Citations-2012-06-25T22:20:00.000Z"></span>Quantum Computing Attack Citations<span data-mw-comment-end="h-Quantum_Computing_Attack_Citations-2012-06-25T22:20:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=16" title="Edit section: Quantum Computing Attack Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-GromXXVII-2012-06-25T22:20:00.000Z-Quantum_Computing_Attack_Citations"></span>In reference to quantum computing attacks the article reads "Elliptic curve cryptography is vulnerable to a modified Shor's algorithm for solving the discrete logarithm problem on elliptic curves" with two citations (<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>). Looking through both of these citations, they both work over fields of prime order, with the latter paper explicitly stating that they did not consider fields of prime power order. If ECC over fields of prime power order is truly vulnerable to QC attacks, I think there should be a citation that references this. <a href="/wiki/User:GromXXVII" title="User:GromXXVII">GromXXVII</a> (<a href="/wiki/User_talk:GromXXVII" title="User talk:GromXXVII">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-GromXXVII-2012-06-25T22:20:00.000Z-Quantum_Computing_Attack_Citations" class="ext-discussiontools-init-timestamplink">22:20, 25 June 2012 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-GromXXVII-2012-06-25T22:20:00.000Z-Quantum_Computing_Attack_Citations"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-GromXXVII-2012-06-25T22:20:00.000Z-Quantum_Computing_Attack_Citations"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Skippydo-2012-06-26T01:58:00.000Z-GromXXVII-2012-06-25T22:20:00.000Z"></span>Good catch! I updated the reference. If you need more information or have any other ideas, please share. <a href="/wiki/User:Skippydo" title="User:Skippydo">Skippydo</a> (<a href="/wiki/User_talk:Skippydo" title="User talk:Skippydo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Skippydo-2012-06-26T01:58:00.000Z-GromXXVII-2012-06-25T22:20:00.000Z" class="ext-discussiontools-init-timestamplink">01:58, 26 June 2012 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Skippydo-2012-06-26T01:58:00.000Z-GromXXVII-2012-06-25T22:20:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Skippydo-2012-06-26T01:58:00.000Z-GromXXVII-2012-06-25T22:20:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Jimw338-2016-09-12T04:18:00.000Z-Skippydo-2012-06-26T01:58:00.000Z"></span>I found a copy of the first 1997 Eicher reference if it is still of use: (cite tag: <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFEicherOpoku1997" class="citation journal cs1">Eicher, Jodie; Opoku, Yaw (July 29, 1997). <a rel="nofollow" class="external text" href="http://web.archive.org/web/20030509012110/http://www.mathcs.richmond.edu/~jad/summerwork/ellipticcurvequantum.pdf">"Using the Quantum Computer to Break Elliptic Curve Cryptosystems"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.mathcs.richmond.edu/~jad/summerwork/ellipticcurvequantum.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2003-05-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Using+the+Quantum+Computer+to+Break+Elliptic+Curve+Cryptosystems&rft.date=1997-07-29&rft.aulast=Eicher&rft.aufirst=Jodie&rft.au=Opoku%2C+Yaw&rft_id=http%3A%2F%2Fwww.mathcs.richmond.edu%2F~jad%2Fsummerwork%2Fellipticcurvequantum.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATalk%3AElliptic-curve+cryptography" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span>; <span class="cs1-visible-error citation-comment">Unknown parameter <code class="cs1-code">|dead-url=</code> ignored (<code class="cs1-code">|url-status=</code> suggested) (<a href="/wiki/Help:CS1_errors#parameter_ignored_suggest" title="Help:CS1 errors">help</a>)</span>). I (probably is) be outdated though, though it might be useful for background info (?). <a href="/wiki/User:Jimw338" title="User:Jimw338">Jimw338</a> (<a href="/wiki/User_talk:Jimw338" title="User talk:Jimw338">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Jimw338-2016-09-12T04:18:00.000Z-Skippydo-2012-06-26T01:58:00.000Z" class="ext-discussiontools-init-timestamplink">04:18, 12 September 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Jimw338-2016-09-12T04:18:00.000Z-Skippydo-2012-06-26T01:58:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Jimw338-2016-09-12T04:18:00.000Z-Skippydo-2012-06-26T01:58:00.000Z"></span></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-Tarcieri-2017-11-03T18:40:00.000Z-Quantum_Computing_Attack_Citations"></span>I