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class="vector-toc-numb">3</span> <span>Factoring algorithms</span> </div> </a> <button aria-controls="toc-Factoring_algorithms-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 Factoring algorithms subsection</span> </button> <ul id="toc-Factoring_algorithms-sublist" class="vector-toc-list"> <li id="toc-Special-purpose" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special-purpose"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Special-purpose</span> </div> </a> <ul id="toc-Special-purpose-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General-purpose" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General-purpose"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>General-purpose</span> </div> </a> <ul id="toc-General-purpose-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_notable_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_notable_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Other notable algorithms</span> </div> </a> <ul id="toc-Other_notable_algorithms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Heuristic_running_time" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Heuristic_running_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Heuristic running time</span> </div> </a> <ul id="toc-Heuristic_running_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigorous_running_time" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rigorous_running_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Rigorous running time</span> </div> </a> <button aria-controls="toc-Rigorous_running_time-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 Rigorous running time subsection</span> </button> <ul id="toc-Rigorous_running_time-sublist" class="vector-toc-list"> <li id="toc-Schnorr–Seysen–Lenstra_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schnorr–Seysen–Lenstra_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Schnorr–Seysen–Lenstra algorithm</span> </div> </a> <ul id="toc-Schnorr–Seysen–Lenstra_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expected_running_time" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expected_running_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Expected running time</span> </div> </a> <ul id="toc-Expected_running_time-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">6</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">7</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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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" > <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">Integer factorization</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" 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Available in 33 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-33" 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">33 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Primfaktorzerlegung" title="Primfaktorzerlegung – Alemannic" lang="gsw" hreflang="gsw" data-title="Primfaktorzerlegung" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%B9%D8%AF%D8%AF_%D8%B5%D8%AD%D9%8A%D8%AD_%D8%A5%D9%84%D9%89_%D8%B9%D9%88%D8%A7%D9%85%D9%84" title="تحليل عدد صحيح إلى عوامل – Arabic" lang="ar" hreflang="ar" data-title="تحليل عدد صحيح إلى عوامل" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Factoritzaci%C3%B3_dels_enters" title="Factorització dels enters – Catalan" lang="ca" hreflang="ca" data-title="Factorització dels enters" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Prvo%C4%8D%C3%ADseln%C3%BD_rozklad" title="Prvočíselný rozklad – Czech" lang="cs" hreflang="cs" data-title="Prvočíselný rozklad" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Primtalsopl%C3%B8sning" title="Primtalsopløsning – Danish" lang="da" hreflang="da" data-title="Primtalsopløsning" 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/Primfaktorzerlegung" title="Primfaktorzerlegung – German" lang="de" hreflang="de" data-title="Primfaktorzerlegung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factorizaci%C3%B3n_de_enteros" title="Factorización de enteros – Spanish" lang="es" hreflang="es" data-title="Factorización de enteros" 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/Faktorado_de_entjero" title="Faktorado de entjero – Esperanto" lang="eo" hreflang="eo" data-title="Faktorado de entjero" 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/Zenbaki_osoen_faktorizazio" title="Zenbaki osoen faktorizazio – Basque" lang="eu" hreflang="eu" data-title="Zenbaki osoen faktorizazio" 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%AA%D8%AC%D8%B2%DB%8C%D9%87_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C" title="تجزیه اعداد طبیعی – Persian" lang="fa" hreflang="fa" data-title="تجزیه اعداد طبیعی" 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/D%C3%A9composition_en_produit_de_facteurs_premiers" title="Décomposition en produit de facteurs premiers – French" lang="fr" hreflang="fr" data-title="Décomposition en produit de facteurs premiers" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%91%D3%80%D0%B0%D1%80%D1%87%D1%87%D0%B0%D1%87%D0%B0_%D1%82%D0%B0%D1%8C%D1%80%D0%B0%D1%85%D1%8C%D0%B8%D0%B9_%D1%84%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8" title="БӀарччача таьрахьий факторизаци – Ingush" lang="inh" hreflang="inh" data-title="БӀарччача таьрахьий факторизаци" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingush" class="interlanguage-link-target"><span>ГӀалгӀай</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%86%8C%EC%9D%B8%EC%88%98%EB%B6%84%ED%95%B4" title="소인수분해 – Korean" lang="ko" hreflang="ko" data-title="소인수분해" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Faktorisasi_prima" title="Faktorisasi prima – Indonesian" lang="id" hreflang="id" data-title="Faktorisasi prima" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Frum%C3%BE%C3%A1ttun" title="Frumþáttun – Icelandic" lang="is" hreflang="is" data-title="Frumþáttun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%99%D7%A8%D7%95%D7%A7_%D7%9C%D7%92%D7%95%D7%A8%D7%9E%D7%99%D7%9D_%D7%A9%D7%9C_%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A9%D7%9C%D7%9D" title="פירוק לגורמים של מספר שלם – Hebrew" lang="he" hreflang="he" data-title="פירוק לגורמים של מספר שלם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Haaptsaz_vun_der_elementarer_Zuelentheorie" title="Haaptsaz vun der elementarer Zuelentheorie – Luxembourgish" lang="lb" hreflang="lb" data-title="Haaptsaz vun der elementarer Zuelentheorie" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pr%C3%ADmfelbont%C3%A1s" title="Prímfelbontás – Hungarian" lang="hu" hreflang="hu" data-title="Prímfelbontás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ontbinden_in_priemfactoren" title="Ontbinden in priemfactoren – Dutch" lang="nl" hreflang="nl" data-title="Ontbinden in priemfactoren" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B4%A0%E5%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3" title="素因数分解 – Japanese" lang="ja" hreflang="ja" data-title="素因数分解" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fatora%C3%A7%C3%A3o_de_inteiros" title="Fatoração de inteiros – Portuguese" lang="pt" hreflang="pt" data-title="Fatoração de inteiros" 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/Descompunerea_%C3%AEn_factori_primi" title="Descompunerea în factori primi – Romanian" lang="ro" hreflang="ro" data-title="Descompunerea în factori primi" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F_%D1%86%D0%B5%D0%BB%D1%8B%D1%85_%D1%87%D0%B8%D1%81%D0%B5%D0%BB" title="Факторизация целых чисел – Russian" lang="ru" hreflang="ru" data-title="Факторизация целых чисел" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Prime_factorization" title="Prime factorization – Simple English" lang="en-simple" hreflang="en-simple" data-title="Prime factorization" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Pra%C5%A1tevilski_razcep" title="Praštevilski razcep – Slovenian" lang="sl" hreflang="sl" data-title="Praštevilski razcep" 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/Rastavljanje_na_faktore" title="Rastavljanje na faktore – Serbian" lang="sr" hreflang="sr" data-title="Rastavljanje na faktore" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kokonaislukujen_tekij%C3%B6ihinjako" title="Kokonaislukujen tekijöihinjako – Finnish" lang="fi" hreflang="fi" data-title="Kokonaislukujen tekijöihinjako" 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/Primtalsfaktorisering" title="Primtalsfaktorisering – Swedish" lang="sv" hreflang="sv" data-title="Primtalsfaktorisering" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%A2%E0%B8%81%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B8%81%E0%B8%AD%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1" title="การแยกตัวประกอบจำนวนเต็ม – Thai" lang="th" hreflang="th" data-title="การแยกตัวประกอบจำนวนเต็ม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Asal_%C3%A7arpanlara_ay%C4%B1rma" title="Asal çarpanlara ayırma – Turkish" lang="tr" hreflang="tr" data-title="Asal çarpanlara ayırma" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D1%96%D1%8F_%D1%86%D1%96%D0%BB%D0%B8%D1%85_%D1%87%D0%B8%D1%81%D0%B5%D0%BB" title="Факторизація цілих чисел – Ukrainian" lang="uk" hreflang="uk" data-title="Факторизація цілих чисел" 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/Ph%C3%A2n_t%C3%ADch_s%E1%BB%91_nguy%C3%AAn" title="Phân tích số nguyên – Vietnamese" lang="vi" hreflang="vi" data-title="Phân tích số nguyên" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B4%E6%95%B0%E5%88%86%E8%A7%A3" title="整数分解 – Chinese" lang="zh" hreflang="zh" data-title="整数分解" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q4846249#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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searchaux" style="display:none">Decomposition of a number into a product</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">"Prime decomposition" redirects here. For the prime decomposition theorem for 3-manifolds, see <a href="/wiki/Prime_decomposition_of_3-manifolds" title="Prime decomposition of 3-manifolds">Prime decomposition of 3-manifolds</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-computer_science" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-computer_science">Unsolved problem in computer science</span>:</div> <div class="unsolved-body">Can integer factorization be solved in polynomial time on a classical computer?