{"id":182311,"date":"2008-10-15T00:00:00","date_gmt":"2009-10-31T09:32:57","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/msr-research-item\/bilinear-complexity-of-the-multiplication-in-a-finite-extention-of-a-finite-field\/"},"modified":"2018-07-19T08:49:10","modified_gmt":"2018-07-19T15:49:10","slug":"bilinear-complexity-of-the-multiplication-in-a-finite-extention-of-a-finite-field","status":"publish","type":"msr-video","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/video\/bilinear-complexity-of-the-multiplication-in-a-finite-extention-of-a-finite-field\/","title":{"rendered":"Bilinear Complexity of the Multiplication in a Finite Extention of a Finite Field"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Let <i>q=p<sup>r<\/sup><\/i> be a prime power and <i>F<sub>q<\/sub><\/i> be the finite field with <i>q<\/i> elements. We study the multiplication of two polynomials in <i>F<sub>q<\/sub> [X]<\/i>, with degree <i>\u2264 n-1<\/i>, modulo an irreducible polynomial of degree <i>n<\/i>, namely the multiplication in the finite field <i>F<sub>q<sup>n<\/sup><\/sub><\/i>.<\/p>\n<p>We want to find a multiplication algorithm involving two variables in <i>F<sub>q<sup>n<\/sup><\/sub><\/i> minimizing the number of bilinear multiplications (i.e. involving two variables) in <i>F<sub>q<\/sub><\/i>. We don&#8217;t take in account multiplications of a variable in <i>F<sub>q<\/sub><\/i> by a constant in <i>F<sub>q<\/sub><\/i> (However these linear operations have a cost).<\/p>\n<p>It turns out that the number of bilinear operations is related to a tensor expression of the multiplication and that the problem is to find the rank of this tensor.<\/p>\n<p>We will give, using interpolation on algebraic curves over the finite field <i>F<sub>q<\/sub><\/i>, a sharp estimate for this bilinear complexity.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let q=pr be a prime power and Fq be the finite field with q elements. We study the multiplication of two polynomials in Fq [X], with degree \u2264 n-1, modulo an irreducible polynomial of degree n, namely the multiplication in the finite field Fqn. We want to find a multiplication algorithm involving two variables in [&hellip;]<\/p>\n","protected":false},"featured_media":194557,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[13558],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-182311","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-research-area-security-privacy-cryptography","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/RfmpDzudtCE","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/182311","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":1,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/182311\/revisions"}],"predecessor-version":[{"id":496157,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/182311\/revisions\/496157"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/media\/194557"}],"wp:attachment":[{"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/media?parent=182311"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=182311"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=182311"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=182311"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=182311"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=182311"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=182311"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=182311"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=182311"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=182311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}