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id="toc-Authentication" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Authentication"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Authentication</span> </div> </a> <ul id="toc-Authentication-sublist" class="vector-toc-list"> <li id="toc-Limitations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Limitations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Limitations</span> </div> </a> <ul id="toc-Limitations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-repudiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-repudiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Non-repudiation</span> </div> </a> <ul id="toc-Non-repudiation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notions_of_security" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notions_of_security"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notions of security</span> </div> </a> <ul id="toc-Notions_of_security-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_security_precautions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Additional_security_precautions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Additional security precautions</span> </div> </a> <button aria-controls="toc-Additional_security_precautions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Additional security precautions subsection</span> </button> <ul id="toc-Additional_security_precautions-sublist" class="vector-toc-list"> <li id="toc-Putting_the_private_key_on_a_smart_card" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Putting_the_private_key_on_a_smart_card"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Putting the private key on a smart card</span> </div> </a> <ul id="toc-Putting_the_private_key_on_a_smart_card-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_smart_card_readers_with_a_separate_keyboard" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_smart_card_readers_with_a_separate_keyboard"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Using smart card readers with a separate keyboard</span> </div> </a> <ul id="toc-Using_smart_card_readers_with_a_separate_keyboard-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_smart_card_designs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_smart_card_designs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Other smart card designs</span> </div> </a> <ul id="toc-Other_smart_card_designs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_digital_signatures_only_with_trusted_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_digital_signatures_only_with_trusted_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Using digital signatures only with trusted applications</span> </div> </a> <ul id="toc-Using_digital_signatures_only_with_trusted_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_a_network_attached_hardware_security_module" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_a_network_attached_hardware_security_module"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Using a network attached hardware security module</span> </div> </a> <ul id="toc-Using_a_network_attached_hardware_security_module-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-WYSIWYS" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#WYSIWYS"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>WYSIWYS</span> </div> </a> <ul id="toc-WYSIWYS-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Digital_signatures_versus_ink_on_paper_signatures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Digital_signatures_versus_ink_on_paper_signatures"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.7</span> <span>Digital signatures versus ink on paper signatures</span> </div> </a> <ul id="toc-Digital_signatures_versus_ink_on_paper_signatures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Some_digital_signature_algorithms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Some_digital_signature_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Some digital signature algorithms</span> </div> </a> <ul id="toc-Some_digital_signature_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_current_state_of_use_–_legal_and_practical" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_current_state_of_use_–_legal_and_practical"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>The current state of use – legal and practical</span> </div> </a> <ul id="toc-The_current_state_of_use_–_legal_and_practical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Industry_standards" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Industry_standards"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Industry standards</span> </div> </a> <button aria-controls="toc-Industry_standards-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Industry standards subsection</span> </button> <ul id="toc-Industry_standards-sublist" class="vector-toc-list"> <li id="toc-Using_separate_key_pairs_for_signing_and_encryption" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_separate_key_pairs_for_signing_and_encryption"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Using separate key pairs for signing and encryption</span> </div> </a> <ul id="toc-Using_separate_key_pairs_for_signing_and_encryption-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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 cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Digital signature</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 53 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-53" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">53 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D9%82%D9%8A%D8%B9_%D8%B1%D9%82%D9%85%D9%8A" title="توقيع رقمي – Arabic" lang="ar" hreflang="ar" data-title="توقيع رقمي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/R%C9%99q%C9%99msal_imza" title="Rəqəmsal imza – Azerbaijani" lang="az" hreflang="az" data-title="Rəqəmsal imza" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BF%E0%A6%9C%E0%A6%BF%E0%A6%9F%E0%A6%BE%E0%A6%B2_%E0%A6%B8%E0%A7%8D%E0%A6%AC%E0%A6%BE%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%B0" title="ডিজিটাল স্বাক্ষর – Bangla" lang="bn" hreflang="bn" data-title="ডিজিটাল স্বাক্ষর" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD_%D2%A1%D1%83%D0%BB%D1%82%D0%B0%D0%BC%D2%93%D0%B0" title="Электрон ҡултамға – Bashkir" lang="ba" hreflang="ba" data-title="Электрон ҡултамға" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Signatura_digital" title="Signatura digital – Catalan" lang="ca" hreflang="ca" data-title="Signatura digital" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A6%D0%B8%D1%84%D1%80%C4%83%D0%BB%D0%BB%D0%B0_%D0%B0%D0%BB%D0%BF%D1%83%D1%81%D0%BD%D0%B8" title="Цифрăлла алпусни – Chuvash" lang="cv" hreflang="cv" data-title="Цифрăлла алпусни" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Digital_signatur" title="Digital signatur – Danish" lang="da" hreflang="da" data-title="Digital signatur" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Digitale_Signatur" title="Digitale Signatur – German" lang="de" hreflang="de" data-title="Digitale Signatur" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Digitaalallkiri" title="Digitaalallkiri – Estonian" lang="et" hreflang="et" data-title="Digitaalallkiri" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A8%CE%B7%CF%86%CE%B9%CE%B1%CE%BA%CE%AE_%CF%85%CF%80%CE%BF%CE%B3%CF%81%CE%B1%CF%86%CE%AE" title="Ψηφιακή υπογραφή – Greek" lang="el" hreflang="el" data-title="Ψηφιακή υπογραφή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Firma_digital" title="Firma digital – Spanish" lang="es" hreflang="es" data-title="Firma digital" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Cifereca_subskribo" title="Cifereca subskribo – Esperanto" lang="eo" hreflang="eo" data-title="Cifereca subskribo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Sinadura_digital" title="Sinadura digital – Basque" lang="eu" hreflang="eu" data-title="Sinadura digital" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%85%D8%B6%D8%A7%DB%8C_%D8%AF%DB%8C%D8%AC%DB%8C%D8%AA%D8%A7%D9%84" title="امضای دیجیتال – Persian" lang="fa" hreflang="fa" data-title="امضای دیجیتال" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Signature_num%C3%A9rique" title="Signature numérique – French" lang="fr" hreflang="fr" data-title="Signature numérique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EC%84%9C%EB%AA%85" title="디지털 서명 – Korean" lang="ko" hreflang="ko" data-title="디지털 서명" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B9%D5%BE%D5%A1%D5%B5%D5%AB%D5%B6_%D5%BD%D5%BF%D5%B8%D6%80%D5%A1%D5%A3%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Թվային ստորագրություն – Armenian" lang="hy" hreflang="hy" data-title="Թվային ստորագրություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A1%E0%A4%BF%E0%A4%9C%E0%A4%BF%E0%A4%9F%E0%A4%B2_%E0%A4%B9%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%BE%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%B0" title="डिजिटल हस्ताक्षर – Hindi" lang="hi" hreflang="hi" data-title="डिजिटल हस्ताक्षर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Bica_signato" title="Bica signato – Ido" lang="io" hreflang="io" data-title="Bica signato" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tanda_tangan_digital" title="Tanda tangan digital – Indonesian" lang="id" hreflang="id" data-title="Tanda tangan digital" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Firma_digitale" title="Firma digitale – Italian" lang="it" hreflang="it" data-title="Firma digitale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%AA%D7%99%D7%9E%D7%94_%D7%93%D7%99%D7%92%D7%99%D7%98%D7%9C%D7%99%D7%AA" title="חתימה דיגיטלית – Hebrew" lang="he" hreflang="he" data-title="חתימה דיגיטלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D0%B4%D1%83%D0%BA_%D0%BA%D0%BE%D0%BB_%D1%82%D0%B0%D0%BC%D0%B3%D0%B0" title="Электрондук кол тамга – Kyrgyz" lang="ky" hreflang="ky" data-title="Электрондук кол тамга" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Elektroniskais_paraksts" title="Elektroniskais paraksts – Latvian" lang="lv" hreflang="lv" data-title="Elektroniskais paraksts" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skaitmeninis_para%C5%A1as" title="Skaitmeninis parašas – Lithuanian" lang="lt" hreflang="lt" data-title="Skaitmeninis parašas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Firma_digital" title="Firma digital – Lombard" lang="lmo" hreflang="lmo" data-title="Firma digital" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Digit%C3%A1lis_al%C3%A1%C3%ADr%C3%A1s" title="Digitális aláírás – Hungarian" lang="hu" hreflang="hu" data-title="Digitális aláírás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A1%E0%B4%BF%E0%B4%9C%E0%B4%BF%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B5%BD_%E0%B4%92%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D" title="ഡിജിറ്റൽ ഒപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="ഡിജിറ്റൽ ഒപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Digitale_handtekening" title="Digitale handtekening – Dutch" lang="nl" hreflang="nl" data-title="Digitale handtekening" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%A1%E0%A4%BF%E0%A4%9C%E0%A5%80%E0%A4%9F%E0%A4%B2_%E0%A4%B8%E0%A4%BF%E0%A4%97%E0%A5%8D%E0%A4%A8%E0%A5%87%E0%A4%9A%E0%A4%B0" title="डिजीटल सिग्नेचर – Nepali" lang="ne" hreflang="ne" data-title="डिजीटल सिग्नेचर" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%82%B8%E3%82%BF%E3%83%AB%E7%BD%B2%E5%90%8D" title="デジタル署名 – Japanese" lang="ja" hreflang="ja" data-title="デジタル署名" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Elektron_raqamli_imzo" title="Elektron raqamli imzo – Uzbek" lang="uz" hreflang="uz" data-title="Elektron raqamli imzo" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Podpis_cyfrowy" title="Podpis cyfrowy – Polish" lang="pl" hreflang="pl" data-title="Podpis cyfrowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Assinatura_digital" title="Assinatura digital – Portuguese" lang="pt" hreflang="pt" data-title="Assinatura digital" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Semn%C4%83tur%C4%83_digital%C4%83" title="Semnătură