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_computer_science" title="List of unsolved problems in computer science">(more unsolved problems in computer science)</a></div> </div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>integer factorization</b> is the decomposition of a <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a> into a <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of integers. Every positive integer greater than 1 is either the product of two or more integer <a href="/wiki/Divisor" title="Divisor">factors</a> greater than 1, in which case it is called a <a href="/wiki/Composite_number" title="Composite number">composite number</a>, or it is not, in which case it is called a <a href="/wiki/Prime_number" title="Prime number">prime number</a>. For example, <span class="texhtml">15</span> is a composite number because <span class="texhtml">15 = 3 · 5</span>, but <span class="texhtml">7</span> is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example <span class="texhtml">60 = 3 · 20 = 3 · (5 · 4)</span>. Continuing this process until every factor is prime is called <b>prime factorization</b>; the result is always unique up to the order of the factors by the <a href="/wiki/Prime_factorization_theorem" class="mw-redirect" title="Prime factorization theorem">prime factorization theorem</a>. </p><p>To factorize a small integer <span class="texhtml mvar" style="font-style:italic;">n</span> using mental or pen-and-paper arithmetic, the simplest method is <a href="/wiki/Trial_division" title="Trial division">trial division</a>: checking if the number is divisible by prime numbers <span class="texhtml">2</span>, <span class="texhtml">3</span>, <span class="texhtml">5</span>, and so on, up to the <a href="/wiki/Square_root" title="Square root">square root</a> of <span class="texhtml mvar" style="font-style:italic;">n</span>. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient. A prime factorization algorithm typically involves <a href="/wiki/Primality_test" title="Primality test">testing whether each factor is prime</a> each time a factor is found. </p><p>When the numbers are sufficiently large, no efficient <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">non-quantum</a> integer factorization <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> is known. However, it has not been proven that such an algorithm does not exist. The presumed <a href="/wiki/Computational_hardness_assumption" title="Computational hardness assumption">difficulty</a> of this problem is important for the algorithms used in <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> such as <a href="/wiki/RSA_(cryptosystem)" title="RSA (cryptosystem)">RSA public-key encryption</a> and the <a href="/wiki/Digital_Signature_Algorithm" title="Digital Signature Algorithm">RSA digital signature</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Many areas of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a> have been brought to bear on the problem, including <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a>, <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, and <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computing</a>. </p><p>Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are <a href="/wiki/Semiprime" title="Semiprime">semiprimes</a>, the product of two prime numbers. When they are both large, for instance more than two thousand <a href="/wiki/Bit" title="Bit">bits</a> long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by <a href="/wiki/Fermat%27s_factorization_method" title="Fermat's factorization method">Fermat's factorization method</a>), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the integer being factored increases, the number of operations required to perform the factorization on any computer increases drastically. </p><p>Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the <a href="/wiki/RSA_problem" title="RSA problem">RSA problem</a>. An algorithm that efficiently factors an arbitrary integer would render <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a>-based <a href="/wiki/Public-key" class="mw-redirect" title="Public-key">public-key</a> cryptography insecure. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Prime_decomposition">Prime decomposition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=1" title="Edit section: Prime decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PrimeDecompositionExample.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/PrimeDecompositionExample.svg/220px-PrimeDecompositionExample.svg.png" decoding="async" width="220" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/PrimeDecompositionExample.svg/330px-PrimeDecompositionExample.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/PrimeDecompositionExample.svg/440px-PrimeDecompositionExample.svg.png 2x" data-file-width="200" data-file-height="115" /></a><figcaption>Prime decomposition of <span class="texhtml"><i>n</i> = 864</span> as <span class="texhtml">2<sup>5</sup> × 3<sup>3</sup></span></figcaption></figure> <p>By the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, every positive integer has a unique <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factorization</a>. (By convention, 1 is the <a href="/wiki/Empty_product" title="Empty product">empty product</a>.) <a href="/wiki/Primality_test" title="Primality test">Testing</a> whether the integer is prime can be done in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>, for example, by the <a href="/wiki/AKS_primality_test" title="AKS primality test">AKS primality test</a>. If composite, however, the polynomial time tests give no insight into how to obtain the factors. </p><p>Given a general algorithm for integer factorization, any integer can be factored into its constituent <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> by repeated application of this algorithm. The situation is more complicated with special-purpose factorization algorithms, whose benefits may not be realized as well or even at all with the factors produced during decomposition. For example, if <span class="texhtml"><i>n</i> = 171 × <i>p</i> × <i>q</i></span> where <span class="texhtml"><i>p</i> < <i>q</i></span> are very large primes, <a href="/wiki/Trial_division" title="Trial division">trial division</a> will quickly produce the factors 3 and 19 but will take <span class="texhtml"><i>p</i></span> divisions to find the next factor. As a contrasting example, if <span class="texhtml"><i>n</i></span> is the product of the primes <span class="texhtml">13729</span>, <span class="texhtml">1372933</span>, and <span class="texhtml">18848997161</span>, where <span class="texhtml">13729 × 1372933 = 18848997157</span>, Fermat's factorization method will begin with <span class="texhtml">⌈<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>n</i></span></span>⌉ = 18848997159</span> which immediately yields <span class="texhtml"><i>b</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>a</i><sup>2</sup> − <i>n</i></span></span> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">4</span></span> = 2</span> and hence the factors <span class="texhtml"><i>a</i> − <i>b</i> = 18848997157</span> and <span class="texhtml"><i>a</i> + <i>b</i> = 18848997161</span>. While these are easily recognized as composite and prime respectively, Fermat's method will take much longer to factor the composite number because the starting value of <span class="texhtml">⌈<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">18848997157</span></span>⌉ = 137292</span> for <span class="texhtml"><i>a</i></span> is a factor of 10 from <span class="texhtml">1372933</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Current_state_of_the_art">Current state of the art</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=2" title="Edit section: Current state of the art"><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">See also: <a href="/wiki/Integer_factorization_records" title="Integer factorization records">Integer factorization records</a></div> <p>Among the <span class="texhtml"><i>b</i></span>-bit numbers, the most difficult to factor in practice using existing algorithms are those <a href="/wiki/Semiprimes" class="mw-redirect" title="Semiprimes">semiprimes</a> whose factors are of similar size. For this reason, these are the integers used in cryptographic applications. </p><p>In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (<a href="/wiki/RSA-240" class="mw-redirect" title="RSA-240">RSA-240</a>) utilizing approximately 900 core-years of computing power.