just rewrote the whole section with an updated citation and what I hope is both clearer wording and a more NPOV. <a href="/wiki/User:Tarcieri" title="User:Tarcieri">Tarcieri</a> (<a href="/wiki/User_talk:Tarcieri" title="User talk:Tarcieri">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Tarcieri-2017-11-03T18:40:00.000Z-Quantum_Computing_Attack_Citations" class="ext-discussiontools-init-timestamplink">18:40, 3 November 2017 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tarcieri-2017-11-03T18:40:00.000Z-Quantum_Computing_Attack_Citations"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tarcieri-2017-11-03T18:40:00.000Z-Quantum_Computing_Attack_Citations"></span> </p> <style data-mw-deduplicate="TemplateStyles:r1232521257">.mw-parser-output .reflist-talk{margin:auto 0;border:1px dashed var(--border-color-base,#a2a9b1);padding:4px;padding-left:1em}.mw-parser-output .reflist-talk-title{font-weight:bold}</style><div class="reflist-talk"> <p class="reflist-talk-title">References</p> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEicherOpoku1997" class="citation journal cs1">Eicher, Jodie; Opoku, Yaw (July 29, 1997). "Using the Quantum Computer to Break Elliptic Curve Cryptosystems".</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Using+the+Quantum+Computer+to+Break+Elliptic+Curve+Cryptosystems&rft.date=1997-07-29&rft.aulast=Eicher&rft.aufirst=Jodie&rft.au=Opoku%2C+Yaw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATalk%3AElliptic-curve+cryptography" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFProosZalka2003" class="citation journal cs1">Proos, John; Zalka, Christof (2003). "Shor's Discrete Logarithm Quantum Algorithm for Elliptic Curves". <i>Quantum Information and Computing</i>. <b>3</b> (4): <span class="nowrap">317–</span>344. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0301141">quant-ph/0301141</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003quant.ph..1141P">2003quant.ph..1141P</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantum+Information+and+Computing&rft.atitle=Shor%27s+Discrete+Logarithm+Quantum+Algorithm+for+Elliptic+Curves&rft.volume=3&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E317-%3C%2Fspan%3E344&rft.date=2003&rft_id=info%3Aarxiv%2Fquant-ph%2F0301141&rft_id=info%3Abibcode%2F2003quant.ph..1141P&rft.aulast=Proos&rft.aufirst=John&rft.au=Zalka%2C+Christof&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATalk%3AElliptic-curve+cryptography" class="Z3988"></span></span> </li> </ol></div></div> </div> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Possible_NSA_backdoor" data-mw-thread-id="h-Possible_NSA_backdoor-2013-09-06T17:29:00.000Z"><span data-mw-comment-start="" id="h-Possible_NSA_backdoor-2013-09-06T17:29:00.000Z"></span>Possible NSA backdoor<span data-mw-comment-end="h-Possible_NSA_backdoor-2013-09-06T17:29:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=17" title="Edit section: Possible NSA backdoor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-24.126.40.36-2013-09-06T17:29:00.000Z-Possible_NSA_backdoor"></span>I don't have the technical competence to write a section about this, but I think it is important to point out that there is serious speculation that the NSA inserted a backdoor into the NIST Special Publication 800-90 Dual_EC_DRBG elliptic curve pseudo random generator. If I understand the concern, it is basically that some defined constants in the standard are related to a second, unknown set of numbers, but whoever originally generated those constants does know those numbers. Cryptographic experts say that whoever knows those numbers can gain encryption keys given only 32 bytes of cyphertext. </p><p>The technical discussion of the issue is found in these sources: </p><p><a rel="nofollow" class="external free" href="http://www.wired.com/politics/security/commentary/securitymatters/2007/11/securitymatters_1115">http://www.wired.com/politics/security/commentary/securitymatters/2007/11/securitymatters_1115</a> <a rel="nofollow" class="external free" href="http://eprint.iacr.org/2007/048.pdf">http://eprint.iacr.org/2007/048.pdf</a> <a rel="nofollow" class="external free" href="http://rump2007.cr.yp.to/15-shumow.pdf">http://rump2007.cr.yp.to/15-shumow.pdf</a> </p><p>And some analysis of the possibility that these concerns are founded, based on leaks from Edward Snowden, is found here: </p><p><a