{"id":183144,"date":"2007-01-29T00:00:00","date_gmt":"2009-10-31T10:21:42","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/a-p-adic-algorithm-to-compute-the-hilbert-class-polynomial\/"},"modified":"2016-09-09T09:52:18","modified_gmt":"2016-09-09T16:52:18","slug":"a-p-adic-algorithm-to-compute-the-hilbert-class-polynomial","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/a-p-adic-algorithm-to-compute-the-hilbert-class-polynomial\/","title":{"rendered":"A p-adic algorithm to compute the Hilbert class polynomial"},"content":{"rendered":"<div class=\"asset-content\">\n<p>A classical approach of constructing elliptic curves that can be used for cryptographic purposes relies on the theory of complex multiplication. A key ingredient in the algorithm is to compute the Hilbert class polynomial P<sub>D<\/sub> for a suitable discriminant D. The polynomial P<sub>D<\/sub> has integer coefficients, and is the minimal polynomial of the modular j-value j(O<sub>D<\/sub>) for the imaginary quadratic order O<sub>D<\/sub> of discriminant D.<\/p>\n<p>The polynomial P<sub>D<\/sub> can be computed using complex analytic techniques. In this talk we present a new p-adic algorithm to compute P<sub>D<\/sub>. One of the advantages of working over a p-adic field is that we do not have to worry about rounding errors, and the p-adic algorithm is the first algorithm with a rigorous run time analysis. When implemented carefully, the p-adic algorithm is very fast in practice and easily competes with the complex analytic approach. Many examples will be given.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A classical approach of constructing elliptic curves that can be used for cryptographic purposes relies on the theory of complex multiplication. A key ingredient in the algorithm is to compute the Hilbert class polynomial PD for a suitable discriminant D. The polynomial PD has integer coefficients, and is the minimal polynomial of the modular j-value [&hellip;]<\/p>\n","protected":false},"featured_media":194919,"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":[],"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-183144","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/m0JOvYTcMjM","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183144","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183144\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/194919"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=183144"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=183144"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=183144"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=183144"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=183144"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=183144"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=183144"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=183144"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=183144"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=183144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}