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.<sup id="cite_ref-rsa768_3-0" class="reference"><a href="#cite_note-rsa768-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The largest such semiprime yet factored was <a href="/wiki/RSA_numbers#RSA-250" title="RSA numbers">RSA-250</a>, an 829-bit number with 250 decimal digits, in February 2020. The total computation time was roughly 2700 core-years of computing using Intel <a href="/wiki/Skylake_(microarchitecture)#Xeon_Gold_(quad_processor)" title="Skylake (microarchitecture)">Xeon Gold</a> 6130 at 2.1 GHz. Like all recent factorization records, this factorization was completed with a highly optimized implementation of the <a href="/wiki/General_number_field_sieve" title="General number field sieve">general number field sieve</a> run on hundreds of machines. </p> <div class="mw-heading mw-heading3"><h3 id="Time_complexity">Time complexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=3" title="Edit section: Time complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>No <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> has been published that can factor all integers in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>, that is, that can factor a <span class="texhtml"><i>b</i></span>-bit number <span class="texhtml"><i>n</i></span> in time <span class="texhtml"><a href="/wiki/Big_O_notation" title="Big O notation">O</a>(<i>b</i><sup><i>k</i></sup>)</span> for some constant <span class="texhtml"><i>k</i></span>. Neither the existence nor non-existence of such algorithms has been proved, but it is generally suspected that they do not exist.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>There are published algorithms that are faster than <span class="texhtml">O((1 + <i>ε</i>)<sup><i>b</i></sup>)</span> for all positive <span class="texhtml"><i>ε</i></span>, that is, <a href="/wiki/Time_complexity#Sub-exponential_time" title="Time complexity">sub-exponential</a>. As of 2022<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Integer_factorization&action=edit">[update]</a></sup>, the algorithm with best theoretical asymptotic running time is the <a href="/wiki/General_number_field_sieve" title="General number field sieve">general number field sieve</a> (GNFS), first published in 1993,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> running on a <span class="texhtml"><i>b</i></span>-bit number <span class="texhtml"><i>n</i></span> in time: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left(\left(\left({\tfrac {8}{3}}\right)^{\frac {2}{3}}+o(1)\right)\left(\log n\right)^{\frac {1}{3}}\left(\log \log n\right)^{\frac {2}{3}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left(\left(\left({\tfrac {8}{3}}\right)^{\frac {2}{3}}+o(1)\right)\left(\log n\right)^{\frac {1}{3}}\left(\log \log n\right)^{\frac {2}{3}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7938373b478c57ee7f904580c2ba1ff355d7b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.014ex; height:7.509ex;" alt="{\displaystyle \exp \left(\left(\left({\tfrac {8}{3}}\right)^{\frac {2}{3}}+o(1)\right)\left(\log n\right)^{\frac {1}{3}}\left(\log \log n\right)^{\frac {2}{3}}\right).}"></span></dd></dl> <p>For current computers, GNFS is the best published algorithm for large <span class="texhtml"><i>n</i></span> (more than about 400 bits). For a <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computer</a>, however, <a href="/wiki/Peter_Shor" title="Peter Shor">Peter Shor</a> discovered an algorithm in 1994 that solves it in polynomial time. <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a> takes only <span class="texhtml">O(<i>b</i><sup>3</sup>)</span> time and <span class="texhtml">O(<i>b</i>)</span> space on <span class="texhtml"><i>b</i></span>-bit number inputs. In 2001, Shor's algorithm was implemented for the first time, by using <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">NMR</a> techniques on molecules that provide seven qubits.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>In order to talk about <a href="/wiki/Complexity_class" title="Complexity class">complexity classes</a> such as P, NP, and co-NP, the problem has to be stated as a <a href="/wiki/Decision_problem" title="Decision problem">decision problem</a>. </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Decision problem</strong> <span class="theorem-note">(Integer factorization)</span><span class="theoreme-tiret"> — </span>For every natural numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, does <span class="texhtml"><i>n</i></span> have a factor smaller than <span class="texhtml"><i>k</i></span> besides 1? </p> </div> <p>It is known to be in both <a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a> and <a href="/wiki/Co-NP" title="Co-NP">co-NP</a>, meaning that both "yes" and "no" answers can be verified in polynomial time. An answer of "yes" can be certified by exhibiting a factorization <span class="texhtml"><i>n</i> = <i>d</i>(<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>d</i></span></span>⁠</span>)</span> with <span class="texhtml"><i>d</i> ≤ <i>k</i></span>. An answer of "no" can be certified by exhibiting the factorization of <span class="texhtml"><i>n</i></span> into distinct primes, all larger than <span class="texhtml"><i>k</i></span>; one can verify their primality using the <a href="/wiki/AKS_primality_test" title="AKS primality test">AKS primality test</a>, and then multiply them to obtain <span class="texhtml"><i>n</i></span>. The <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a> guarantees that there is only one possible string of increasing primes that will be accepted, which shows that the problem is in both <a href="/wiki/UP_(complexity)" title="UP (complexity)">UP</a> and co-UP.<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> It is known to be in <a href="/wiki/BQP" title="BQP">BQP</a> because of Shor's algorithm. </p><p>The problem is suspected to be outside all three of the complexity classes P, NP-complete,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Co-NP-complete" title="Co-NP-complete">co-NP-complete</a>. It is therefore a candidate for the <a href="/wiki/NP-intermediate" title="NP-intermediate">NP-intermediate</a> complexity class. </p><p>In contrast, the decision problem "Is <span class="texhtml"><i>n</i></span> a composite number?" (or equivalently: "Is <span class="texhtml"><i>n</i></span> a prime number?") appears to be much easier than the problem of specifying factors of <span class="texhtml"><i>n</i></span>. The composite/prime problem can be solved in polynomial time (in the number <span class="texhtml"><i>b</i></span> of digits of <span class="texhtml"><i>n</i></span>) with the <a href="/wiki/AKS_primality_test" title="AKS primality test">AKS primality test</a>. In addition, there are several <a href="/wiki/Randomized_algorithm" title="Randomized algorithm">probabilistic algorithms</a> that can test primality very quickly in practice if one is willing to accept a vanishingly small possibility of error. The ease of <a href="/wiki/Primality_test" title="Primality test">primality testing</a> is a crucial part of the <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a> algorithm, as it is necessary to find large prime numbers to start with. </p> <div class="mw-heading mw-heading2"><h2 id="Factoring_algorithms">Factoring algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=4" title="Edit section: Factoring algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Special-purpose">Special-purpose</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=5" title="Edit section: Special-purpose"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A special-purpose factoring algorithm's running time depends on the properties of the number to be factored or on one of its unknown factors: size, special form, etc. The parameters which determine the running time vary among algorithms. </p><p>An important subclass of special-purpose factoring algorithms is the <i>Category 1</i> or <i>First Category</i> algorithms, whose running time depends on the size of smallest prime factor. Given an integer of unknown form, these methods are usually applied before general-purpose methods to remove small factors.<sup id="cite_ref-Bressoud_and_Wagon_10-0" class="reference"><a href="#cite_note-Bressoud_and_Wagon-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> For example, naive <a href="/wiki/Trial_division" title="Trial division">trial division</a> is a Category 1 algorithm. </p> <ul><li><a href="/wiki/Trial_division" title="Trial division">Trial division</a></li> <li><a href="/wiki/Wheel_factorization" title="Wheel factorization">Wheel factorization</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm" title="Pollard's rho algorithm">Pollard's rho algorithm</a>, which has two common flavors to <a href="/wiki/Cycle_detection" title="Cycle detection">identify group cycles</a>: one by Floyd and one by Brent.