digitală – Romanian" lang="ro" hreflang="ro" data-title="Semnătură digitală" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D0%BD%D0%B0%D1%8F_%D0%BF%D0%BE%D0%B4%D0%BF%D0%B8%D1%81%D1%8C" title="Электронная подпись – Russian" lang="ru" hreflang="ru" data-title="Электронная подпись" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/N%C3%ABnshkrimi_dixhital" title="Nënshkrimi dixhital – Albanian" lang="sq" hreflang="sq" data-title="Nënshkrimi dixhital" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Digital_signature" title="Digital signature – Simple English" lang="en-simple" hreflang="en-simple" data-title="Digital signature" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Digit%C3%A1lny_podpis" title="Digitálny podpis – Slovak" lang="sk" hreflang="sk" data-title="Digitálny podpis" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Digitalni_podpis" title="Digitalni podpis – Slovenian" lang="sl" hreflang="sl" data-title="Digitalni podpis" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D1%81%D0%BA%D0%B8_%D0%BF%D0%BE%D1%82%D0%BF%D0%B8%D1%81" title="Електронски потпис – Serbian" lang="sr" hreflang="sr" data-title="Електронски потпис" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Elektronski_potpis" title="Elektronski potpis – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Elektronski potpis" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/S%C3%A4hk%C3%B6inen_allekirjoitus" title="Sähköinen allekirjoitus – Finnish" lang="fi" hreflang="fi" data-title="Sähköinen allekirjoitus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Digital_signatur" title="Digital signatur – Swedish" lang="sv" hreflang="sv" data-title="Digital signatur" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%88%E0%AE%AF%E0%AF%8A%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%AE%E0%AF%8D" title="எண்முறைக் கையொப்பம் – Tamil" lang="ta" hreflang="ta" data-title="எண்முறைக் கையொப்பம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A1%E0%B0%BF%E0%B0%9C%E0%B0%BF%E0%B0%9F%E0%B0%B2%E0%B1%8D_%E0%B0%B8%E0%B0%82%E0%B0%A4%E0%B0%95%E0%B0%82" title="డిజిటల్ సంతకం – Telugu" lang="te" hreflang="te" data-title="డిజిటల్ సంతకం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dijital_imza" title="Dijital imza – Turkish" lang="tr" hreflang="tr" data-title="Dijital imza" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D0%BD%D0%B8%D0%B9_%D1%86%D0%B8%D1%84%D1%80%D0%BE%D0%B2%D0%B8%D0%B9_%D0%BF%D1%96%D0%B4%D0%BF%D0%B8%D1%81" title="Електронний цифровий підпис – Ukrainian" lang="uk" hreflang="uk" data-title="Електронний цифровий підпис" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ch%E1%BB%AF_k%C3%BD_s%E1%BB%91" title="Chữ ký số – Vietnamese" lang="vi" hreflang="vi" data-title="Chữ ký số" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%97%E7%AD%BE%E5%90%8D" title="数字签名 – Wu" lang="wuu" hreflang="wuu" data-title="数字签名" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%93%D7%99%D7%92%D7%99%D7%98%D7%90%D7%9C%D7%99%D7%A9%D7%A2_%D7%90%D7%95%D7%A0%D7%98%D7%A2%D7%A8%D7%A9%D7%A8%D7%99%D7%A4%D7%98" title="דיגיטאלישע אונטערשריפט – Yiddish" lang="yi" hreflang="yi" data-title="דיגיטאלישע אונטערשריפט" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical scheme for verifying the authenticity of digital documents</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about a cryptographic construct derived from a mathematical scheme which is hard to forge. For a data record not secured by cryptographic scheme, see <a href="/wiki/Electronic_signature" title="Electronic signature">Electronic signature</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Private_key_signing.svg" class="mw-file-description"><img alt="Alice signs a message—"Hello Bob!"—by appending a signature which is computed from the message and her private key. Bob receives the message, including the signature, and using Alice's public key, verifies the authenticity of the signed message." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Private_key_signing.svg/220px-Private_key_signing.svg.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Private_key_signing.svg/330px-Private_key_signing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Private_key_signing.svg/440px-Private_key_signing.svg.png 2x" data-file-width="525" data-file-height="513" /></a><figcaption>Alice signs a message—"Hello Bob!"—by appending a signature which is computed from the message and her private key. Bob receives both the message and signature. He uses Alice's public key to verify the authenticity of the signed message.</figcaption></figure> <p>A <b>digital signature</b> is a mathematical scheme for verifying the authenticity of digital messages or documents. A valid digital signature on a message gives a recipient confidence that the message came from a sender known to the recipient.<sup id="cite_ref-bellare-goldwasser2008digsigs_1-0" class="reference"><a href="#cite_note-bellare-goldwasser2008digsigs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-katz-lindell2007digsigs_2-0" class="reference"><a href="#cite_note-katz-lindell2007digsigs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Digital signatures are a standard element of most <a href="/wiki/Cryptographic_protocol" title="Cryptographic protocol">cryptographic protocol</a> suites, and are commonly used for software distribution, financial transactions, <a href="/wiki/Contract_management_software" title="Contract management software">contract management software</a>, and in other cases where it is important to detect forgery or <a href="/wiki/Tampering_(crime)" title="Tampering (crime)">tampering</a>. </p><p>Digital signatures are often used to implement <a href="/wiki/Electronic_signature" title="Electronic signature">electronic signatures</a>, which include any electronic data that carries the intent of a signature,<sup id="cite_ref-cEMCl_3-0" class="reference"><a href="#cite_note-cEMCl-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but not all electronic signatures use digital signatures.<sup id="cite_ref-6TaHP_4-0" class="reference"><a href="#cite_note-6TaHP-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-uExGJ_5-0" class="reference"><a href="#cite_note-uExGJ-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Electronic signatures have legal significance in some countries, including <a href="/wiki/Brazil" title="Brazil">Brazil</a>, <a href="/wiki/Canada" title="Canada">Canada</a>,<sup id="cite_ref-iwENG_6-0" class="reference"><a href="#cite_note-iwENG-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/South_Africa" title="South Africa">South Africa</a>,<sup id="cite_ref-TZv4Q_7-0" class="reference"><a href="#cite_note-TZv4Q-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Russia" title="Russia">Russia</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/United_States" title="United States">United States</a>, <a href="/wiki/Algeria" title="Algeria">Algeria</a>,<sup id="cite_ref-9wrdi_9-0" class="reference"><a href="#cite_note-9wrdi-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Turkey" title="Turkey">Turkey</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="/wiki/India" title="India">India</a>,<sup id="cite_ref-fN2Ic_11-0" class="reference"><a href="#cite_note-fN2Ic-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Indonesia" title="Indonesia">Indonesia</a>, <a href="/wiki/Mexico" title="Mexico">Mexico</a>, <a href="/wiki/Saudi_Arabia" title="Saudi Arabia">Saudi Arabia</a>,<sup id="cite_ref-LplND_12-0" class="reference"><a href="#cite_note-LplND-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Uruguay" title="Uruguay">Uruguay</a>,<sup id="cite_ref-Xf3lW_13-0" class="reference"><a href="#cite_note-Xf3lW-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Switzerland" title="Switzerland">Switzerland</a>, <a href="/wiki/Chile" title="Chile">Chile</a><sup id="cite_ref-4h0yN_14-0" class="reference"><a href="#cite_note-4h0yN-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and the countries of the <a href="/wiki/European_Union" title="European Union">European Union</a>.<sup id="cite_ref-Cryptomathic_MajorStandardsDigSig_15-0" class="reference"><a href="#cite_note-Cryptomathic_MajorStandardsDigSig-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CryptomathicDigSigServicesAshiqJA_16-0" class="reference"><a href="#cite_note-CryptomathicDigSigServicesAshiqJA-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Digital signatures employ <a href="/wiki/Asymmetric_key_algorithm" class="mw-redirect" title="Asymmetric key algorithm">asymmetric cryptography</a>. In many instances, they provide a layer of validation and security to messages sent through a non-secure channel: Properly implemented, a digital signature gives the receiver reason to believe the message was sent by the claimed sender. Digital signatures are equivalent to traditional handwritten signatures in many respects, but properly implemented digital signatures are more difficult to forge than the handwritten type. Digital signature schemes, in the sense used here, are cryptographically based, and must be implemented properly to be effective. They can also provide <a href="/wiki/Non-repudiation" title="Non-repudiation">non-repudiation</a>, meaning that the signer cannot successfully claim they did not sign a message, while also claiming their <a href="/wiki/Private_key" class="mw-redirect" title="Private key">private key</a> remains secret.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Further, some non-repudiation schemes offer a timestamp for the digital signature, so that even if the private key is exposed, the signature is valid.<sup id="cite_ref-G18bg_18-0" class="reference"><a href="#cite_note-G18bg-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-U844i_19-0" class="reference"><a href="#cite_note-U844i-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Digitally signed messages may be anything representable as a <a href="/wiki/Bitstring" class="mw-redirect" title="Bitstring">bitstring</a>: examples include electronic mail, contracts, or a message sent via some other cryptographic protocol. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">Public-key cryptography</a></div> <p>A digital signature scheme typically consists of three algorithms: </p> <ul><li>A <i><a href="/wiki/Key_generation" title="Key generation">key generation</a></i> algorithm that selects a <i>private key</i> <a href="/wiki/Uniform_distribution_(discrete)" class="mw-redirect" title="Uniform distribution (discrete)">uniformly at random</a> from a set of possible private keys. The algorithm outputs the private key and a corresponding <i>public key</i>.</li> <li>A <i>signing</i> algorithm that, given a message and a private key, produces a signature.</li> <li>A <i>signature verifying</i> algorithm that, given the message, public key and signature, either accepts or rejects the message's claim to authenticity.</li></ul> <p>Two main properties are required: </p><p>First, the authenticity of a signature generated from a fixed message and fixed private key can be verified by using the corresponding public key. </p><p>Secondly, it should be computationally infeasible to generate a valid signature for a party without knowing that party's private key. A digital signature is an authentication mechanism that enables the creator of the message to attach a code that acts as a signature. The <a href="/wiki/Digital_Signature_Algorithm" title="Digital Signature Algorithm">Digital Signature Algorithm</a> (DSA), developed by the <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>, is one of <a class="mw-selflink-fragment" href="#Some_digital_signature_algorithms">many examples</a> of a signing algorithm. </p><p>In the following discussion, 1<sup><i>n</i></sup> refers to a <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary number</a>. </p><p>Formally, a <b>digital signature scheme</b> is a triple of probabilistic polynomial time algorithms, (<i>G</i>, <i>S</i>, <i>V</i>), satisfying: </p> <ul><li><i>G</i> (key-generator) generates a public key (<i>pk</i>), and a corresponding private key (<i>sk</i>), on input 1<sup><i>n</i></sup>, where <i>n</i> is the security parameter.