rel="nofollow" class="external free" href="http://arstechnica.com/security/2013/09/the-nsas-work-to-make-crypto-worse-and-better/">http://arstechnica.com/security/2013/09/the-nsas-work-to-make-crypto-worse-and-better/</a> <span style="font-size: smaller;" class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/24.126.40.36" title="Special:Contributions/24.126.40.36">24.126.40.36</a> (<a href="/w/index.php?title=User_talk:24.126.40.36&action=edit&redlink=1" class="new" title="User talk:24.126.40.36 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-24.126.40.36-2013-09-06T17:29:00.000Z-Possible_NSA_backdoor" class="ext-discussiontools-init-timestamplink">17:29, 6 September 2013 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-24.126.40.36-2013-09-06T17:29:00.000Z-Possible_NSA_backdoor"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-24.126.40.36-2013-09-06T17:29:00.000Z-Possible_NSA_backdoor"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Mightyname-2013-09-06T20:39:00.000Z-24.126.40.36-2013-09-06T17:29:00.000Z"></span>Wrong place here. Go to the <a href="/wiki/Dual_EC_DRBG" title="Dual EC DRBG">Dual_EC_DRBG</a> article. Besides this is about a specific random number generator. It's about a possible weakness on the practicality of the technique based on geometrical identity of elliptic curves, I think. Something like finding a twin to get the answer. <a href="/w/index.php?title=User:Mightyname&action=edit&redlink=1" class="new" title="User:Mightyname (page does not exist)">Mightyname</a> (<a href="/wiki/User_talk:Mightyname" title="User talk:Mightyname">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Mightyname-2013-09-06T20:39:00.000Z-24.126.40.36-2013-09-06T17:29:00.000Z" class="ext-discussiontools-init-timestamplink">20:39, 6 September 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mightyname-2013-09-06T20:39:00.000Z-24.126.40.36-2013-09-06T17:29:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Mightyname-2013-09-06T20:39:00.000Z-24.126.40.36-2013-09-06T17:29:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Yakatz-2013-09-11T15:35:00.000Z-Mightyname-2013-09-06T20:39:00.000Z"></span>I think it might be appropriate to mention it in the section about <b>NIST-recommended elliptic curves</b>. <br /> I think there are good sources in the Slashdot summary: <a rel="nofollow" class="external free" href="http://it.slashdot.org/firehose.pl?op=view&type=story&sid=13/09/11/1224252">http://it.slashdot.org/firehose.pl?op=view&type=story&sid=13/09/11/1224252</a> <br /> <a href="/wiki/User:Yakatz" title="User:Yakatz">Yakatz</a> (<a href="/wiki/User_talk:Yakatz" title="User talk:Yakatz">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Yakatz-2013-09-11T15:35:00.000Z-Mightyname-2013-09-06T20:39:00.000Z" class="ext-discussiontools-init-timestamplink">15:35, 11 September 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Yakatz-2013-09-11T15:35:00.000Z-Mightyname-2013-09-06T20:39:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Yakatz-2013-09-11T15:35:00.000Z-Mightyname-2013-09-06T20:39:00.000Z"></span></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-Dbachmann-2013-09-21T10:10:00.000Z-Yakatz-2013-09-11T15:35:00.000Z"></span>considering the gravity of the scandal, there should obviously be a paragraph dedicated to it here. Details can still go to <a href="/wiki/Dual_EC_DRBG" title="Dual EC DRBG">Dual_EC_DRBG</a>, but the topic needs to be given <a href="/wiki/Wikipedia:SS" class="mw-redirect" title="Wikipedia:SS">WP:SS</a> treatment on this page, because this page is the first people will come to when they read about the "NSA ECC backdoor". --<a href="/wiki/User:Dbachmann" class="mw-redirect" title="User:Dbachmann">dab</a> <small><a href="/wiki/User_talk:Dbachmann" title="User talk:Dbachmann">(𒁳)</a></small> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Dbachmann-2013-09-21T10:10:00.000Z-Yakatz-2013-09-11T15:35:00.000Z" class="ext-discussiontools-init-timestamplink">10:10, 21 September 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dbachmann-2013-09-21T10:10:00.000Z-Yakatz-2013-09-11T15:35:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Dbachmann-2013-09-21T10:10:00.000Z-Yakatz-2013-09-11T15:35:00.000Z"></span></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-DokReggar-2014-01-03T12:46:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z"></span>I think this is quite out of topic and should be removed. Even though there are links between random number generation and cryptography, Dual_EC_DBRG is a random number generator, based on elliptic curves (the mathematical objects). This article should be about the cryptographic primitives. <code>#!