</li> <li><a href="/wiki/Algebraic-group_factorisation_algorithms" class="mw-redirect" title="Algebraic-group factorisation algorithms">Algebraic-group factorization algorithms</a>, among which are <a href="/wiki/Pollard%27s_p_%E2%88%92_1_algorithm" title="Pollard's p − 1 algorithm">Pollard's <span class="texhtml"><i>p</i> − 1</span> algorithm</a>, <a href="/wiki/Williams%27_p_%2B_1_algorithm" class="mw-redirect" title="Williams' p + 1 algorithm">Williams' <span class="texhtml"><i>p</i> + 1</span> algorithm</a>, and <a href="/wiki/Lenstra_elliptic_curve_factorization" class="mw-redirect" title="Lenstra elliptic curve factorization">Lenstra elliptic curve factorization</a></li> <li><a href="/wiki/Fermat%27s_factorization_method" title="Fermat's factorization method">Fermat's factorization method</a></li> <li><a href="/wiki/Euler%27s_factorization_method" title="Euler's factorization method">Euler's factorization method</a></li> <li><a href="/wiki/Special_number_field_sieve" title="Special number field sieve">Special number field sieve</a></li> <li><a href="/wiki/Difference_of_two_squares" title="Difference of two squares">Difference of two squares</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="General-purpose">General-purpose</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=6" title="Edit section: General-purpose"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A general-purpose factoring algorithm, also known as a <i>Category 2</i>, <i>Second Category</i>, or <a href="/wiki/Maurice_Kraitchik" title="Maurice Kraitchik"><i>Kraitchik</i></a> <i>family</i> algorithm,<sup id="cite_ref-Bressoud_and_Wagon_10-1" class="reference"><a href="#cite_note-Bressoud_and_Wagon-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor <a href="/wiki/RSA_number" class="mw-redirect" title="RSA number">RSA numbers</a>. Most general-purpose factoring algorithms are based on the <a href="/wiki/Congruence_of_squares" title="Congruence of squares">congruence of squares</a> method. </p> <ul><li><a href="/wiki/Dixon%27s_factorization_method" title="Dixon's factorization method">Dixon's factorization method</a></li> <li><a href="/wiki/Continued_fraction_factorization" title="Continued fraction factorization">Continued fraction factorization</a> (CFRAC)</li> <li><a href="/wiki/Quadratic_sieve" title="Quadratic sieve">Quadratic sieve</a></li> <li><a href="/wiki/Rational_sieve" title="Rational sieve">Rational sieve</a></li> <li><a href="/wiki/General_number_field_sieve" title="General number field sieve">General number field sieve</a></li> <li><a href="/wiki/Shanks%27s_square_forms_factorization" title="Shanks's square forms factorization">Shanks's square forms factorization</a> (SQUFOF)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_notable_algorithms">Other notable algorithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=7" title="Edit section: Other notable algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a>, for quantum computers</li></ul> <div class="mw-heading mw-heading2"><h2 id="Heuristic_running_time">Heuristic running time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=8" title="Edit section: Heuristic running time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In number theory, there are many integer factoring algorithms that heuristically have expected <a href="/wiki/Time_complexity" title="Time complexity">running time</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}\left[{\tfrac {1}{2}},1+o(1)\right]=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}\left[{\tfrac {1}{2}},1+o(1)\right]=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518dd095a6d940c8fa4c133fb2dfd510e9004f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.254ex; height:4.009ex;" alt="{\displaystyle L_{n}\left[{\tfrac {1}{2}},1+o(1)\right]=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}}"></span></dd></dl> <p>in <a href="/wiki/Big_O_notation" title="Big O notation">little-o</a> and <a href="/wiki/L-notation" title="L-notation">L-notation</a>. Some examples of those algorithms are the <a href="/wiki/Elliptic_curve_method" class="mw-redirect" title="Elliptic curve method">elliptic curve method</a> and the <a href="/wiki/Quadratic_sieve" title="Quadratic sieve">quadratic sieve</a>. Another such algorithm is the <b>class group relations method</b> proposed by Schnorr,<sup id="cite_ref-1982-schnorr_11-0" class="reference"><a href="#cite_note-1982-schnorr-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Seysen,<sup id="cite_ref-1987-seysen_12-0" class="reference"><a href="#cite_note-1987-seysen-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and Lenstra,<sup id="cite_ref-1988-lenstra_13-0" class="reference"><a href="#cite_note-1988-lenstra-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> which they proved only assuming the unproved <a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">generalized Riemann hypothesis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Rigorous_running_time">Rigorous running time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=9" title="Edit section: Rigorous running time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Schnorr–Seysen–Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance<sup id="cite_ref-lenstra-pomerance_14-0" class="reference"><a href="#cite_note-lenstra-pomerance-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> to have expected running time <span class="texhtml"><i>L<sub>n</sub></i>[<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, 1+<i>o</i>(1)]</span> by replacing the GRH assumption with the use of multipliers. The algorithm uses the <a href="/wiki/Ideal_class_group" title="Ideal class group">class group</a> of positive binary <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> of <a href="/wiki/Discriminant_of_a_quadratic_form" class="mw-redirect" title="Discriminant of a quadratic form">discriminant</a> <span class="texhtml">Δ</span> denoted by <span class="texhtml"><i>G</i><sub>Δ</sub></span>. <span class="texhtml"><i>G</i><sub>Δ</sub></span> is the set of triples of integers <span class="texhtml">(<i>a</i>, <i>b</i>, <i>c</i>)</span> in which those integers are relative prime. </p> <div class="mw-heading mw-heading3"><h3 id="Schnorr–Seysen–Lenstra_algorithm"><span id="Schnorr.E2.80.93Seysen.E2.80.93Lenstra_algorithm"></span>Schnorr–Seysen–Lenstra algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=10" title="Edit section: Schnorr–Seysen–Lenstra algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given an integer <span class="texhtml mvar" style="font-style:italic;">n</span> that will be factored, where <span class="texhtml mvar" style="font-style:italic;">n</span> is an odd positive integer greater than a certain constant. In this factoring algorithm the discriminant <span class="texhtml">Δ</span> is chosen as a multiple of <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml">Δ = −<i>dn</i></span>, where <span class="texhtml mvar" style="font-style:italic;">d</span> is some positive multiplier. The algorithm expects that for one <span class="texhtml mvar" style="font-style:italic;">d</span> there exist enough <a href="/wiki/Smooth_number" title="Smooth number">smooth</a> forms in <span class="texhtml"><i>G</i><sub>Δ</sub></span>. Lenstra and Pomerance show that the choice of <span class="texhtml mvar" style="font-style:italic;">d</span> can be restricted to a small set to guarantee the smoothness result. </p><p>Denote by <span class="texhtml"><i>P</i><sub>Δ</sub></span> the set of all primes <span class="texhtml mvar" style="font-style:italic;">q</span> with <a href="/wiki/Kronecker_symbol" title="Kronecker symbol">Kronecker symbol</a> <span class="texhtml"><span style="font-size:150%;">(</span><span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">Δ</span><span class="sr-only">/</span><span class="den"><i>q</i></span></span>⁠</span></span><span style="font-size:150%;">)</span> = 1</span>. By constructing a set of <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generators</a> of <span class="texhtml"><i>G</i><sub>Δ</sub></span> and prime forms <span class="texhtml"><i>f</i><sub><i>q</i></sub></span> of <span class="texhtml"><i>G</i><sub>Δ</sub></span> with <span class="texhtml mvar" style="font-style:italic;">q</span> in <span class="texhtml"><i>P</i><sub>Δ</sub></span> a sequence of relations between the set of generators and <span class="texhtml"><i>f</i><sub><i>q</i></sub></span> are produced. The size of <span class="texhtml mvar" style="font-style:italic;">q</span> can be bounded by <span class="texhtml"><i>c</i><sub>0</sub>(log|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Δ</span>|)<sup>2</sup></span> for some constant <span class="texhtml"><i>c</i><sub>0</sub></span>. </p><p>The relation that will be used is a relation between the product of powers that is equal to the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">neutral element</a> of <span class="texhtml"><i>G</i><sub>Δ</sub></span>. These relations will be used to construct a so-called ambiguous form of <span class="texhtml"><i>G</i><sub>Δ</sub></span>, which is an element of <span class="texhtml"><i>G</i><sub>Δ</sub></span> of order dividing 2. By calculating the corresponding factorization of <span class="texhtml">Δ</span> and by taking a <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>, this ambiguous form provides the complete prime factorization of <span class="texhtml mvar" style="font-style:italic;">n</span>. This algorithm has these main steps: </p><p>Let <span class="texhtml mvar" style="font-style:italic;">n</span> be the number to be factored. </p> <div><ol><li>Let <span class="texhtml">Δ</span> be a negative integer with <span class="texhtml">Δ = −<i>dn</i></span>, where <span class="texhtml mvar" style="font-style:italic;">d</span> is a multiplier and <span class="texhtml">Δ</span> is the negative discriminant of some quadratic form.