</li> <li><i>S</i> (signing) returns a tag, <i>t</i>, on the inputs: the private key (<i>sk</i>), and a string (<i>x</i>).</li> <li><i>V</i> (verifying) outputs <i>accepted</i> or <i>rejected</i> on the inputs: the public key (<i>pk</i>), a string (<i>x</i>), and a tag (<i>t</i>).</li></ul> <p>For correctness, <i>S</i> and <i>V</i> must satisfy </p> <dl><dd>Pr [ (<i>pk</i>, <i>sk</i>) ← <i>G</i>(1<sup><i>n</i></sup>), <i>V</i>( <i>pk</i>, <i>x</i>, <i>S</i>(<i>sk</i>, <i>x</i>) ) = <i>accepted</i> ] = 1.<sup id="cite_ref-IjhTrs_20-0" class="reference"><a href="#cite_note-IjhTrs-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></dd></dl> <p>A digital signature scheme is <b>secure</b> if for every non-uniform probabilistic polynomial time <a href="/wiki/Adversary_(cryptography)" title="Adversary (cryptography)">adversary</a>, <i>A</i> </p> <dl><dd>Pr [ (<i>pk</i>, <i>sk</i>) ← <i>G</i>(1<sup><i>n</i></sup>), (<i>x</i>, <i>t</i>) ← <i>A</i><sup><i>S</i>(<i>sk</i>, · )</sup>(<i>pk</i>, 1<sup><i>n</i></sup>), <i>x</i> ∉ <i>Q</i>, <i>V</i>(<i>pk</i>, <i>x</i>, <i>t</i>) = <i>accepted</i>] < <a href="/wiki/Negligible_function" title="Negligible function">negl</a>(<i>n</i>),</dd></dl> <p>where <i>A</i><sup><i>S</i>(<i>sk</i>, · )</sup> denotes that <i>A</i> has access to the <a href="/wiki/Oracle_machine" title="Oracle machine">oracle</a>, <i>S</i>(<i>sk</i>, · ), <i>Q</i> denotes the set of the queries on <i>S</i> made by <i>A</i>, which knows the public key, <i>pk</i>, and the security parameter, <i>n</i>, and <i>x</i> ∉ <i>Q</i> denotes that the adversary may not directly query the string, <i>x</i>, on <i>S</i>.<sup id="cite_ref-IjhTrs_20-1" class="reference"><a href="#cite_note-IjhTrs-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-nOvvy_21-0" class="reference"><a href="#cite_note-nOvvy-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1976, <a href="/wiki/Whitfield_Diffie" title="Whitfield Diffie">Whitfield Diffie</a> and <a href="/wiki/Martin_Hellman" title="Martin Hellman">Martin Hellman</a> first described the notion of a digital signature scheme, although they only conjectured that such schemes existed based on functions that are trapdoor one-way permutations.<sup id="cite_ref-ikWoF_22-0" class="reference"><a href="#cite_note-ikWoF-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lysythesis_23-0" class="reference"><a href="#cite_note-lysythesis-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Soon afterwards, <a href="/wiki/Ronald_Rivest" class="mw-redirect" title="Ronald Rivest">Ronald Rivest</a>, <a href="/wiki/Adi_Shamir" title="Adi Shamir">Adi Shamir</a>, and <a href="/wiki/Len_Adleman" class="mw-redirect" title="Len Adleman">Len Adleman</a> invented the <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a> algorithm, which could be used to produce primitive digital signatures<sup id="cite_ref-rsa_24-0" class="reference"><a href="#cite_note-rsa-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> (although only as a proof-of-concept – "plain" RSA signatures are not secure<sup id="cite_ref-7aSJZ_25-0" class="reference"><a href="#cite_note-7aSJZ-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>). The first widely marketed software package to offer digital signature was <a href="/wiki/Lotus_Notes" class="mw-redirect" title="Lotus Notes">Lotus Notes</a> 1.0, released in 1989, which used the RSA algorithm.<sup id="cite_ref-CKgyC_26-0" class="reference"><a href="#cite_note-CKgyC-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Other digital signature schemes were soon developed after RSA, the earliest being <a href="/wiki/Lamport_signature" title="Lamport signature">Lamport signatures</a>,<sup id="cite_ref-ejiWf_27-0" class="reference"><a href="#cite_note-ejiWf-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Merkle_signature_scheme" title="Merkle signature scheme">Merkle signatures</a> (also known as "Merkle trees" or simply "Hash trees"),<sup id="cite_ref-jl9LD_28-0" class="reference"><a href="#cite_note-jl9LD-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Rabin_signature" class="mw-redirect" title="Rabin signature">Rabin signatures</a>.<sup id="cite_ref-HgBx0_29-0" class="reference"><a href="#cite_note-HgBx0-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>In 1988, <a href="/wiki/Shafi_Goldwasser" title="Shafi Goldwasser">Shafi Goldwasser</a>, <a href="/wiki/Silvio_Micali" title="Silvio Micali">Silvio Micali</a>, and <a href="/wiki/Ronald_Rivest" class="mw-redirect" title="Ronald Rivest">Ronald Rivest</a> became the first to rigorously define the security requirements of digital signature schemes.<sup id="cite_ref-SJC_17(2)_30-0" class="reference"><a href="#cite_note-SJC_17(2)-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> They described a hierarchy of attack models for signature schemes, and also presented the <a href="/wiki/GMR_(cryptography)" title="GMR (cryptography)">GMR signature scheme</a>, the first that could be proved to prevent even an existential forgery against a chosen message attack, which is the currently accepted security definition for signature schemes.<sup id="cite_ref-SJC_17(2)_30-1" class="reference"><a href="#cite_note-SJC_17(2)-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> The first such scheme which is not built on trapdoor functions but rather on a family of function with a much weaker required property of one-way permutation was presented by <a href="/wiki/Moni_Naor" title="Moni Naor">Moni Naor</a> and <a href="/wiki/Moti_Yung" title="Moti Yung">Moti Yung</a>.<sup id="cite_ref-Z2zaX_31-0" class="reference"><a href="#cite_note-Z2zaX-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Method">Method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=3" title="Edit section: Method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This Method section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Digital_signature" title="Special:EditPage/Digital signature">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this Method section. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Digital+signature%22">"Digital signature"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Digital+signature%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Digital+signature%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Digital+signature%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Digital+signature%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Digital+signature%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">January 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>One digital signature scheme (of many) is based on <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a>. To create signature keys, generate an RSA key pair containing a modulus, <i>N</i>, that is the product of two random secret distinct large primes, along with integers, <i>e</i> and <i>d</i>, such that <i>e</i> <i>d</i> <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">≡</a> 1 (mod <i>φ</i>(<i>N</i>)), where <i>φ</i> is <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a>. The signer's public key consists of <i>N</i> and <i>e</i>, and the signer's secret key contains <i>d</i>. </p><p>Used directly, this type of signature scheme is vulnerable to key-only existential forgery attack. To create a forgery, the attacker picks a random signature σ and uses the verification procedure to determine the message, <i>m</i>, corresponding to that signature.<sup id="cite_ref-2Zzbo_32-0" class="reference"><a href="#cite_note-2Zzbo-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> In practice, however, this type of signature is not used directly, but rather, the message to be signed is first <a href="/wiki/Cryptographic_hash_function" title="Cryptographic hash function">hashed</a> to produce a short digest, that is then <a href="/wiki/Padding_(cryptography)#Public_key_cryptography" title="Padding (cryptography)">padded</a> to larger width comparable to <i>N</i>, then signed with the reverse <a href="/wiki/Trapdoor_function" title="Trapdoor function">trapdoor function</a>.<sup id="cite_ref-Jwm4G_33-0" class="reference"><a href="#cite_note-Jwm4G-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> This forgery attack, then, only produces the padded hash function output that corresponds to σ, but not a message that leads to that value, which does not lead to an attack. In the random oracle model, <a href="/wiki/Full_domain_hash" class="mw-redirect" title="Full domain hash">hash-then-sign</a> (an idealized version of that practice where hash and padding combined have close to <i>N</i> possible outputs), this form of signature is existentially unforgeable, even against a <a href="/wiki/Chosen-plaintext_attack" title="Chosen-plaintext attack">chosen-plaintext attack</a>.<sup id="cite_ref-lysythesis_23-1" class="reference"><a href="#cite_note-lysythesis-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Please give a page number or theorem number. (September 2010)">clarification needed</span></a></i>]</sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>There are several reasons to sign such a hash (or message digest) instead of the whole document. </p> <dl><dt>For efficiency</dt> <dd>The signature will be much shorter and thus save time since hashing is generally much faster than signing in practice.</dd> <dt>For compatibility</dt> <dd>Messages are typically bit strings, but some signature schemes operate on other domains (such as, in the case of RSA, numbers modulo a composite number <i>N</i>). A hash function can be used to convert an arbitrary input into the proper format.</dd> <dt>For integrity</dt> <dd>Without the hash function, the text "to be signed" may have to be split (separated) in blocks small enough for the signature scheme to act on them directly. However, the receiver of the signed blocks is not able to recognize if all the blocks are present and in the appropriate order.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As organizations move away from paper documents with ink signatures or authenticity stamps, digital signatures can provide added assurances of the evidence to provenance, identity, and status of an <a href="/wiki/Electronic_document" title="Electronic document">electronic document</a> as well as acknowledging informed consent and approval by a signatory. The United States Government Printing Office (GPO) publishes electronic versions of the budget, public and private laws, and congressional bills with digital signatures. Universities including Penn State, <a href="/wiki/University_of_Chicago" title="University of Chicago">University of Chicago</a>, and Stanford are publishing electronic student transcripts with digital signatures. </p><p>Below are some common reasons for applying a digital signature to communications: </p> <div class="mw-heading mw-heading3"><h3 id="Authentication">Authentication</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=5" title="Edit section: Authentication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A message may have letterhead or a handwritten signature identifying its sender, but letterheads and handwritten signatures can be copied and pasted onto forged messages. Even legitimate messages may be modified in transit.<sup id="cite_ref-stinson3e2006digsigs_35-0" class="reference"><a href="#cite_note-stinson3e2006digsigs-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>If a bank's central office receives a letter claiming to be from a branch office with instructions to change the balance of an account, the central bankers need to be sure, before acting on the instructions, that they were actually sent by a branch banker, and not forged—whether a forger fabricated the whole letter, or just modified an existing letter in transit by adding some digits. </p><p>With a digital signature scheme, the central office can arrange beforehand to have a public key on file whose private key is known only to the branch office. The branch office can later sign a message and the central office can use the public key to verify the signed message was not a forgery before acting on it. A forger who <i>doesn't</i> know the sender's private key can't sign a different message, or even change a single digit in an existing message without making the recipient's signature verification fail.