We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z\/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also consider several modular functions g for which Phi_n^g is smaller than Phi_n, allowing us to handle n over 60000.<\/p>\n","protected":false},"excerpt":{"rendered":"
We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z\/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem […]<\/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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13558],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-163870","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-security-privacy-cryptography","msr-locale-en_us"],"msr_publishername":"American Mathematical Society","msr_edition":"Mathematics of Computation","msr_affiliation":"","msr_published_date":"2012-01-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"Mathematics of Computation 81 (2012), 1201-1231","msr_volume":"81","msr_number":"278","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":"457320","msr_publicationurl":"http:\/\/arxiv.org\/abs\/1001.0402","msr_doi":"10.1090\/S0025-5718-2011-02508-1","msr_publication_uploader":[{"type":"file","title":"Modular Polynomials via Isogeny Volcanoes","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/uploads\/prod\/2012\/01\/1001.0402v2.pdf","id":457320,"label_id":0},{"type":"url","title":"http:\/\/arxiv.org\/abs\/1001.0402","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.1090\/S0025-5718-2011-02508-1","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"http:\/\/arxiv.org\/abs\/1001.0402"}],"msr-author-ordering":[{"type":"user_nicename","value":"reinierb","user_id":33377,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=reinierb"},{"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"},{"type":"text","value":"Andrew V. 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