</li><li>Take the <span class="texhtml mvar" style="font-style:italic;">t</span> first primes <span class="texhtml"><i>p</i><sub>1</sub> = 2, <i>p</i><sub>2</sub> = 3, <i>p</i><sub>3</sub> = 5, ..., <i>p</i><sub><i>t</i></sub></span>, for some <span class="texhtml"><i>t</i> ∈ <b>N</b></span>.</li><li>Let <span class="texhtml"><i>f</i><sub><i>q</i></sub></span> be a random prime form of <span class="texhtml"><i>G</i><sub>Δ</sub></span> with <span class="texhtml"><span style="font-size:150%;">(</span><span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">Δ</span><span class="sr-only">/</span><span class="den"><i>q</i></span></span>⁠</span></span><span style="font-size:150%;">)</span> = 1</span>.</li><li>Find a generating set <span class="texhtml mvar" style="font-style:italic;">X</span> of <span class="texhtml"><i>G</i><sub>Δ</sub></span>.</li><li>Collect a sequence of relations between set <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml">{<i>f</i><sub><i>q</i></sub> : <i>q</i> ∈ <i>P</i><sub>Δ</sub>}</span> satisfying: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\prod _{x\in X_{}}x^{r(x)}\right).\left(\prod _{q\in P_{\Delta }}f_{q}^{t(q)}\right)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msub> </mrow> </munder> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>∈</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </msub> </mrow> </munder> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\prod _{x\in X_{}}x^{r(x)}\right).\left(\prod _{q\in P_{\Delta }}f_{q}^{t(q)}\right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf163246a6d632cc54541e523589f0817de26e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.791ex; height:7.843ex;" alt="{\displaystyle \left(\prod _{x\in X_{}}x^{r(x)}\right).\left(\prod _{q\in P_{\Delta }}f_{q}^{t(q)}\right)=1.}"></span></dd></dl></li><li>Construct an ambiguous form <span class="texhtml">(<i>a</i>, <i>b</i>, <i>c</i>)</span> that is an element <span class="texhtml"><i>f</i> ∈ <i>G</i><sub>Δ</sub></span> of order dividing 2 to obtain a coprime factorization of the largest odd divisor of <span class="texhtml">Δ</span> in which <span class="texhtml">Δ = −4<i>ac</i></span> or <span class="texhtml">Δ = <i>a</i>(<i>a</i> − 4<i>c</i>)</span> or <span class="texhtml">Δ = (<i>b</i> − 2<i>a</i>)(<i>b</i> + 2<i>a</i>)</span>.</li><li>If the ambiguous form provides a factorization of <span class="texhtml mvar" style="font-style:italic;">n</span> then stop, otherwise find another ambiguous form until the factorization of <span class="texhtml mvar" style="font-style:italic;">n</span> is found. In order to prevent useless ambiguous forms from generating, build up the <a href="/wiki/Sylow_theorems" title="Sylow theorems">2-Sylow</a> group <span class="texhtml">Sll<sub>2</sub>(Δ)</span> of <span class="texhtml"><i>G</i>(Δ)</span>.</li></ol></div> <p>To obtain an algorithm for factoring any positive integer, it is necessary to add a few steps to this algorithm such as trial division, and the <a href="/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test" title="Adleman–Pomerance–Rumely primality test">Jacobi sum test</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Expected_running_time">Expected running time</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=11" title="Edit section: Expected running time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The algorithm as stated is a <a href="/wiki/Probabilistic_algorithm" class="mw-redirect" title="Probabilistic algorithm">probabilistic algorithm</a> as it makes random choices. Its expected running time is at most <span class="texhtml"><i>L<sub>n</sub></i>[<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, 1+<i>o</i>(1)]</span>.<sup id="cite_ref-lenstra-pomerance_14-1" class="reference"><a href="#cite_note-lenstra-pomerance-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </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=Integer_factorization&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Aurifeuillean_factorization" title="Aurifeuillean factorization">Aurifeuillean factorization</a></li> <li><a href="/wiki/Bach%27s_algorithm" title="Bach's algorithm">Bach's algorithm</a> for generating random numbers with their factorizations</li> <li><a href="/wiki/Canonical_representation_of_a_positive_integer" class="mw-redirect" title="Canonical representation of a positive integer">Canonical representation of a positive integer</a></li> <li><a href="/wiki/Factorization" title="Factorization">Factorization</a></li> <li><a href="/wiki/Multiplicative_partition" title="Multiplicative partition">Multiplicative partition</a></li> <li><a href="/wiki/P-adic_valuation" title="P-adic valuation"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic valuation</a></li> <li><a href="/wiki/Integer_partition" title="Integer partition">Integer partition</a> – a way of writing a number as a sum of positive integers.</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=Integer_factorization&action=edit&section=13" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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 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J.; Timofeev, Andrey; Zimmermann, Paul (2010). <a rel="nofollow" class="external text" href="https://eprint.iacr.org/2010/006.pdf">"Factorization of a 768-Bit RSA Modulus"</a> <span class="cs1-format">(PDF)</span>. In Rabin, Tal (ed.). <i>Advances in Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings</i>. Lecture Notes in Computer Science. Vol. 6223. 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Krantz">Krantz, Steven G.</a> (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mMZBtxVZiQoC&pg=PA203"><i>The Proof is in the Pudding: The Changing Nature of Mathematical Proof</i></a>, New York: Springer, p. 203, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-48744-1">10.1007/978-0-387-48744-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-48908-7" title="Special:BookSources/978-0-387-48908-7"><bdi>978-0-387-48908-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2789493">2789493</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Proof+is+in+the+Pudding%3A+The+Changing+Nature+of+Mathematical+Proof&rft.place=New+York&rft.pages=203&rft.pub=Springer&rft.date=2011&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2789493%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-0-387-48744-1&rft.isbn=978-0-387-48908-7&rft.aulast=Krantz&rft.aufirst=Steven+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmMZBtxVZiQoC%26pg%3DPA203&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAroraBarak2009" class="citation cs2"><a href="/wiki/Sanjeev_Arora" title="Sanjeev Arora">Arora, Sanjeev</a>; Barak, Boaz (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGvI7cOuOOQC&pg=PA230"><i>Computational complexity</i></a>, Cambridge: Cambridge University Press, p. 230, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511804090">10.1017/CBO9780511804090</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-42426-4" title="Special:BookSources/978-0-521-42426-4"><bdi>978-0-521-42426-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2500087">2500087</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:215746906">215746906</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+complexity&rft.place=Cambridge&rft.pages=230&rft.pub=Cambridge+University+Press&rft.date=2009&rft_id=info%3Adoi%2F10.1017%2FCBO9780511804090&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2500087%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A215746906%23id-name%3DS2CID&rft.isbn=978-0-521-42426-4&rft.aulast=Arora&rft.aufirst=Sanjeev&rft.au=Barak%2C+Boaz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnGvI7cOuOOQC%26pg%3DPA230&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuhlerLenstraPomerance1993" class="citation book cs1">Buhler, J. P.; Lenstra, H. W. Jr.; Pomerance, Carl (1993). "Factoring integers with the number field sieve". <a rel="nofollow" class="external text" href="https://doi.org/10.1007/BFb0091539"><i>The development of the number field sieve</i></a>. Lecture Notes in Mathematics. Vol. 1554. Springer. pp. 50–94. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0091539">10.1007/BFb0091539</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1887%2F2149">1887/2149</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-57013-4" title="Special:BookSources/978-3-540-57013-4"><bdi>978-3-540-57013-4</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">12 March</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Factoring+integers+with+the+number+field+sieve&rft.btitle=The+development+of+the+number+field+sieve&rft.series=Lecture+Notes+in+Mathematics&rft.pages=50-94&rft.pub=Springer&rft.date=1993&rft_id=info%3Ahdl%2F1887%2F2149&rft_id=info%3Adoi%2F10.1007%2FBFb0091539&rft.isbn=978-3-540-57013-4&rft.aulast=Buhler&rft.aufirst=J.+P.&rft.au=Lenstra%2C+H.+W.+Jr.