<sup id="cite_ref-stinson3e2006digsigs_35-1" class="reference"><a href="#cite_note-stinson3e2006digsigs-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bellare-goldwasser2008digsigs_1-1" class="reference"><a href="#cite_note-bellare-goldwasser2008digsigs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-katz-lindell2007digsigs_2-1" class="reference"><a href="#cite_note-katz-lindell2007digsigs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Encryption" title="Encryption">Encryption</a> can hide the content of the message from an eavesdropper, but encryption on its own may not let recipient verify the message's authenticity, or even detect <a href="/wiki/Malleability_(cryptography)" title="Malleability (cryptography)">selective modifications like changing a digit</a>—if the bank's offices simply encrypted the messages they exchange, they could still be vulnerable to forgery. In other applications, such as software updates, the messages are not secret—when a software author publishes a patch for all existing installations of the software to apply, the patch itself is not secret, but computers running the software must verify the authenticity of the patch before applying it, lest they become victims to malware.<sup id="cite_ref-katz-lindell2007digsigs_2-2" class="reference"><a href="#cite_note-katz-lindell2007digsigs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Limitations">Limitations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=6" title="Edit section: Limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Replays.</b> A digital signature scheme on its own does not prevent a valid signed message from being recorded and then maliciously reused in a <a href="/wiki/Replay_attack" title="Replay attack">replay attack</a>. For example, the branch office may legitimately request that bank transfer be issued once in a signed message. If the bank doesn't use a system of transaction IDs in their messages to detect which transfers have already happened, someone could illegitimately reuse the same signed message many times to drain an account.<sup id="cite_ref-stinson3e2006digsigs_35-2" class="reference"><a href="#cite_note-stinson3e2006digsigs-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p><b>Uniqueness and malleability of signatures.</b> A signature itself cannot be used to uniquely identify the message it signs—in some signature schemes, every message has a large number of possible valid signatures from the same signer, and it may be easy, even without knowledge of the private key, to transform one valid signature into another.<sup id="cite_ref-bcjz2020ed25519sectheorypractice_36-0" class="reference"><a href="#cite_note-bcjz2020ed25519sectheorypractice-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> If signatures are misused as transaction IDs in an attempt by a bank-like system such as a <a href="/wiki/Bitcoin" title="Bitcoin">Bitcoin</a> exchange to detect replays, this can be exploited to replay transactions.<sup id="cite_ref-decker-wattenhofer2014btcmalleablemtgox_37-0" class="reference"><a href="#cite_note-decker-wattenhofer2014btcmalleablemtgox-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p><b>Authenticating a public key.</b> Prior knowledge of a <i>public key</i> can be used to verify authenticity of a <i>signed message</i>, but not the other way around—prior knowledge of a <i>signed message</i> cannot be used to verify authenticity of a <i>public key</i>. In some signature schemes, given a signed message, it is easy to construct a public key under which the signed message will pass verification, even without knowledge of the private key that was used to make the signed message in the first place.<sup id="cite_ref-ayer2015sigmisuseacme_38-0" class="reference"><a href="#cite_note-ayer2015sigmisuseacme-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Non-repudiation">Non-repudiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=7" title="Edit section: Non-repudiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Non-repudiation" title="Non-repudiation">Non-repudiation</a>,<sup id="cite_ref-Cryptomathic_MajorStandardsDigSig_15-1" class="reference"><a href="#cite_note-Cryptomathic_MajorStandardsDigSig-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> or more specifically non-repudiation of origin, is an important aspect of digital signatures. By this property, an entity that has signed some information cannot at a later time deny having signed it. Similarly, access to the public key only does not enable a fraudulent party to fake a valid signature. </p><p>Note that these authentication, non-repudiation etc. properties rely on the secret key <em>not having been revoked</em> prior to its usage. Public <a href="/wiki/Certificate_revocation" title="Certificate revocation">revocation</a> of a key-pair is a required ability, else leaked secret keys would continue to implicate the claimed owner of the key-pair. Checking revocation status requires an "online" check; e.g., checking a <a href="/wiki/Certificate_revocation_list" title="Certificate revocation list">certificate revocation list</a> or via the <a href="/wiki/Online_Certificate_Status_Protocol" title="Online Certificate Status Protocol">Online Certificate Status Protocol</a>.<sup id="cite_ref-CryptomathicDigSigServicesAshiqJA_16-1" class="reference"><a href="#cite_note-CryptomathicDigSigServicesAshiqJA-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Very roughly this is analogous to a vendor who receives credit-cards first checking online with the credit-card issuer to find if a given card has been reported lost or stolen. Of course, with stolen key pairs, the theft is often discovered only after the secret key's use, e.g., to sign a bogus certificate for espionage purpose. </p> <div class="mw-heading mw-heading2"><h2 id="Notions_of_security">Notions of security</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=8" title="Edit section: Notions of security"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In their foundational paper, Goldwasser, Micali, and Rivest lay out a hierarchy of attack models against digital signatures:<sup id="cite_ref-SJC_17(2)_30-2" class="reference"><a href="#cite_note-SJC_17(2)-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <ol><li>In a <i>key-only</i> attack, the attacker is only given the public verification key.</li> <li>In a <i>known message</i> attack, the attacker is given valid signatures for a variety of messages known by the attacker but not chosen by the attacker.</li> <li>In an <i>adaptive chosen message</i> attack, the attacker first learns signatures on arbitrary messages of the attacker's choice.</li></ol> <p>They also describe a hierarchy of attack results:<sup id="cite_ref-SJC_17(2)_30-3" class="reference"><a href="#cite_note-SJC_17(2)-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <ol><li>A <i>total break</i> results in the recovery of the signing key.</li> <li>A <a href="/wiki/Universal_forgery" class="mw-redirect" title="Universal forgery">universal forgery</a> attack results in the ability to forge signatures for any message.</li> <li>A <a href="/wiki/Selective_forgery" class="mw-redirect" title="Selective forgery">selective forgery</a> attack results in a signature on a message of the adversary's choice.</li> <li>An <a href="/wiki/Existential_forgery" class="mw-redirect" title="Existential forgery">existential forgery</a> merely results in some valid message/signature pair not already known to the adversary.</li></ol> <p>The strongest notion of security, therefore, is security against existential forgery under an adaptive chosen message attack. </p> <div class="mw-heading mw-heading2"><h2 id="Additional_security_precautions">Additional security precautions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=9" title="Edit section: Additional security precautions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Putting_the_private_key_on_a_smart_card">Putting the private key on a smart card</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=10" title="Edit section: Putting the private key on a smart card"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All public key / private key cryptosystems depend entirely on keeping the private key secret. A private key can be stored on a user's computer, and protected by a local password, but this has two disadvantages: </p> <ul><li>the user can only sign documents on that particular computer</li> <li>the security of the private key depends entirely on the <a href="/wiki/Computer_insecurity" class="mw-redirect" title="Computer insecurity">security</a> of the computer</li></ul> <p>A more secure alternative is to store the private key on a <a href="/wiki/Smart_card" title="Smart card">smart card</a>. Many smart cards are designed to be tamper-resistant (although some designs have been broken, notably by <a href="/wiki/Ross_J._Anderson" title="Ross J. Anderson">Ross Anderson</a> and his students<sup id="cite_ref-jKyz7_39-0" class="reference"><a href="#cite_note-jKyz7-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup>). In a typical digital signature implementation, the hash calculated from the document is sent to the smart card, whose CPU signs the hash using the stored private key of the user, and then returns the signed hash. Typically, a user must activate their smart card by entering a <a href="/wiki/Personal_identification_number" title="Personal identification number">personal identification number</a> or PIN code (thus providing <a href="/wiki/Two-factor_authentication" class="mw-redirect" title="Two-factor authentication">two-factor authentication</a>). It can be arranged that the private key never leaves the smart card, although this is not always implemented. If the smart card is stolen, the thief will still need the PIN code to generate a digital signature. This reduces the security of the scheme to that of the PIN system, although it still requires an attacker to possess the card. A mitigating factor is that private keys, if generated and stored on smart cards, are usually regarded as difficult to copy, and are assumed to exist in exactly one copy. Thus, the loss of the smart card may be detected by the owner and the corresponding certificate can be immediately revoked. Private keys that are protected by software only may be easier to copy, and such compromises are far more difficult to detect. </p> <div class="mw-heading mw-heading3"><h3 id="Using_smart_card_readers_with_a_separate_keyboard">Using smart card readers with a separate keyboard</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=11" title="Edit section: Using smart card readers with a separate keyboard"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Entering a PIN code to activate the smart card commonly requires a <a href="/wiki/Numeric_keypad" title="Numeric keypad">numeric keypad</a>. Some card readers have their own numeric keypad. This is safer than using a card reader integrated into a PC, and then entering the PIN using that computer's keyboard. Readers with a numeric keypad are meant to circumvent the eavesdropping threat where the computer might be running a <a href="/wiki/Keystroke_logging" title="Keystroke logging">keystroke logger</a>, potentially compromising the PIN code. Specialized card readers are also less vulnerable to tampering with their software or hardware and are often <a href="/wiki/Evaluation_Assurance_Level" title="Evaluation Assurance Level">EAL3</a> certified. </p> <div class="mw-heading mw-heading3"><h3 id="Other_smart_card_designs">Other smart card designs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=12" title="Edit section: Other smart card designs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Smart card design is an active field, and there are smart card schemes which are intended to avoid these particular problems, despite having few security proofs so far. </p> <div class="mw-heading mw-heading3"><h3 id="Using_digital_signatures_only_with_trusted_applications">Using