/bin/<a href="/wiki/User:DokReggar" title="User:DokReggar">DokReggar</a> <a href="/wiki/User_talk:DokReggar" title="User talk:DokReggar">-talk</a></code> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-DokReggar-2014-01-03T12:46:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z" class="ext-discussiontools-init-timestamplink">12:46, 3 January 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-DokReggar-2014-01-03T12:46:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-DokReggar-2014-01-03T12:46:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z"></span></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-130.89.106.70-2014-03-17T15:22:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z"></span>I strongly disagree. There is no such thing as an NSA ECC backdoor (that we know of), there is an NSA Dual_EC_DBRG backdoor. Furthermore, the fact that elliptic curves were used as the construct for this RNG is a mere detail; the NSA could have just as easily based this upon modular exponentiation in integer fields. Elliptic curves just <i>happen</i> to be used by this backdoored construct, but this coincidence is only of interest in other articles. It is irrelevant here, especially when positioned <i>actual implementation issues specific to elliptic curve cryptography</i>. Please remove it here. <small><span class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:130.89.106.70&action=edit&redlink=1" class="new" title="User:130.89.106.70 (page does not exist)">130.89.106.70</a> (<a href="/w/index.php?title=User_talk:130.89.106.70&action=edit&redlink=1" class="new" title="User talk:130.89.106.70 (page does not exist)">talk</a> • <a href="/wiki/Special:Contributions/130.89.106.70" title="Special:Contributions/130.89.106.70">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-130.89.106.70-2014-03-17T15:22:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z" class="ext-discussiontools-init-timestamplink">15:22, 17 March 2014 (UTC)</a></span><span data-mw-comment-end="c-130.89.106.70-2014-03-17T15:22:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z"></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-130.89.106.70-2014-03-17T15:22:00.000Z-Dbachmann-2013-09-21T10:10:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-Quondum-2014-03-17T18:09:00.000Z-130.89.106.70-2014-03-17T15:22:00.000Z"></span>You should clearly read the text a little more closely. The fact that quotable sources are making comments about the possible untrustworthiness of the NSA-recommended elliptic curves, and hence on their use in ECC, is relevant in the section. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum"><i>Quondum</i></a> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Quondum-2014-03-17T18:09:00.000Z-130.89.106.70-2014-03-17T15:22:00.000Z" class="ext-discussiontools-init-timestamplink">18:09, 17 March 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-2014-03-17T18:09:00.000Z-130.89.106.70-2014-03-17T15:22:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Quondum-2014-03-17T18:09:00.000Z-130.89.106.70-2014-03-17T15:22:00.000Z"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-68.45.155.10-2015-01-22T15:42:00.000Z-Possible_NSA_backdoor"></span>Just have to agree with people saying it should be mentioned if only to disambiguate the issue from this page. I too expected to see something about it and had to read the talk page to understand that not all ECC was compromised. Only a very small number of potential readers here know enough to make the distinction required; the vast majority simply think ECC->NSA->backdoored. It's just the reality of the situation. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/68.45.155.10" title="Special:Contributions/68.45.155.10">68.45.155.10</a> (<a href="/w/index.php?title=User_talk:68.45.155.10&action=edit&redlink=1" class="new" title="User talk:68.45.155.10 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-68.45.155.10-2015-01-22T15:42:00.000Z-Possible_NSA_backdoor" class="ext-discussiontools-init-timestamplink">15:42, 22 January 2015 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-68.45.155.10-2015-01-22T15:42:00.000Z-Possible_NSA_backdoor"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-68.45.155.10-2015-01-22T15:42:00.000Z-Possible_NSA_backdoor"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Group_Order" data-mw-thread-id="h-Group_Order-2014-11-03T15:37:00.000Z"><span data-mw-comment-start="" id="h-Group_Order-2014-11-03T15:37:00.000Z"></span>Group Order<span data-mw-comment-end="h-Group_Order-2014-11-03T15:37:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=18" title="Edit section: Group Order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-John_Palkovic-2014-11-03T15:37:00.000Z-Group_Order"></span>The article states, under the heading "Domain Parameters," that </p> <dl><dd>For cryptographic application the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> of <i>G</i>, that is the smallest positive number <i>n</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nG=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>G</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nG=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f80988ce7f5ba73e0d7c45b8600933fa78af25a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.644ex; height:2.176ex;" alt="{\displaystyle nG=\infty }" /></span>, is normally prime.</dd></dl> <p>The <i>order</i> of an element <i>G</i> in an additive group is the smallest positive integer <i>n</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nG=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>G</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nG=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30e7ba74017ea6ab9da0fff0c2383122a501274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.482ex; height:2.176ex;" alt="{\displaystyle nG=0}" /></span>, not ∞ (Gallian, <i>Contemporary Abstract Algebra</i>, ch. 4). This needs to be fixed. </p><p><a href="/wiki/User:John_Palkovic" title="User:John Palkovic">John</a> (<a href="/wiki/User_talk:John_Palkovic" title="User talk:John Palkovic">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-John_Palkovic-2014-11-03T15:37:00.000Z-Group_Order" class="ext-discussiontools-init-timestamplink">15:37, 3 November 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-John_Palkovic-2014-11-03T15:37:00.000Z-Group_Order"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-John_Palkovic-2014-11-03T15:37:00.000Z-Group_Order"></span> </p> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Gurnec-2015-01-15T21:33:00.000Z-John_Palkovic-2014-11-03T15:37:00.000Z"></span>I believe either is accurate -- it's equal to the identity element of the group (denoted 0 (zero), O (uppercase o), or e), which is a <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>, specifically [0:1:0]. I suspect this is what the original author was trying to convey.... <a href="/wiki/User:Gurnec" title="User:Gurnec">gurnec</a> (<a href="/wiki/User_talk:Gurnec" title="User talk:Gurnec">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-Gurnec-2015-01-15T21:33:00.000Z-John_Palkovic-2014-11-03T15:37:00.000Z" class="ext-discussiontools-init-timestamplink">21:33, 15 January 2015 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Gurnec-2015-01-15T21:33:00.000Z-John_Palkovic-2014-11-03T15:37:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Gurnec-2015-01-15T21:33:00.000Z-John_Palkovic-2014-11-03T15:37:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="External_links_modified" data-mw-thread-id="h-External_links_modified-2016-09-11T20:15:00.000Z"><span data-mw-comment-start="" id="h-External_links_modified-2016-09-11T20:15:00.000Z"></span>External links modified<span data-mw-comment-end="h-External_links_modified-2016-09-11T20:15:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=19" title="Edit section: External links modified"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-InternetArchiveBot-2016-09-11T20:15:00.000Z-External_links_modified"></span>Hello fellow Wikipedians, </p><p>I have just modified 2 external links on <a href="/wiki/Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">Elliptic curve cryptography</a>. 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This message is updated dynamically through the template <code>{{<a href="/wiki/Template:Source_check" title="Template:Source check">source check</a>}}</code> <small>(last update: 5 June 2024)</small>. </p> <ul><li>If you have discovered URLs which were erroneously considered dead by the bot, you can report them with <a rel="nofollow" class="external text" href="https://iabot.wmcloud.org/index.php?page=reportfalsepositive">this tool</a>.