&rft.au=Pomerance%2C+Carl&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2FBFb0091539&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVandersypen2001" class="citation journal cs1">Vandersypen, Lieven M. K.; et al. (2001). "Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance". <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>414</b> (6866): 883–887. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0112176">quant-ph/0112176</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001Natur.414..883V">2001Natur.414..883V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F414883a">10.1038/414883a</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11780055">11780055</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4400832">4400832</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Experimental+realization+of+Shor%27s+quantum+factoring+algorithm+using+nuclear+magnetic+resonance&rft.volume=414&rft.issue=6866&rft.pages=883-887&rft.date=2001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4400832%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2001Natur.414..883V&rft_id=info%3Aarxiv%2Fquant-ph%2F0112176&rft_id=info%3Apmid%2F11780055&rft_id=info%3Adoi%2F10.1038%2F414883a&rft.aulast=Vandersypen&rft.aufirst=Lieven+M.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLance_Fortnow2002" class="citation web cs1">Lance Fortnow (2002-09-13). <a rel="nofollow" class="external text" href="http://weblog.fortnow.com/2002/09/complexity-class-of-week-factoring.html">"Computational Complexity Blog: Complexity Class of the Week: Factoring"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Computational+Complexity+Blog%3A+Complexity+Class+of+the+Week%3A+Factoring&rft.date=2002-09-13&rft.au=Lance+Fortnow&rft_id=http%3A%2F%2Fweblog.fortnow.com%2F2002%2F09%2Fcomplexity-class-of-week-factoring.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldreichWigderson2008" class="citation cs2"><a href="/wiki/Oded_Goldreich" title="Oded Goldreich">Goldreich, Oded</a>; <a href="/wiki/Avi_Wigderson" title="Avi Wigderson">Wigderson, Avi</a> (2008), "IV.20 Computational Complexity", in <a href="/wiki/Timothy_Gowers" title="Timothy Gowers">Gowers, Timothy</a>; <a href="/wiki/June_Barrow-Green" title="June Barrow-Green">Barrow-Green, June</a>; <a href="/wiki/Imre_Leader" title="Imre Leader">Leader, Imre</a> (eds.), <i>The Princeton Companion to Mathematics</i>, Princeton, New Jersey: Princeton University Press, pp. 575–604, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11880-2" title="Special:BookSources/978-0-691-11880-2"><bdi>978-0-691-11880-2</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2467561">2467561</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=IV.20+Computational+Complexity&rft.btitle=The+Princeton+Companion+to+Mathematics&rft.place=Princeton%2C+New+Jersey&rft.pages=575-604&rft.pub=Princeton+University+Press&rft.date=2008&rft.isbn=978-0-691-11880-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2467561%23id-name%3DMR&rft.aulast=Goldreich&rft.aufirst=Oded&rft.au=Wigderson%2C+Avi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span>. See in particular <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA583">p. 583</a>.</span> </li> <li id="cite_note-Bressoud_and_Wagon-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bressoud_and_Wagon_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bressoud_and_Wagon_10-1"><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="CITEREFDavid_Bressoud_and_Stan_Wagon2000" class="citation book cs1"><a href="/wiki/David_Bressoud" title="David Bressoud">David Bressoud</a> and <a href="/wiki/Stan_Wagon" title="Stan Wagon">Stan Wagon</a> (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/courseincomputat0000bres/page/168"><i>A Course in Computational Number Theory</i></a></span>. Key College Publishing/Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/courseincomputat0000bres/page/168">168–69</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-930190-10-8" title="Special:BookSources/978-1-930190-10-8"><bdi>978-1-930190-10-8</bdi></a>.</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+Computational+Number+Theory&rft.pages=168-69&rft.pub=Key+College+Publishing%2FSpringer&rft.date=2000&rft.isbn=978-1-930190-10-8&rft.au=David+Bressoud+and+Stan+Wagon&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcourseincomputat0000bres%2Fpage%2F168&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-1982-schnorr-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-1982-schnorr_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchnorr1982" class="citation journal cs1">Schnorr, Claus P. (1982). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170924140543/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA096348">"Refined analysis and improvements on some factoring algorithms"</a>. <i>Journal of Algorithms</i>. <b>3</b> (2): 101–127. <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%2F0196-6774%2882%2990012-8">10.1016/0196-6774(82)90012-8</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0657269">0657269</a>. Archived from <a rel="nofollow" class="external text" href="http://www.dtic.mil/get-tr-doc/pdf?AD=ADA096348">the original</a> on September 24, 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Algorithms&rft.atitle=Refined+analysis+and+improvements+on+some+factoring+algorithms&rft.volume=3&rft.issue=2&rft.pages=101-127&rft.date=1982&rft_id=info%3Adoi%2F10.1016%2F0196-6774%2882%2990012-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0657269%23id-name%3DMR&rft.aulast=Schnorr&rft.aufirst=Claus+P.&rft_id=http%3A%2F%2Fwww.dtic.mil%2Fget-tr-doc%2Fpdf%3FAD%3DADA096348&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-1987-seysen-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-1987-seysen_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeysen1987" class="citation journal cs1">Seysen, Martin (1987). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1987-0878705-X">"A probabilistic factorization algorithm with quadratic forms of negative discriminant"</a>. <i>Mathematics of Computation</i>. <b>48</b> (178): 757–780. <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/10.1090%2FS0025-5718-1987-0878705-X">10.1090/S0025-5718-1987-0878705-X</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0878705">0878705</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=A+probabilistic+factorization+algorithm+with+quadratic+forms+of+negative+discriminant&rft.volume=48&rft.issue=178&rft.pages=757-780&rft.date=1987&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-1987-0878705-X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0878705%23id-name%3DMR&rft.aulast=Seysen&rft.aufirst=Martin&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0025-5718-1987-0878705-X&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-1988-lenstra-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-1988-lenstra_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLenstra1988" class="citation journal cs1">Lenstra, Arjen K (1988). <a rel="nofollow" class="external text" href="https://infoscience.epfl.ch/record/164491/files/nscan9.PDF">"Fast and rigorous factorization under the generalized Riemann hypothesis"</a> <span class="cs1-format">(PDF)</span>. <i>Indagationes Mathematicae</i>. <b>50</b> (4): 443–454. <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%2FS1385-7258%2888%2980022-2">10.1016/S1385-7258(88)80022-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indagationes+Mathematicae&rft.atitle=Fast+and+rigorous+factorization+under+the+generalized+Riemann+hypothesis&rft.volume=50&rft.issue=4&rft.pages=443-454&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2FS1385-7258%2888%2980022-2&rft.aulast=Lenstra&rft.aufirst=Arjen+K&rft_id=https%3A%2F%2Finfoscience.epfl.ch%2Frecord%2F164491%2Ffiles%2Fnscan9.PDF&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></span> </li> <li id="cite_note-lenstra-pomerance-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-lenstra-pomerance_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lenstra-pomerance_14-1"><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="CITEREFLenstraPomerance1992" class="citation journal cs1">Lenstra, H. W.; Pomerance, Carl (July 1992). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/jams/1992-05-03/S0894-0347-1992-1137100-0/S0894-0347-1992-1137100-0.pdf">"A Rigorous Time Bound for Factoring Integers"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of the American Mathematical Society</i>. <b>5</b> (3): 483–516. <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/10.1090%2FS0894-0347-1992-1137100-0">10.1090/S0894-0347-1992-1137100-0</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1137100">1137100</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Mathematical+Society&rft.atitle=A+Rigorous+Time+Bound+for+Factoring+Integers&rft.volume=5&rft.issue=3&rft.pages=483-516&rft.date=1992-07&rft_id=info%3Adoi%2F10.1090%2FS0894-0347-1992-1137100-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1137100%23id-name%3DMR&rft.aulast=Lenstra&rft.aufirst=H.+W.