digital signatures only with trusted applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=13" title="Edit section: Using digital signatures only with trusted applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the main differences between a digital signature and a written signature is that the user does not "see" what they sign. The user application presents a hash code to be signed by the digital signing algorithm using the private key. An attacker who gains control of the user's PC can possibly replace the user application with a foreign substitute, in effect replacing the user's own communications with those of the attacker. This could allow a malicious application to trick a user into signing any document by displaying the user's original on-screen, but presenting the attacker's own documents to the signing application. </p><p>To protect against this scenario, an authentication system can be set up between the user's application (word processor, email client, etc.) and the signing application. The general idea is to provide some means for both the user application and signing application to verify each other's integrity. For example, the signing application may require all requests to come from digitally signed binaries. </p> <div class="mw-heading mw-heading3"><h3 id="Using_a_network_attached_hardware_security_module">Using a network attached hardware security module</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=14" title="Edit section: Using a network attached hardware security module"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the main differences between a <a href="/wiki/Cloud_computing" title="Cloud computing">cloud</a> based digital signature service and a locally provided one is risk. Many risk averse companies, including governments, financial and medical institutions, and payment processors require more secure standards, like <a href="/wiki/FIPS_140-2" title="FIPS 140-2">FIPS 140-2</a> level 3 and <a href="/wiki/FIPS_201" title="FIPS 201">FIPS 201</a> certification, to ensure the signature is validated and secure. </p> <div class="mw-heading mw-heading3"><h3 id="WYSIWYS">WYSIWYS</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=15" title="Edit section: WYSIWYS"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/WYSIWYS" title="WYSIWYS">WYSIWYS</a></div> <p>Technically speaking, a digital signature applies to a string of bits, whereas humans and applications "believe" that they sign the semantic interpretation of those bits. In order to be semantically interpreted, the bit string must be transformed into a form that is meaningful for humans and applications, and this is done through a combination of hardware and software based processes on a computer system. The problem is that the semantic interpretation of bits can change as a function of the processes used to transform the bits into semantic content. It is relatively easy to change the interpretation of a digital document by implementing changes on the computer system where the document is being processed. From a semantic perspective this creates uncertainty about what exactly has been signed. WYSIWYS (What You See Is What You Sign)<sup id="cite_ref-WYSIWYS_SeminalPaper_40-0" class="reference"><a href="#cite_note-WYSIWYS_SeminalPaper-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> means that the semantic interpretation of a signed message cannot be changed. In particular this also means that a message cannot contain hidden information that the signer is unaware of, and that can be revealed after the signature has been applied. WYSIWYS is a requirement for the validity of digital signatures, but this requirement is difficult to guarantee because of the increasing complexity of modern computer systems. The term WYSIWYS was coined by <a href="/wiki/Peter_Landrock" title="Peter Landrock">Peter Landrock</a> and <a href="/wiki/Cryptomathic" title="Cryptomathic">Torben Pedersen</a> to describe some of the principles in delivering secure and legally binding digital signatures for Pan-European projects.<sup id="cite_ref-WYSIWYS_SeminalPaper_40-1" class="reference"><a href="#cite_note-WYSIWYS_SeminalPaper-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Digital_signatures_versus_ink_on_paper_signatures">Digital signatures versus ink on paper signatures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=16" title="Edit section: Digital signatures versus ink on paper signatures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An ink signature could be replicated from one document to another by copying the image manually or digitally, but to have credible signature copies that can resist some scrutiny is a significant manual or technical skill, and to produce ink signature copies that resist professional scrutiny is very difficult. </p><p>Digital signatures cryptographically bind an electronic identity to an electronic document and the digital signature cannot be copied to another document. Paper contracts sometimes have the ink signature block on the last page, and the previous pages may be replaced after a signature is applied. Digital signatures can be applied to an entire document, such that the digital signature on the last page will indicate tampering if any data on any of the pages have been altered, but this can also be achieved by signing with ink and numbering all pages of the contract. </p> <div class="mw-heading mw-heading2"><h2 id="Some_digital_signature_algorithms">Some digital signature algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=17" title="Edit section: Some digital signature algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a></li> <li><a href="/wiki/Digital_Signature_Algorithm" title="Digital Signature Algorithm">DSA</a></li> <li><a href="/wiki/Elliptic_Curve_Digital_Signature_Algorithm" title="Elliptic Curve Digital Signature Algorithm">ECDSA</a></li> <li><a href="/wiki/EdDSA" title="EdDSA">EdDSA</a></li> <li>RSA with <a href="/wiki/Secure_Hash_Algorithm" class="mw-redirect" title="Secure Hash Algorithm">SHA</a></li> <li>ECDSA with SHA<sup id="cite_ref-4zta8_41-0" class="reference"><a href="#cite_note-4zta8-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/ElGamal_signature_scheme" title="ElGamal signature scheme">ElGamal signature scheme</a> as the predecessor to DSA, and variants <a href="/wiki/Schnorr_signature" title="Schnorr signature">Schnorr signature</a> and <a href="/wiki/Pointcheval%E2%80%93Stern_signature_algorithm" title="Pointcheval–Stern signature algorithm">Pointcheval–Stern signature algorithm</a></li> <li><a href="/wiki/Rabin_signature_algorithm" title="Rabin signature algorithm">Rabin signature algorithm</a></li> <li><a href="/wiki/Pairing" title="Pairing">Pairing</a>-based schemes such as <a href="/wiki/Boneh%E2%80%93Lynn%E2%80%93Shacham" class="mw-redirect" title="Boneh–Lynn–Shacham">BLS</a></li> <li>CRYSTALS-Dilithium, a <a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">quantum-resistant scheme</a> based on <a href="/wiki/Learning_with_errors" title="Learning with errors">LWE</a> in <a href="/wiki/Lattice-based_cryptography" title="Lattice-based cryptography">lattices</a></li> <li><a href="/wiki/Falcon_(signature_scheme)" title="Falcon (signature scheme)">Falcon</a>, a <a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">quantum-resistant scheme</a> based on <a href="/wiki/Closest_vector_problem" class="mw-redirect" title="Closest vector problem">CVP</a> in <a href="/wiki/Lattice-based_cryptography" title="Lattice-based cryptography">lattices</a></li> <li>SPHINCS+, a <a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">quantum-resistant scheme</a> based on <a href="/wiki/Hash_function" title="Hash function">hash functions</a></li> <li><a href="/wiki/Undeniable_signature" title="Undeniable signature">Undeniable signatures</a></li> <li><a href="/w/index.php?title=Aggregate_signature&action=edit&redlink=1" class="new" title="Aggregate signature (page does not exist)">Aggregate signature</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D0%BF%D0%BE%D0%B4%D0%BF%D0%B8%D1%81%D1%8C" class="extiw" title="ru:Множественная подпись">ru</a>]</span> – a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users. This single signature will convince the verifier that the n users did indeed sign the n original messages. A scheme by <a href="/wiki/Mihir_Bellare" title="Mihir Bellare">Mihir Bellare</a> and <a href="/w/index.php?title=Gregory_Neven&action=edit&redlink=1" class="new" title="Gregory Neven (page does not exist)">Gregory Neven</a> may be used with <a href="/wiki/Bitcoin" title="Bitcoin">Bitcoin</a>.<sup id="cite_ref-DdBGf_42-0" class="reference"><a href="#cite_note-DdBGf-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Signatures_with_efficient_protocols" title="Signatures with efficient protocols">Signatures with efficient protocols</a> – are signature schemes that facilitate efficient cryptographic protocols such as <a href="/wiki/Zero-knowledge_proofs" class="mw-redirect" title="Zero-knowledge proofs">zero-knowledge proofs</a> or <a href="/wiki/Secure_computation" class="mw-redirect" title="Secure computation">secure computation</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="The_current_state_of_use_–_legal_and_practical"><span id="The_current_state_of_use_.E2.80.93_legal_and_practical"></span>The current state of use – legal and practical</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=18" title="Edit section: The current state of use – legal and practical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For International uses, see <a href="/wiki/Electronic_signatures_and_law" title="Electronic signatures and law">Electronic signatures and law</a></div> <p>Most digital signature schemes share the following goals regardless of cryptographic theory or legal provision: </p> <ol><li>Quality algorithms: Some public-key algorithms are known to be insecure, as practical attacks against them have been discovered.</li> <li class="mw-empty-elt"></li> <li>Quality implementations: An implementation of a good algorithm (or <a href="/wiki/Cryptographic_protocol" title="Cryptographic protocol">protocol</a>) with mistake(s) will not work.</li> <li class="mw-empty-elt"></li> <li>Users (and their software) must carry out the signature protocol properly.</li> <li class="mw-empty-elt"></li> <li>The private key must remain private: If the private key becomes known to any other party, that party can produce <i>perfect</i> digital signatures of anything.</li> <li class="mw-empty-elt"></li> <li>The public key owner must be verifiable: A public key associated with Bob actually came from Bob. This is commonly done using a <a href="/wiki/Public_key_infrastructure" title="Public key infrastructure">public key infrastructure</a> (PKI) and the public key↔user association is attested by the operator of the PKI (called a <a href="/wiki/Certificate_authority" title="Certificate authority">certificate authority</a>). For 'open' PKIs in which anyone can request such an attestation (universally embodied in a cryptographically protected <a href="/wiki/Public_key_certificate" title="Public key certificate">public key certificate</a>), the possibility of mistaken attestation is non-trivial. Commercial PKI operators have suffered several publicly known problems. Such mistakes could lead to falsely signed, and thus wrongly attributed, documents. 'Closed' PKI systems are more expensive, but less easily subverted in this way.