</li> <li>If you found an error with any archives or the URLs themselves, you can fix them with <a rel="nofollow" class="external text" href="https://iabot.wmcloud.org/index.php?page=manageurlsingle">this tool</a>.</li></ul> <p>Cheers.—<a href="/wiki/User:InternetArchiveBot" title="User:InternetArchiveBot"><b><span style="color:darkgrey;font-family:monospace">InternetArchiveBot</span></b></a> <span style="color:green;font-family:Rockwell">(<a href="/wiki/User_talk:InternetArchiveBot" title="User talk:InternetArchiveBot">Report bug</a>)</span> <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-InternetArchiveBot-2017-09-19T21:14:00.000Z-External_links_modified_3" class="ext-discussiontools-init-timestamplink">21:14, 19 September 2017 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-InternetArchiveBot-2017-09-19T21:14:00.000Z-External_links_modified_3"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-InternetArchiveBot-2017-09-19T21:14:00.000Z-External_links_modified_3"></span> </p><p><br /> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Algorithm_needed" data-mw-thread-id="h-Algorithm_needed-2019-12-06T20:30:00.000Z"><span data-mw-comment-start="" id="h-Algorithm_needed-2019-12-06T20:30:00.000Z"></span>Algorithm needed<span data-mw-comment-end="h-Algorithm_needed-2019-12-06T20:30:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=22" title="Edit section: Algorithm needed"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-198.52.160.180-2019-12-06T20:30:00.000Z-Algorithm_needed"></span>This article doesn't contain the algorithm for ECC like the RSA article does. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/198.52.160.180" title="Special:Contributions/198.52.160.180">198.52.160.180</a> (<a href="/wiki/User_talk:198.52.160.180#top" title="User talk:198.52.160.180">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-198.52.160.180-2019-12-06T20:30:00.000Z-Algorithm_needed" class="ext-discussiontools-init-timestamplink">20:30, 6 December 2019 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-198.52.160.180-2019-12-06T20:30:00.000Z-Algorithm_needed"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-198.52.160.180-2019-12-06T20:30:00.000Z-Algorithm_needed"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="The_first_paragraph_seems_self-contradictory" data-mw-thread-id="h-The_first_paragraph_seems_self-contradictory-2018-10-10T04:13:00.000Z"><span data-mw-comment-start="" id="h-The_first_paragraph_seems_self-contradictory-2018-10-10T04:13:00.000Z"></span>The first paragraph seems self-contradictory<span data-mw-comment-end="h-The_first_paragraph_seems_self-contradictory-2018-10-10T04:13:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=23" title="Edit section: The first paragraph seems self-contradictory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-VecLuci-2018-10-10T04:13:00.000Z-The_first_paragraph_seems_self-contradictory"></span>The first paragraph states that ECC is based on finite fields, as opposed to non-EC cryptography, which is based on plain Galois fields. However, the referenced article on <a href="/wiki/Finite_field" title="Finite field">finite fields</a> explains that finite fields and Galois fields are one and the same. I suspect the intended meaning is that non-EC crypto is based structures over finite fields which are not elliptic curves. If so, this is not clear from the text. I won't change the formulation myself, since I'm not an expert in the field. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:VecLuci&action=edit&redlink=1" class="new" title="User:VecLuci (page does not exist)">VecLuci</a> (<a href="/wiki/User_talk:VecLuci#top" title="User talk:VecLuci">talk</a> • <a href="/wiki/Special:Contributions/VecLuci" title="Special:Contributions/VecLuci">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-VecLuci-2018-10-10T04:13:00.000Z-The_first_paragraph_seems_self-contradictory" class="ext-discussiontools-init-timestamplink">04:13, 10 October 2018 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-VecLuci-2018-10-10T04:13:00.000Z-The_first_paragraph_seems_self-contradictory"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-VecLuci-2018-10-10T04:13:00.000Z-The_first_paragraph_seems_self-contradictory"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Not_a_typo._521,_not_512." data-mw-thread-id="h-Not_a_typo._521,_not_512.-2021-12-28T04:00:00.000Z"><span id="Not_a_typo._521.2C_not_512."></span><span data-mw-comment-start="" id="h-Not_a_typo._521,_not_512.