&rft.au=Pomerance%2C+Carl&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fjams%2F1992-05-03%2FS0894-0347-1992-1137100-0%2FS0894-0347-1992-1137100-0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" 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=Integer_factorization&action=edit&section=14" 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="CITEREFRichard_Crandall_and_Carl_Pomerance2001" class="citation book cs1"><a href="/wiki/Richard_Crandall" title="Richard Crandall">Richard Crandall</a> and <a href="/wiki/Carl_Pomerance" title="Carl Pomerance">Carl Pomerance</a> (2001). <i>Prime Numbers: A Computational Perspective</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94777-9" title="Special:BookSources/0-387-94777-9"><bdi>0-387-94777-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Numbers%3A+A+Computational+Perspective&rft.pub=Springer&rft.date=2001&rft.isbn=0-387-94777-9&rft.au=Richard+Crandall+and+Carl+Pomerance&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span> Chapter 5: Exponential Factoring Algorithms, pp. 191–226. Chapter 6: Subexponential Factoring Algorithms, pp. 227–284. Section 7.4: Elliptic curve method, pp. 301–313.</li> <li><a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>. <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i>, Volume 2: <i>Seminumerical Algorithms</i>, Third Edition. Addison-Wesley, 1997. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-89684-2" title="Special:BookSources/0-201-89684-2">0-201-89684-2</a>. Section 4.5.4: Factoring into Primes, pp. 379–417.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamuel_S._Wagstaff_Jr.2013" class="citation book cs1"><a href="/wiki/Samuel_S._Wagstaff_Jr." title="Samuel S. Wagstaff Jr.">Samuel S. Wagstaff Jr.</a> (2013). <a rel="nofollow" class="external text" href="https://www.ams.org/bookpages/stml-68"><i>The Joy of Factoring</i></a>. Providence, RI: American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-1048-3" title="Special:BookSources/978-1-4704-1048-3"><bdi>978-1-4704-1048-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Joy+of+Factoring&rft.place=Providence%2C+RI&rft.pub=American+Mathematical+Society&rft.date=2013&rft.isbn=978-1-4704-1048-3&rft.au=Samuel+S.+Wagstaff+Jr.&rft_id=https%3A%2F%2Fwww.ams.org%2Fbookpages%2Fstml-68&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarren2013" class="citation book cs1">Warren, Henry S. Jr. (2013). <i><a href="/wiki/Hacker%27s_Delight" title="Hacker's Delight">Hacker's Delight</a></i> (2 ed.). <a href="/wiki/Addison_Wesley" class="mw-redirect" title="Addison Wesley">Addison Wesley</a> - <a href="/wiki/Pearson_Education,_Inc." class="mw-redirect" title="Pearson Education, Inc.">Pearson Education, Inc.</a> <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-84268-8" title="Special:BookSources/978-0-321-84268-8"><bdi>978-0-321-84268-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hacker%27s+Delight&rft.edition=2&rft.pub=Addison+Wesley+-+Pearson+Education%2C+Inc.&rft.date=2013&rft.isbn=978-0-321-84268-8&rft.aulast=Warren&rft.aufirst=Henry+S.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInteger+factorization" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integer_factorization&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://sourceforge.net/projects/msieve/">msieve</a> – SIQS and NFS – has helped complete some of the largest public factorizations known</li> <li>Richard P. Brent, "Recent Progress and Prospects for Integer Factorisation Algorithms", <i>Computing and Combinatorics"</i>, 2000, pp. 3–22. <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/327036.html">download</a></li> <li><a href="/wiki/Manindra_Agrawal" title="Manindra Agrawal">Manindra Agrawal</a>, Neeraj Kayal, Nitin Saxena, "PRIMES is in P." Annals of Mathematics 160(2): 781–793 (2004). <a rel="nofollow" class="external text" href="http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf">August 2005 version PDF</a></li> <li>Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/news/2005-11-08/rsa-640/">“RSA-640 Factored” <i>MathWorld Headline News</i>, November 8, 2005</a></li> <li><a rel="nofollow" class="external text" href="https://www.alpertron.com.ar/ECM.HTM">Dario Alpern's Integer factorization calculator</a> – A web app for factoring large integers</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 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assumption">Strong RSA</a></li> <li><a href="/wiki/Quadratic_residuosity_problem" title="Quadratic residuosity problem">Quadratic residuosity</a></li> <li><a href="/wiki/Decisional_composite_residuosity_assumption" title="Decisional composite residuosity assumption">Decisional composite residuosity</a></li> <li><a href="/wiki/Higher_residuosity_problem" title="Higher residuosity problem">Higher residuosity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Group theoretic</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/Discrete_logarithm#Cryptography" title="Discrete logarithm">Discrete logarithm</a></li> <li><a href="/wiki/Diffie%E2%80%93Hellman_problem" title="Diffie–Hellman problem">Diffie-Hellman</a></li> <li><a href="/wiki/Decisional_Diffie%E2%80%93Hellman_assumption" title="Decisional Diffie–Hellman assumption">Decisional Diffie–Hellman</a></li> <li><a href="/wiki/Computational_Diffie%E2%80%93Hellman_assumption" title="Computational Diffie–Hellman assumption">Computational Diffie–Hellman</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Pairings</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/XDH_assumption" title="XDH assumption">External Diffie–Hellman</a></li> <li><a href="/wiki/Sub-group_hiding" title="Sub-group hiding">Sub-group hiding</a></li> <li><a href="/wiki/Decision_Linear_assumption" title="Decision Linear assumption">Decision linear</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lattices</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/Lattice_problem#Shortest_vector_problem_(SVP)" title="Lattice problem">Shortest vector problem</a> (<a href="/wiki/Lattice_problem#GapSVP" title="Lattice problem">gap</a>)</li> <li><a href="/wiki/Lattice_problem#Closest_vector_problem_(CVP)" title="Lattice problem">Closest vector problem</a> (<a href="/wiki/Lattice_problem#GapCVP" title="Lattice problem">gap</a>)</li> <li><a href="/wiki/Learning_with_errors" title="Learning with errors">Learning with errors</a></li> <li><a href="/wiki/Ring_learning_with_errors" title="Ring learning with errors">Ring learning with errors</a></li> <li><a href="/wiki/Short_integer_solution_problem" title="Short integer solution problem">Short integer solution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-cryptographic</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/Exponential_time_hypothesis" title="Exponential time hypothesis">Exponential time hypothesis</a></li> <li><a href="/wiki/Unique_games_conjecture" title="Unique games conjecture">Unique games conjecture</a></li> <li><a href="/wiki/Planted_clique" title="Planted clique">Planted clique conjecture</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Number-theoretic_algorithms" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number-theoretic_algorithms" title="Template:Number-theoretic algorithms"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number-theoretic_algorithms" title="Template talk:Number-theoretic algorithms"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number-theoretic_algorithms" title="Special:EditPage/Template:Number-theoretic algorithms"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number-theoretic_algorithms" style="font-size:114%;margin:0 4em"><a href="/wiki/Number_theory" title="Number theory">Number-theoretic</a> <a href="/wiki/Algorithm" title="Algorithm">algorithms</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primality_test" title="Primality test">Primality tests</a></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/AKS_primality_test" title="AKS primality test">AKS</a></li> <li><a href="/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test" title="Adleman–Pomerance–Rumely primality test">APR</a></li> <li><a href="/wiki/Baillie%E2%80%93PSW_primality_test" title="Baillie–PSW primality test">Baillie–PSW</a></li> <li><a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">Elliptic curve</a></li> <li><a href="/wiki/Pocklington_primality_test" title="Pocklington primality test">Pocklington</a></li> <li><a href="/wiki/Fermat_primality_test" title="Fermat primality test">Fermat</a></li> <li><a href="/wiki/Lucas_primality_test" title="Lucas primality test">Lucas</a></li> <li><i><a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer</a></i></li> <li><i><a href="/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test" title="Lucas–Lehmer–Riesel test">Lucas–Lehmer–Riesel</a></i></li> <li><i><a href="/wiki/Proth%27s_theorem" title="Proth's theorem">Proth's theorem</a></i></li> <li><i><a href="/wiki/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's</a></i></li> <li><a href="/wiki/Quadratic_Frobenius_test" title="Quadratic Frobenius test">Quadratic Frobenius</a></li> <li><a href="/wiki/Solovay%E2%80%93Strassen_primality_test" title="Solovay–Strassen primality test">Solovay–Strassen</a></li> <li><a href="/wiki/Miller%E2%80%93Rabin_primality_test" title="Miller–Rabin primality test">Miller–Rabin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Generating_primes" class="mw-redirect" title="Generating primes">Prime-generating</a></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/Sieve_of_Atkin" title="Sieve of Atkin">Sieve of Atkin</a></li> <li><a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a></li> <li><a href="/wiki/Sieve_of_Pritchard" title="Sieve of Pritchard">Sieve of Pritchard</a></li> <li><a href="/wiki/Sieve_of_Sundaram" title="Sieve of