</li></ol> <p>Only if all of these conditions are met will a digital signature actually be any evidence of who sent the message, and therefore of their assent to its contents. Legal enactment cannot change this reality of the existing engineering possibilities, though some such have not reflected this actuality. </p><p>Legislatures, being importuned by businesses expecting to profit from operating a PKI, or by the technological avant-garde advocating new solutions to old problems, have enacted statutes and/or regulations in many jurisdictions authorizing, endorsing, encouraging, or permitting digital signatures and providing for (or limiting) their legal effect. The first appears to have been in <a href="/wiki/Utah" title="Utah">Utah</a> in the United States, followed closely by the states <a href="/wiki/Massachusetts" title="Massachusetts">Massachusetts</a> and <a href="/wiki/California" title="California">California</a>. Other countries have also passed statutes or issued regulations in this area as well and the UN has had an active model law project for some time. These enactments (or proposed enactments) vary from place to place, have typically embodied expectations at variance (optimistically or pessimistically) with the state of the underlying cryptographic engineering, and have had the net effect of confusing potential users and specifiers, nearly all of whom are not cryptographically knowledgeable. </p><p>Adoption of technical standards for digital signatures have lagged behind much of the legislation, delaying a more or less unified engineering position on <a href="/wiki/Interoperability" title="Interoperability">interoperability</a>, <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> choice, <a href="/wiki/Key_length" class="mw-redirect" title="Key length">key lengths</a>, and so on what the engineering is attempting to provide. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/ABA_digital_signature_guidelines" title="ABA digital signature guidelines">ABA digital signature guidelines</a></div> <div class="mw-heading mw-heading2"><h2 id="Industry_standards">Industry standards</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=19" title="Edit section: Industry standards"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Digital_signature" title="Special:EditPage/Digital signature">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">January 2015</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Some industries have established common interoperability standards for the use of digital signatures between members of the industry and with regulators. These include the <a href="/wiki/Automotive_Network_Exchange" title="Automotive Network Exchange">Automotive Network Exchange</a> for the automobile industry and the SAFE-BioPharma Association for the <a href="/wiki/Healthcare_industry" title="Healthcare industry">healthcare industry</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Using_separate_key_pairs_for_signing_and_encryption">Using separate key pairs for signing and encryption</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=20" title="Edit section: Using separate key pairs for signing and encryption"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In several countries, a digital signature has a status somewhat like that of a traditional pen and paper signature, as in the <a href="/wiki/Electronic_Signatures_Directive" title="Electronic Signatures Directive">1999 EU digital signature directive</a> and <a href="/wiki/EIDAS" title="EIDAS">2014 EU follow-on legislation</a>.<sup id="cite_ref-Cryptomathic_MajorStandardsDigSig_15-2" class="reference"><a href="#cite_note-Cryptomathic_MajorStandardsDigSig-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Generally, these provisions mean that anything digitally signed legally binds the signer of the document to the terms therein. For that reason, it is often thought best to use separate key pairs for encrypting and signing. Using the encryption key pair, a person can engage in an encrypted conversation (e.g., regarding a real estate transaction), but the encryption does not legally sign every message he or she sends. Only when both parties come to an agreement do they sign a contract with their signing keys, and only then are they legally bound by the terms of a specific document. After signing, the document can be sent over the encrypted link. If a signing key is lost or compromised, it can be revoked to mitigate any future transactions. If an encryption key is lost, a backup or <a href="/wiki/Key_escrow" title="Key escrow">key escrow</a> should be utilized to continue viewing encrypted content. Signing keys should never be backed up or escrowed unless the backup destination is securely encrypted. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/21_CFR_11" class="mw-redirect" title="21 CFR 11">21 CFR 11</a></li> <li><a href="/wiki/X.509" title="X.509">X.509</a></li> <li><a href="/wiki/Advanced_electronic_signature" title="Advanced electronic signature">Advanced electronic signature</a></li> <li><a href="/wiki/Blind_signature" title="Blind signature">Blind signature</a></li> <li><a href="/wiki/Detached_signature" title="Detached signature">Detached signature</a></li> <li><a href="/wiki/Public_key_certificate" title="Public key certificate">Public key certificate</a></li> <li><a href="/wiki/Digital_signature_in_Estonia" title="Digital signature in Estonia">Digital signature in Estonia</a></li> <li><a href="/wiki/Electronic_lab_notebook" title="Electronic lab notebook">Electronic lab notebook</a></li> <li><a href="/wiki/Electronic_signature" title="Electronic signature">Electronic signature</a></li> <li><a href="/wiki/Electronic_signatures_and_law" title="Electronic signatures and law">Electronic signatures and law</a></li> <li><a href="/wiki/ESign_(India)" title="ESign (India)">eSign (India)</a></li> <li><a href="/wiki/GNU_Privacy_Guard" title="GNU Privacy Guard">GNU Privacy Guard</a></li> <li><a href="/wiki/Public_key_infrastructure" title="Public key infrastructure">Public key infrastructure</a></li> <li><a href="/wiki/Public_key_fingerprint" title="Public key fingerprint">Public key fingerprint</a></li> <li><a href="/wiki/Server-based_signatures" title="Server-based signatures">Server-based signatures</a></li> <li><a href="/wiki/Probabilistic_signature_scheme" title="Probabilistic signature scheme">Probabilistic signature scheme</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=22" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 mw-references-columns"><ol class="references"> <li id="cite_note-bellare-goldwasser2008digsigs-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-bellare-goldwasser2008digsigs_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bellare-goldwasser2008digsigs_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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="CITEREFBellareGoldwasser2008" class="citation book cs1"><a href="/wiki/Mihir_Bellare" title="Mihir Bellare">Bellare, Mihir</a>; <a href="/wiki/Shafi_Goldwasser" title="Shafi Goldwasser">Goldwasser, Shafi</a> (July 2008). "Chapter 10: Digital signatures". <a rel="nofollow" class="external text" href="https://cseweb.ucsd.edu/~mihir/papers/gb.pdf#page=168"><i>Lecture Notes on Cryptography</i></a> <span class="cs1-format">(PDF)</span>. p. 168. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220420003617/https://cseweb.ucsd.edu/~mihir/papers/gb.pdf#page=168">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-04-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-06-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+10%3A+Digital+signatures&rft.btitle=Lecture+Notes+on+Cryptography&rft.pages=168&rft.date=2008-07&rft.aulast=Bellare&rft.aufirst=Mihir&rft.au=Goldwasser%2C+Shafi&rft_id=https%3A%2F%2Fcseweb.ucsd.edu%2F~mihir%2Fpapers%2Fgb.pdf%23page%3D168&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-katz-lindell2007digsigs-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-katz-lindell2007digsigs_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-katz-lindell2007digsigs_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-katz-lindell2007digsigs_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatzLindell2007" class="citation book cs1"><a href="/wiki/Jonathan_Katz_(computer_scientist)" title="Jonathan Katz (computer scientist)">Katz, Jonathan</a>; <a href="/wiki/Yehuda_Lindell" title="Yehuda Lindell">Lindell, Yehuda</a> (2007). "Chapter 12: Digital Signature Schemes". <i>Introduction to Modern Cryptography</i>. p. 399.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+12%3A+Digital+Signature+Schemes&rft.btitle=Introduction+to+Modern+Cryptography&rft.pages=399&rft.date=2007&rft.aulast=Katz&rft.aufirst=Jonathan&rft.au=Lindell%2C+Yehuda&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-cEMCl-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-cEMCl_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://frwebgate.access.gpo.gov/cgi-bin/getdoc.cgi?dbname=106_cong_public_laws&docid=f:publ229.106.pdf">"US ESIGN Act of 2000"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110522212411/http://frwebgate.access.gpo.gov/cgi-bin/getdoc.cgi?dbname=106_cong_public_laws&docid=f:publ229.106.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2011-05-22<span class="reference-accessdate">. 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Retrieved <span class="nowrap">2020-10-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Communications&rft.atitle=Securing+digital+signatures+for+non-repudiation&rft.volume=22&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E710-%3C%2Fspan%3E716&rft.date=1999-05&rft_id=info%3Adoi%2F10.1016%2Fs0140-3664%2899%2900031-6&rft.issn=0140-3664&rft.aulast=Zhou&rft.aufirst=J.&rft.au=Lam%2C+K.Y.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1016%2Fs0140-3664%2899%2900031-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-IjhTrs-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-IjhTrs_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IjhTrs_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Pass, def 135.1</span> </li> <li id="cite_note-nOvvy-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-nOvvy_21-0">^</a></b></span> <span class="reference-text">Goldreich's FoC, vol. 2, def 6.1.2. Pass, def 135.2</span> </li> <li id="cite_note-ikWoF-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-ikWoF_22-0">^</a></b></span> <span class="reference-text">"New Directions in Cryptography", IEEE Transactions on Information Theory, IT-22(6):644–654, Nov. 1976.</span> </li> <li id="cite_note-lysythesis-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-lysythesis_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lysythesis_23-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://dspace.mit.edu/handle/1721.1/29271">Signature Schemes and Applications to Cryptographic Protocol Design</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220908083823/https://dspace.mit.edu/handle/1721.1/29271">Archived</a> 2022-09-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>", <a href="/wiki/Anna_Lysyanskaya" title="Anna Lysyanskaya">Anna Lysyanskaya</a>, PhD thesis, <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">MIT</a>, 2002.</span> </li> <li id="cite_note-rsa-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-rsa_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRivestShamirAdleman1978" class="citation journal cs1">Rivest, R.; Shamir, A.; Adleman, L. 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Archived from <a rel="nofollow" class="external text" href="http://people.csail.mit.edu/rivest/Rsapaper.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2008-12-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-11-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=A+Method+for+Obtaining+Digital+Signatures+and+Public-Key+Cryptosystems&rft.volume=21&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E120-%3C%2Fspan%3E126&rft.date=1978&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.607.2677%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2873616%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F359340.359342&rft.aulast=Rivest&rft.aufirst=R.&rft.au=Shamir%2C+A.&rft.au=Adleman%2C+L.&rft_id=http%3A%2F%2Fpeople.csail.mit.edu%2Frivest%2FRsapaper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-7aSJZ-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-7aSJZ_25-0">^</a></b></span> <span class="reference-text">For example any integer, <i>r</i>, "signs" <i>m</i>=<i>r</i><sup><i>e</i></sup> and the product, <i>s</i><sub>1</sub><i>s</i><sub>2</sub>, of any two valid signatures, <i>s</i><sub>1</sub>, <i>s</i><sub>2</sub> of <i>m</i><sub>1</sub>, <i>m</i><sub>2</sub> is a valid signature of the product, <i>m</i><sub>1</sub><i>m</i><sub>2</sub>.