{"id":159780,"date":"2010-01-01T00:00:00","date_gmt":"2010-01-01T00:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/msr-research-item\/a-cm-construction-for-curves-of-genus-2-and-p-rank-1\/"},"modified":"2018-10-16T19:57:23","modified_gmt":"2018-10-17T02:57:23","slug":"a-cm-construction-for-curves-of-genus-2-and-p-rank-1","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/a-cm-construction-for-curves-of-genus-2-and-p-rank-1\/","title":{"rendered":"A CM construction for curves of genus 2 and p-rank 1"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field F<sub>p<sup>2<\/sup><\/sub> of p<sup>2<\/sup> elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of F<sub>p<sup>2<\/sup><\/sub>-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over F<sub>p<sup>2<\/sup><\/sub> out of necessity: we show that curves of p-rank 1 over F<sub>p<\/sub> for large p cannot be efficiently constructed using explicit CM constructions.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field Fp2 of p2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of Fp2-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Elsevier","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Journal of Number Theory","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"Copyright \u00a9 2007 Elsevier B.V. All rights reserved.","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"Laura Hitt O'Connor, Gary McGuire","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2010-01-01","msr_highlight_text":"","msr_notes":"doi:10.1016\/j.jnt.2010.05.002 \r\n","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":2010,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-159780","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"Elsevier","msr_edition":"","msr_affiliation":"","msr_published_date":"2010-01-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"Journal of Number Theory","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"doi:10.1016\/j.jnt.2010.05.002","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"207347","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"hitt-mcguire-naehrig-streng-2010-eprint.pdf","viewUrl":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-content\/uploads\/2016\/02\/hitt-mcguire-naehrig-streng-2010-eprint.pdf","id":207347,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[],"msr-author-ordering":[{"type":"text","value":"Laura Hitt O'Connor","user_id":0,"rest_url":false},{"type":"text","value":"Gary McGuire","user_id":0,"rest_url":false},{"type":"user_nicename","value":"mnaehrig","user_id":32976,"rest_url":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=mnaehrig"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[239792],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"article","related_content":{"projects":[{"ID":239792,"post_title":"Elliptic Curve Cryptography (ECC)","post_name":"elliptic-curve-cryptography-ecc","post_type":"msr-project","post_date":"2016-06-29 20:49:17","post_modified":"2020-03-31 12:25:10","post_status":"publish","permalink":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/project\/elliptic-curve-cryptography-ecc\/","post_excerpt":"In the last 25 years, Elliptic Curve Cryptography (ECC) has become a mainstream primitive for cryptographic protocols and applications. ECC has been standardized for use in key exchange and digital signatures. This project focuses on efficient generation of parameters and implementation of ECC and pairing-based crypto primitives, across architectures and platforms.","_links":{"self":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-project\/239792"}]}}]},"_links":{"self":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/159780","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":1,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/159780\/revisions"}],"predecessor-version":[{"id":409505,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/159780\/revisions\/409505"}],"wp:attachment":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/media?parent=159780"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=159780"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=159780"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=159780"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=159780"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=159780"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=159780"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=159780"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=159780"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=159780"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=159780"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=159780"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=159780"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}