Sundaram">Sieve of Sundaram</a></li> <li><a href="/wiki/Wheel_factorization" title="Wheel factorization">Wheel factorization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Integer factorization</a></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/Continued_fraction_factorization" title="Continued fraction factorization">Continued fraction (CFRAC)</a></li> <li><a href="/wiki/Dixon%27s_factorization_method" title="Dixon's factorization method">Dixon's</a></li> <li><a href="/wiki/Lenstra_elliptic-curve_factorization" title="Lenstra elliptic-curve factorization">Lenstra elliptic curve (ECM)</a></li> <li><a href="/wiki/Euler%27s_factorization_method" title="Euler's factorization method">Euler's</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm" title="Pollard's rho algorithm">Pollard's rho</a></li> <li><a href="/wiki/Pollard%27s_p_%E2%88%92_1_algorithm" title="Pollard's p − 1 algorithm"><i>p</i> − 1</a></li> <li><a href="/wiki/Williams%27s_p_%2B_1_algorithm" title="Williams's p + 1 algorithm"><i>p</i> + 1</a></li> <li><a href="/wiki/Quadratic_sieve" title="Quadratic sieve">Quadratic sieve (QS)</a></li> <li><a href="/wiki/General_number_field_sieve" title="General number field sieve">General number field sieve (GNFS)</a></li> <li><i><a href="/wiki/Special_number_field_sieve" title="Special number field sieve">Special number field sieve (SNFS)</a></i></li> <li><a href="/wiki/Rational_sieve" title="Rational sieve">Rational sieve</a></li> <li><a href="/wiki/Fermat%27s_factorization_method" title="Fermat's factorization method">Fermat's</a></li> <li><a href="/wiki/Shanks%27s_square_forms_factorization" title="Shanks's square forms factorization">Shanks's square forms</a></li> <li><a href="/wiki/Trial_division" title="Trial division">Trial division</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multiplication_algorithm" title="Multiplication algorithm">Multiplication</a></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/Ancient_Egyptian_multiplication" title="Ancient Egyptian multiplication">Ancient Egyptian</a></li> <li><a href="/wiki/Long_multiplication" class="mw-redirect" title="Long multiplication">Long</a></li> <li><a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba</a></li> <li><a href="/wiki/Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook</a></li> <li><a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen</a></li> <li><a href="/wiki/F%C3%BCrer%27s_algorithm" class="mw-redirect" title="Fürer's algorithm">Fürer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean</a> <a href="/wiki/Division_algorithm" title="Division algorithm">division</a></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/Binary_division" class="mw-redirect" title="Binary division">Binary</a></li> <li><a href="/wiki/Chunking_(division)" title="Chunking (division)">Chunking</a></li> <li><a href="/wiki/Fourier_division" title="Fourier division">Fourier</a></li> <li><a href="/wiki/Goldschmidt_division" class="mw-redirect" title="Goldschmidt division">Goldschmidt</a></li> <li><a href="/wiki/Newton%E2%80%93Raphson_division" class="mw-redirect" title="Newton–Raphson division">Newton-Raphson</a></li> <li><a href="/wiki/Long_division" title="Long division">Long</a></li> <li><a href="/wiki/Short_division" title="Short division">Short</a></li> <li><a href="/wiki/SRT_division" class="mw-redirect" title="SRT division">SRT</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" 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 hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baby-step_giant-step" title="Baby-step giant-step">Baby-step giant-step</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm_for_logarithms" title="Pollard's rho algorithm for logarithms">Pollard rho</a></li> <li><a href="/wiki/Pollard%27s_kangaroo_algorithm" title="Pollard's kangaroo algorithm">Pollard kangaroo</a></li> <li><a href="/wiki/Pohlig%E2%80%93Hellman_algorithm" title="Pohlig–Hellman algorithm">Pohlig–Hellman</a></li> <li><a href="/wiki/Index_calculus_algorithm" title="Index calculus algorithm">Index calculus</a></li> <li><a href="/wiki/Function_field_sieve" title="Function field sieve">Function field sieve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">Greatest common divisor</a></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/Binary_GCD_algorithm" title="Binary GCD algorithm">Binary</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean</a></li> <li><a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">Extended Euclidean</a></li> <li><a href="/wiki/Lehmer%27s_GCD_algorithm" title="Lehmer's GCD algorithm">Lehmer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadratic_residue" title="Quadratic residue">Modular square root</a></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/Cipolla%27s_algorithm" title="Cipolla's algorithm">Cipolla</a></li> <li><a href="/wiki/Pocklington%27s_algorithm" title="Pocklington's algorithm">Pocklington's</a></li> <li><a href="/wiki/Tonelli%E2%80%93Shanks_algorithm" title="Tonelli–Shanks algorithm">Tonelli–Shanks</a></li> <li><a href="/wiki/Berlekamp%E2%80%93Rabin_algorithm" title="Berlekamp–Rabin algorithm">Berlekamp</a></li> <li><a href="/wiki/Kunerth%27s_algorithm" title="Kunerth's algorithm">Kunerth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other algorithms</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/Chakravala_method" title="Chakravala method">Chakravala</a></li> <li><a href="/wiki/Cornacchia%27s_algorithm" title="Cornacchia's algorithm">Cornacchia</a></li> <li><a href="/wiki/Exponentiation_by_squaring" title="Exponentiation by squaring">Exponentiation by squaring</a></li> <li><a href="/wiki/Integer_square_root" title="Integer square root">Integer square root</a></li> <li><a href="/wiki/Integer_relation_algorithm" title="Integer relation algorithm">Integer relation</a> (<a href="/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm" title="Lenstra–Lenstra–Lovász lattice basis reduction algorithm">LLL</a>; <a href="/wiki/Korkine%E2%80%93Zolotarev_lattice_basis_reduction_algorithm" title="Korkine–Zolotarev lattice basis reduction algorithm">KZ</a>)</li> <li><a href="/wiki/Modular_exponentiation" title="Modular exponentiation">Modular exponentiation</a></li> <li><a href="/wiki/Montgomery_reduction" class="mw-redirect" title="Montgomery reduction">Montgomery reduction</a></li> <li><a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof</a></li> <li><a href="/wiki/Trachtenberg_system" title="Trachtenberg system">Trachtenberg system</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><i>Italics</i> indicate that algorithm is for numbers of special forms</li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Divisibility-based_sets_of_integers" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Divisor_classes" title="Template:Divisor classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Divisor_classes" title="Template talk:Divisor classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Divisor_classes" title="Special:EditPage/Template:Divisor classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Divisibility-based_sets_of_integers" style="font-size:114%;margin:0 4em">Divisibility-based sets of integers</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</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">Integer factorization</a></li> <li><a href="/wiki/Divisor" title="Divisor">Divisor</a></li> <li><a href="/wiki/Unitary_divisor" title="Unitary divisor">Unitary divisor</a></li> <li><a href="/wiki/Divisor_function" title="Divisor function">Divisor function</a></li> <li><a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">Prime factor</a></li> <li><a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Factorization forms</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/Prime_number" title="Prime number">Prime</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/Square-free_integer" title="Square-free integer">Square-free</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Unusual_number" title="Unusual number">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constrained divisor sums</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/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li> <li><a href="/wiki/Unitary_perfect_number" title="Unitary perfect number">Unitary perfect</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With many divisors</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/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Weird_number" title="Weird number">Weird</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequence</a>-related</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/Untouchable_number" title="Untouchable number">Untouchable</a></li> <li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a> (<a href="/wiki/Amicable_triple" title="Amicable triple">Triple</a>)</li> <li><a href="/wiki/Sociable_number" title="Sociable number">Sociable</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</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/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other sets</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/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Friendly_number#Solitary_numbers" title="Friendly number">Solitary</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" 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