-2021-12-28T04:00:00.000Z"></span>Not a typo. 521, not 512.<span data-mw-comment-end="h-Not_a_typo._521,_not_512.-2021-12-28T04:00:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Elliptic-curve_cryptography&action=edit&section=24" title="Edit section: Not a typo. 521, not 512."><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-76.216.220.191-2021-12-28T04:00:00.000Z-Not_a_typo._521,_not_512."></span>"Five prime fields <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35035371db7bee93733c68c1802114c17d8bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.479ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p}}" /></span> for certain primes <i>p</i> of sizes 192, 224, 256, 384, and {{Not a typo|521}} bits. For each of the prime fields, one elliptic curve is recommended." </p><p>Should there be a footnote about that 521 not being a typo? It really is 521 (see <a rel="nofollow" class="external autonumber" href="https://crypto.stackexchange.com/questions/62083/why-would-diffie-hellman-group-21-be-521-bits-rather-than-512">[2]</a> among many, many sources) but it really looks like someone mis-typed "512". <a href="/wiki/Special:Contributions/76.216.220.191" title="Special:Contributions/76.216.220.191">76.216.220.191</a> (<a href="/wiki/User_talk:76.216.220.191" title="User talk:76.216.220.191">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Elliptic-curve_cryptography#c-76.216.220.191-2021-12-28T04:00:00.000Z-Not_a_typo._521,_not_512." class="ext-discussiontools-init-timestamplink">04:00, 28 December 2021 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-76.216.220.191-2021-12-28T04:00:00.000Z-Not_a_typo._521,_not_512."><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-76.216.220.191-2021-12-28T04:00:00.000Z-Not_a_typo._521,_not_512."></span> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Talk:Elliptic-curve_cryptography&oldid=1243174542">https://en.wikipedia.org/w/index.php?title=Talk:Elliptic-curve_cryptography&oldid=1243174542</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:C-Class_Cryptography_articles" title="Category:C-Class Cryptography articles">C-Class Cryptography articles</a></li><li><a href="/wiki/Category:High-importance_Cryptography_articles" title="Category:High-importance Cryptography articles">High-importance Cryptography articles</a></li><li><a href="/wiki/Category:C-Class_Computer_science_articles" title="Category:C-Class Computer science articles">C-Class Computer science articles</a></li><li><a href="/wiki/Category:High-importance_Computer_science_articles" title="Category:High-importance Computer science articles">High-importance Computer science articles</a></li><li><a href="/wiki/Category:WikiProject_Computer_science_articles" title="Category:WikiProject Computer science articles">WikiProject Computer science articles</a></li><li><a href="/wiki/Category:WikiProject_Cryptography_articles" title="Category:WikiProject Cryptography articles">WikiProject Cryptography articles</a></li><li><a href="/wiki/Category:C-Class_numismatic_articles" title="Category:C-Class numismatic articles">C-Class numismatic articles</a></li><li><a href="/wiki/Category:Low-importance_numismatic_articles" title="Category:Low-importance numismatic articles">Low-importance numismatic articles</a></li><li><a href="/wiki/Category:WikiProject_Numismatics_articles" title="Category:WikiProject Numismatics articles">WikiProject Numismatics articles</a></li><li><a href="/wiki/Category:C-Class_WikiProject_Cryptocurrency_articles" title="Category:C-Class WikiProject Cryptocurrency articles">C-Class WikiProject Cryptocurrency articles</a></li><li><a href="/wiki/Category:Mid-importance_WikiProject_Cryptocurrency_articles" title="Category:Mid-importance WikiProject Cryptocurrency articles">Mid-importance WikiProject Cryptocurrency articles</a></li><li><a href="/wiki/Category:WikiProject_Cryptocurrency_articles" title="Category:WikiProject Cryptocurrency articles">WikiProject Cryptocurrency articles</a></li><li><a href="/wiki/Category:C-Class_mathematics_articles" title="Category:C-Class mathematics articles">C-Class mathematics articles</a></li><li><a href="/wiki/Category:Mid-priority_mathematics_articles" title="Category:Mid-priority mathematics articles">Mid-priority mathematics articles</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 30 August 2024, at 23:20<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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