</span> </li> <li id="cite_note-CKgyC-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-CKgyC_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.ibm.com/developerworks/lotus/library/ls-NDHistory/">"The History of Notes and Domino"</a>. <i>developerWorks</i>. 2007-11-14. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130305042623/http://www.ibm.com/developerworks/lotus/library/ls-NDHistory/">Archived</a> from the original on 2013-03-05<span class="reference-accessdate">. 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Rabin">Michael O. Rabin</a>, Technical Report MIT/LCS/TR-212, MIT Laboratory for Computer Science, Jan. 1979</span> </li> <li id="cite_note-SJC_17(2)-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-SJC_17(2)_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SJC_17(2)_30-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-SJC_17(2)_30-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-SJC_17(2)_30-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">"A digital signature scheme secure against adaptive chosen-message attacks.", Shafi Goldwasser, Silvio Micali, and Ronald Rivest. SIAM Journal on Computing, 17(2):281–308, Apr. 1988.</span> </li> <li id="cite_note-Z2zaX-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Z2zaX_31-0">^</a></b></span> <span class="reference-text">Moni Naor, Moti Yung: Universal One-Way Hash Functions and their Cryptographic Applications. 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Chapman & Hall/CRC. p. 281. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-508-5" title="Special:BookSources/978-1-58488-508-5"><bdi>978-1-58488-508-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=7%3A+Signature+Schemes&rft.btitle=Cryptography%3A+Theory+and+Practice&rft.pages=281&rft.edition=3rd&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2006&rft.isbn=978-1-58488-508-5&rft.aulast=Stinson&rft.aufirst=Douglas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-bcjz2020ed25519sectheorypractice-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-bcjz2020ed25519sectheorypractice_36-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrendelCremersJacksonZhao2020" class="citation techreport cs1">Brendel, Jacqueline; Cremers, Cas; Jackson, Dennis; Zhao, Meng (2020-10-14). <a rel="nofollow" class="external text" href="https://eprint.iacr.org/2020/823"><i>The Provable Security of Ed25519: Theory and Practice</i></a> (Technical report). IACR Cryptology ePrint Archive. 2020/823.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=The+Provable+Security+of+Ed25519%3A+Theory+and+Practice&rft.pub=IACR+Cryptology+ePrint+Archive&rft.date=2020-10-14&rft.aulast=Brendel&rft.aufirst=Jacqueline&rft.au=Cremers%2C+Cas&rft.au=Jackson%2C+Dennis&rft.au=Zhao%2C+Meng&rft_id=https%3A%2F%2Feprint.iacr.org%2F2020%2F823&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-decker-wattenhofer2014btcmalleablemtgox-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-decker-wattenhofer2014btcmalleablemtgox_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeckerWattenhofer2014" class="citation conference cs1">Decker, Christian; Wattenhofer, Roger (2014). Kutyłowski, Mirosław; Vaidya, Jaideep (eds.). <i>Bitcoin Transaction Malleability and MtGox</i>. European Symposium on Research in Computer Security—ESORICS. Lecture Notes in Computer Science. Vol. 8713. Springer. pp. <span class="nowrap">313–</span>326. <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/1403.6676">1403.6676</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-11212-1_18">10.1007/978-3-319-11212-1_18</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-11212-1" title="Special:BookSources/978-3-319-11212-1"><bdi>978-3-319-11212-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Bitcoin+Transaction+Malleability+and+MtGox&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E313-%3C%2Fspan%3E326&rft.pub=Springer&rft.date=2014&rft_id=info%3Aarxiv%2F1403.6676&rft_id=info%3Adoi%2F10.1007%2F978-3-319-11212-1_18&rft.isbn=978-3-319-11212-1&rft.aulast=Decker&rft.aufirst=Christian&rft.au=Wattenhofer%2C+Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-ayer2015sigmisuseacme-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-ayer2015sigmisuseacme_38-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAyer2015" class="citation mailinglist cs1">Ayer, Andrew (2015-08-11). <a rel="nofollow" class="external text" href="https://mailarchive.ietf.org/arch/msg/acme/F71iz6qq1o_QPVhJCV4dqWf-4Yc/">"Signature misuse vulnerability in draft-barnes-acme-04"</a>. <i>acme@ietf.org</i> (Mailing list)<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-06-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Signature+misuse+vulnerability+in+draft-barnes-acme-04&rft.date=2015-08-11&rft.aulast=Ayer&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fmailarchive.ietf.org%2Farch%2Fmsg%2Facme%2FF71iz6qq1o_QPVhJCV4dqWf-4Yc%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-jKyz7-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-jKyz7_39-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180516203624/http://sec.cs.ucl.ac.uk/users/smurdoch/papers/oakland14chipandskim.pdf">"Chip and Skim: cloning EMV cards with the pre-play attack"</a> <span class="cs1-format">(PDF)</span>. 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Retrieved <span class="nowrap">2018-04-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Chip+and+Skim%3A+cloning+EMV+cards+with+the+pre-play+attack&rft_id=http%3A%2F%2Fsec.cs.ucl.ac.uk%2Fusers%2Fsmurdoch%2Fpapers%2Foakland14chipandskim.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-WYSIWYS_SeminalPaper-40"><span class="mw-cite-backlink">^ <a href="#cite_ref-WYSIWYS_SeminalPaper_40-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-WYSIWYS_SeminalPaper_40-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandrockPedersen1998" class="citation journal cs1">Landrock, Peter; Pedersen, Torben (1998). "WYSIWYS? – What you see is what you sign?". <i>Information Security Technical Report</i>. <b>3</b> (2): <span class="nowrap">55–</span>61. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0167-4048%2898%2980005-8">10.1016/S0167-4048(98)80005-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+Security+Technical+Report&rft.atitle=WYSIWYS%3F+%E2%80%93+What+you+see+is+what+you+sign%3F&rft.volume=3&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E55-%3C%2Fspan%3E61&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2FS0167-4048%2898%2980005-8&rft.aulast=Landrock&rft.aufirst=Peter&rft.au=Pedersen%2C+Torben&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> <li id="cite_note-4zta8-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-4zta8_41-0">^</a></b></span> <span class="reference-text">RFC 5758</span> </li> <li id="cite_note-DdBGf-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-DdBGf_42-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://bitcoincore.org/en/2017/03/23/schnorr-signature-aggregation/">"Technology roadmap – Schnorr signatures and signature aggregation"</a>. <i>bitcoincore.org</i>. Bitcoin Core. 23 March 2017. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180324211259/https://bitcoincore.org/en/2017/03/23/schnorr-signature-aggregation/">Archived</a> from the original on 24 March 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">1 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=bitcoincore.org&rft.atitle=Technology+roadmap+%E2%80%93+Schnorr+signatures+and+signature+aggregation&rft.date=2017-03-23&rft_id=https%3A%2F%2Fbitcoincore.org%2Fen%2F2017%2F03%2F23%2Fschnorr-signature-aggregation%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldreich2001" class="citation cs2">Goldreich, Oded (2001), <i>Foundations of cryptography I: Basic Tools</i>, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-511-54689-1" title="Special:BookSources/978-0-511-54689-1"><bdi>978-0-511-54689-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+cryptography+I%3A+Basic+Tools&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2001&rft.isbn=978-0-511-54689-1&rft.aulast=Goldreich&rft.aufirst=Oded&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldreich2004" class="citation cs2">Goldreich, Oded (2004), <i>Foundations of cryptography II: Basic Applications</i> (1. publ. ed.), Cambridge [u.a.]: Cambridge Univ. Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83084-3" title="Special:BookSources/978-0-521-83084-3"><bdi>978-0-521-83084-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+cryptography+II%3A+Basic+Applications&rft.place=Cambridge+%5Bu.a.%5D&rft.edition=1.+publ.&rft.pub=Cambridge+Univ.+Press&rft.date=2004&rft.isbn=978-0-521-83084-3&rft.aulast=Goldreich&rft.aufirst=Oded&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPass" class="citation cs2">Pass, Rafael, <a rel="nofollow" class="external text" href="https://www.cs.cornell.edu/courses/cs4830/2010fa/lecnotes.pdf"><i>A Course in Cryptography</i></a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">, retrieved <span class="nowrap">31 December</span> 2015</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Cryptography&rft.aulast=Pass&rft.aufirst=Rafael&rft_id=https%3A%2F%2Fwww.cs.cornell.edu%2Fcourses%2Fcs4830%2F2010fa%2Flecnotes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADigital+signature" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Digital_signature&action=edit&section=24" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>J. Katz and Y. Lindell, "Introduction to Modern Cryptography" (Chapman & Hall/CRC Press, 2007)</li> <li>Lorna Brazell, Electronic Signatures and Identities Law and Regulation (2nd edn, London: Sweet & Maxwell, 2008)</li> <li>Dennis Campbell, editor, E-Commerce and the Law of Digital Signatures (Oceana Publications, 2005)</li> <li>M. H. M Schellenkens, Electronic Signatures Authentication Technology from a Legal Perspective, (TMC Asser Press, 2004)</li> <li>Jeremiah S. Buckley, John P. Kromer, Margo H. K. Tank, and R. David Whitaker, The Law of Electronic Signatures (3rd Edition, West Publishing, 2010).</li> <li><a rel="nofollow" class="external text" href="http://journals.sas.ac.uk/deeslr/"><i>Digital Evidence and Electronic Signature Law Review</i></a> Free open source</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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href="/wiki/Template:Cryptography_public-key" title="Template:Cryptography public-key"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Cryptography_public-key" title="Template talk:Cryptography public-key"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Cryptography_public-key" title="Special:EditPage/Template:Cryptography public-key"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Public-key_cryptography64" style="font-size:114%;margin:0 4em"><a href="/wiki/Public-key_cryptography" title="Public-key cryptography">Public-key cryptography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group wraplinks" style="width:1%"><a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benaloh_cryptosystem" title="Benaloh cryptosystem">Benaloh</a></li> <li><a href="/wiki/Blum%E2%80%93Goldwasser_cryptosystem" title="Blum–Goldwasser cryptosystem">Blum–Goldwasser</a></li> <li><a href="/wiki/Cayley%E2%80%93Purser_algorithm" title="Cayley–Purser algorithm">Cayley–Purser</a></li> <li><a href="/wiki/Damg%C3%A5rd%E2%80%93Jurik_cryptosystem" title="Damgård–Jurik cryptosystem">Damgård–Jurik</a></li> <li><a href="/wiki/GMR_(cryptography)" title="GMR (cryptography)">GMR</a></li> <li><a href="/wiki/Goldwasser%E2%80%93Micali_cryptosystem" title="Goldwasser–Micali cryptosystem">Goldwasser–Micali</a></li> <li><a href="/wiki/Naccache%E2%80%93Stern_cryptosystem" title="Naccache–Stern cryptosystem">Naccache–Stern</a></li> <li><a href="/wiki/Paillier_cryptosystem" title="Paillier cryptosystem">Paillier</a></li> <li><a href="/wiki/Rabin_cryptosystem" title="Rabin cryptosystem">Rabin</a></li> <li><a href="/wiki/RSA_(cryptosystem)" title="RSA (cryptosystem)">RSA</a></li> <li><a href="/wiki/Okamoto%E2%80%93Uchiyama_cryptosystem" title="Okamoto–Uchiyama cryptosystem">Okamoto–Uchiyama</a></li> <li><a href="/wiki/Schmidt-Samoa_cryptosystem" title="Schmidt-Samoa