{"id":163967,"date":"2007-01-01T00:00:00","date_gmt":"2007-01-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/computing-endomorphism-rings-of-jacobians-of-genus-2-curves-over-finite-fields\/"},"modified":"2018-10-16T20:04:25","modified_gmt":"2018-10-17T03:04:25","slug":"computing-endomorphism-rings-of-jacobians-of-genus-2-curves-over-finite-fields","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/computing-endomorphism-rings-of-jacobians-of-genus-2-curves-over-finite-fields\/","title":{"rendered":"Computing endomorphism rings of Jacobians of genus 2 curves over finite fields"},"content":{"rendered":"

We present algorithms which, given a genus 2 curve C<\/span><\/span><\/span><\/span> defined over a finite field and a quartic CM field K<\/span><\/span><\/span><\/span>, determine whether the endomorphism ring of the Jacobian J<\/span><\/span><\/span><\/span> of C<\/span><\/span><\/span><\/span> is the full ring of integers in K<\/span><\/span><\/span><\/span>. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J<\/span>[<\/span>\u2113<\/span>d<\/span><\/span>]<\/span><\/span><\/span><\/span> for prime powers \u2113<\/span>d<\/span><\/span><\/span><\/span><\/span>. We use these algorithms to create the first implementation of Eisentr\\”ager and Lauter’s algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem \\cite{el}, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute p<\/span>3<\/span><\/span><\/span><\/span><\/span> curves for many small primes p<\/span><\/span><\/span><\/span>.<\/p>\n","protected":false},"excerpt":{"rendered":"

We present algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present probabilistic algorithms for computing the field of definition of, and the […]<\/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":"World Scientific Publishing","msr_publisher_other":"","msr_booktitle":"Number Theory and Its Applications","msr_chapter":"","msr_edition":"Proceedings of SAGA 2007, Number Theory and Its Applications","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"10","msr_page_range_start":"10","msr_page_range_end":"","msr_series":"","msr_volume":"2007","msr_copyright":"","msr_conference_name":"Proceedings of SAGA 2007, Number Theory and Its Applications","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"David Freeman","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":"2007-01-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/arxiv.org\/abs\/math\/0701305","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":2007,"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":[13558],"msr-publication-type":[193716],"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-163967","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-security-privacy-cryptography","msr-locale-en_us"],"msr_publishername":"World Scientific Publishing","msr_edition":"Proceedings of SAGA 2007, Number Theory and Its Applications","msr_affiliation":"","msr_published_date":"2007-01-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"Number Theory and Its Applications","msr_pages_string":"10","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"2007","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","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":"","msr_publicationurl":"http:\/\/arxiv.org\/abs\/math\/0701305","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"http:\/\/arxiv.org\/abs\/math\/0701305","viewUrl":false,"id":false,"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":[{"id":0,"url":"http:\/\/arxiv.org\/abs\/math\/0701305"}],"msr-author-ordering":[{"type":"text","value":"David Freeman","user_id":0,"rest_url":false},{"type":"user_nicename","value":"klauter","user_id":32558,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=klauter"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[239792],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"inproceedings","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:\/\/www.microsoft.com\/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. 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