cryptosystem">Schmidt–Samoa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%"><a href="/wiki/Discrete_logarithm" title="Discrete logarithm">Discrete logarithm</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boneh%E2%80%93Lynn%E2%80%93Shacham" class="mw-redirect" title="Boneh–Lynn–Shacham">BLS</a></li> <li><a href="/wiki/Cramer%E2%80%93Shoup_cryptosystem" title="Cramer–Shoup cryptosystem">Cramer–Shoup</a></li> <li><a href="/wiki/Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">DH</a></li> <li><a href="/wiki/Digital_Signature_Algorithm" title="Digital Signature Algorithm">DSA</a></li> <li><a href="/wiki/Elliptic-curve_Diffie%E2%80%93Hellman" title="Elliptic-curve Diffie–Hellman">ECDH</a> <ul><li><a href="/wiki/Curve25519" title="Curve25519">X25519</a></li> <li><a href="/wiki/Curve448" title="Curve448">X448</a></li></ul></li> <li><a href="/wiki/Elliptic_Curve_Digital_Signature_Algorithm" title="Elliptic Curve Digital Signature Algorithm">ECDSA</a></li> <li><a href="/wiki/EdDSA" title="EdDSA">EdDSA</a> <ul><li><a href="/wiki/EdDSA#Ed25519" title="EdDSA">Ed25519</a></li> <li><a href="/wiki/EdDSA#Ed448" title="EdDSA">Ed448</a></li></ul></li> <li><a href="/wiki/ECMQV" class="mw-redirect" title="ECMQV">ECMQV</a></li> <li><a href="/wiki/Encrypted_key_exchange" title="Encrypted key exchange">EKE</a></li> <li><a href="/wiki/ElGamal_encryption" title="ElGamal encryption">ElGamal</a> <ul><li><a href="/wiki/ElGamal_signature_scheme" title="ElGamal signature scheme">signature scheme</a></li></ul></li> <li><a href="/wiki/MQV" title="MQV">MQV</a></li> <li><a href="/wiki/Schnorr_signature" title="Schnorr signature">Schnorr</a></li> <li><a href="/wiki/SPEKE" title="SPEKE">SPEKE</a></li> <li><a href="/wiki/Secure_Remote_Password_protocol" title="Secure Remote Password protocol">SRP</a></li> <li><a href="/wiki/Station-to-Station_protocol" title="Station-to-Station protocol">STS</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%"><a href="/wiki/Lattice-based_cryptography" title="Lattice-based cryptography">Lattice/SVP/CVP</a>/<wbr /><a href="/wiki/Learning_with_errors" title="Learning with errors">LWE</a>/<wbr /><a href="/wiki/Short_integer_solution_problem" title="Short integer solution problem">SIS</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BLISS_signature_scheme" title="BLISS signature scheme">BLISS</a></li> <li><a href="/wiki/Kyber" title="Kyber">Kyber</a></li> <li><a href="/wiki/NewHope" title="NewHope">NewHope</a></li> <li><a href="/wiki/NTRUEncrypt" title="NTRUEncrypt">NTRUEncrypt</a></li> <li><a href="/wiki/NTRUSign" title="NTRUSign">NTRUSign</a></li> <li><a href="/wiki/RLWE-KEX" class="mw-redirect" title="RLWE-KEX">RLWE-KEX</a></li> <li><a href="/wiki/RLWE-SIG" class="mw-redirect" title="RLWE-SIG">RLWE-SIG</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group wraplinks" style="width:1%">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_Eraser" title="Algebraic Eraser">AE</a></li> <li><a href="/wiki/CEILIDH" title="CEILIDH">CEILIDH</a></li> <li><a href="/wiki/Efficient_Probabilistic_Public-Key_Encryption_Scheme" title="Efficient Probabilistic Public-Key Encryption Scheme">EPOC</a></li> <li><a href="/wiki/Hidden_Field_Equations" title="Hidden Field Equations">HFE</a></li> <li><a href="/wiki/Integrated_Encryption_Scheme" title="Integrated Encryption Scheme">IES</a></li> <li><a href="/wiki/Lamport_signature" title="Lamport signature">Lamport</a></li> <li><a href="/wiki/McEliece_cryptosystem" title="McEliece cryptosystem">McEliece</a></li> <li><a href="/wiki/Merkle%E2%80%93Hellman_knapsack_cryptosystem" title="Merkle–Hellman knapsack cryptosystem">Merkle–Hellman</a></li> <li><span class="wraplinks"><a href="/wiki/Naccache%E2%80%93Stern_knapsack_cryptosystem" title="Naccache–Stern knapsack cryptosystem">Naccache–Stern knapsack cryptosystem</a></span></li> <li><a href="/wiki/Three-pass_protocol" title="Three-pass protocol">Three-pass protocol</a></li> <li><a href="/wiki/XTR" title="XTR">XTR</a></li> <li><a href="/wiki/SQIsign" title="SQIsign">SQIsign</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_logarithm#Cryptography" title="Discrete logarithm">Discrete logarithm cryptography</a></li> <li><a href="/wiki/Elliptic-curve_cryptography" title="Elliptic-curve cryptography">Elliptic-curve cryptography</a></li> <li><a href="/wiki/Hash-based_cryptography" title="Hash-based cryptography">Hash-based cryptography</a></li> <li><a href="/wiki/Non-commutative_cryptography" title="Non-commutative cryptography">Non-commutative cryptography</a></li> <li><a href="/wiki/RSA_problem" title="RSA problem">RSA problem</a></li> <li><a href="/wiki/Trapdoor_function" title="Trapdoor function">Trapdoor function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Standardization</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/CRYPTREC" title="CRYPTREC">CRYPTREC</a></li> <li><a href="/wiki/IEEE_P1363" title="IEEE P1363">IEEE P1363</a></li> <li><a href="/wiki/NESSIE" title="NESSIE">NESSIE</a></li> <li><a href="/wiki/NSA_Suite_B_Cryptography" title="NSA Suite B Cryptography">NSA Suite B</a></li> <li><a href="/wiki/Commercial_National_Security_Algorithm_Suite" title="Commercial National Security Algorithm Suite">CNSA</a></li> <li><a href="/wiki/NIST_Post-Quantum_Cryptography_Standardization" title="NIST Post-Quantum Cryptography Standardization">Post-Quantum Cryptography</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Digital signature</a></li> <li><a href="/wiki/Optimal_asymmetric_encryption_padding" title="Optimal asymmetric encryption padding">OAEP</a></li> <li><a href="/wiki/Public_key_fingerprint" title="Public key fingerprint">Fingerprint</a></li> <li><a href="/wiki/Public_key_infrastructure" title="Public key infrastructure">PKI</a></li> <li><a href="/wiki/Web_of_trust" title="Web of trust">Web of trust</a></li> <li><a href="/wiki/Key_size" title="Key size">Key size</a></li> <li><a href="/wiki/Identity-based_cryptography" title="Identity-based cryptography">Identity-based cryptography</a></li> <li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/OpenPGP_card" title="OpenPGP card">OpenPGP card</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Cryptography_navbox" title="Template:Cryptography navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Cryptography_navbox" title="Template talk:Cryptography navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Cryptography_navbox" title="Special:EditPage/Template:Cryptography navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Cryptography149" style="font-size:114%;margin:0 4em"><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_cryptography" title="History of cryptography">History of cryptography</a></li> <li><a href="/wiki/Outline_of_cryptography" title="Outline of cryptography">Outline of cryptography</a></li> <li><a href="/wiki/Classical_cipher" title="Classical cipher">Classical cipher</a></li> <li><a href="/wiki/Cryptographic_protocol" title="Cryptographic protocol">Cryptographic protocol</a> <ul><li><a href="/wiki/Authentication_protocol" title="Authentication protocol">Authentication protocol</a></li></ul></li> <li><a href="/wiki/Cryptographic_primitive" title="Cryptographic primitive">Cryptographic primitive</a></li> <li><a href="/wiki/Cryptanalysis" title="Cryptanalysis">Cryptanalysis</a></li> <li><a href="/wiki/Cryptocurrency" title="Cryptocurrency">Cryptocurrency</a></li> <li><a href="/wiki/Cryptosystem" title="Cryptosystem">Cryptosystem</a></li> <li><a href="/wiki/Cryptographic_nonce" title="Cryptographic nonce">Cryptographic nonce</a></li> <li><a href="/wiki/Cryptovirology" title="Cryptovirology">Cryptovirology</a></li> <li><a href="/wiki/Hash_function" title="Hash function">Hash function</a> <ul><li><a href="/wiki/Cryptographic_hash_function" title="Cryptographic hash function">Cryptographic hash function</a></li> <li><a href="/wiki/Key_derivation_function" title="Key derivation function">Key derivation function</a></li> <li><a href="/wiki/Secure_Hash_Algorithms" title="Secure Hash Algorithms">Secure Hash Algorithms</a></li></ul></li> <li><a class="mw-selflink selflink">Digital signature</a></li> <li><a href="/wiki/Kleptography" title="Kleptography">Kleptography</a></li> <li><a href="/wiki/Key_(cryptography)" title="Key (cryptography)">Key (cryptography)</a></li> <li><a href="/wiki/Key_exchange" title="Key exchange">Key exchange</a></li> <li><a href="/wiki/Key_generator" title="Key generator">Key generator</a></li> <li><a href="/wiki/Key_schedule" title="Key schedule">Key schedule</a></li> <li><a href="/wiki/Key_stretching" title="Key stretching">Key stretching</a></li> <li><a href="/wiki/Keygen" title="Keygen">Keygen</a></li> <li><a href="/wiki/Template:Cryptography_machines" title="Template:Cryptography machines">Machines</a></li> <li><a href="/wiki/Cryptojacking_malware" class="mw-redirect" title="Cryptojacking malware">Cryptojacking malware</a></li> <li><a href="/wiki/Ransomware" title="Ransomware">Ransomware</a></li> <li><a href="/wiki/Random_number_generation" title="Random number generation">Random number generation</a> <ul><li><a href="/wiki/Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">Cryptographically secure pseudorandom number generator</a> (CSPRNG)</li></ul></li> <li><a href="/wiki/Pseudorandom_noise" title="Pseudorandom noise">Pseudorandom noise</a> (PRN)</li> <li><a href="/wiki/Secure_channel" title="Secure channel">Secure channel</a></li> <li><a href="/wiki/Insecure_channel" class="mw-redirect" title="Insecure channel">Insecure channel</a></li> <li><a href="/wiki/Subliminal_channel" title="Subliminal channel">Subliminal channel</a></li> <li><a href="/wiki/Encryption" title="Encryption">Encryption</a></li> <li><a href="/wiki/Decryption" class="mw-redirect" title="Decryption">Decryption</a></li> <li><a href="/wiki/End-to-end_encryption" title="End-to-end encryption">End-to-end encryption</a></li> <li><a href="/wiki/Harvest_now,_decrypt_later" title="Harvest now, decrypt later">Harvest now, decrypt later</a></li> <li><a href="/wiki/Information-theoretic_security" title="Information-theoretic security">Information-theoretic security</a></li> <li><a href="/wiki/Plaintext" title="Plaintext">Plaintext</a></li> <li><a href="/wiki/Codetext" class="mw-redirect" title="Codetext">Codetext</a></li> <li><a href="/wiki/Ciphertext" title="Ciphertext">Ciphertext</a></li> <li><a href="/wiki/Shared_secret" title="Shared secret">Shared secret</a></li> <li><a href="/wiki/Trapdoor_function" title="Trapdoor function">Trapdoor function</a></li> <li><a href="/wiki/Trusted_timestamping" title="Trusted timestamping">Trusted timestamping</a></li> <li><a href="/wiki/Key-based_routing" title="Key-based routing">Key-based routing</a></li> <li><a href="/wiki/Onion_routing" title="Onion routing">Onion routing</a></li> <li><a href="/wiki/Garlic_routing" title="Garlic routing">Garlic routing</a></li> <li><a href="/wiki/Kademlia" title="Kademlia">Kademlia</a></li> <li><a href="/wiki/Mix_network" title="Mix network">Mix network</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cryptographic_hash_function" title="Cryptographic hash function">Cryptographic hash function</a></li> <li><a href="/wiki/Block_cipher" title="Block cipher">Block cipher</a></li> <li><a href="/wiki/Stream_cipher" title="Stream cipher">Stream cipher</a></li> <li><a href="/wiki/Symmetric-key_algorithm" title="Symmetric-key algorithm">Symmetric-key algorithm</a></li> <li><a href="/wiki/Authenticated_encryption" title="Authenticated encryption">Authenticated encryption</a></li> <li><a href="/wiki/Public-key_cryptography" title="Public-key cryptography">Public-key cryptography</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/Message_authentication_code" title="Message authentication code">Message authentication code</a></li> <li><a href="/wiki/Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">Random numbers</a></li> <li><a href="/wiki/Steganography" title="Steganography">Steganography</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Cryptography" title="Category:Cryptography">Category</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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