{"id":"https://openalex.org/W2007615168","doi":"https://doi.org/10.1090/s0025-5718-09-02199-1","title":"Computing modular polynomials in quasi-linear time","display_name":"Computing modular polynomials in quasi-linear time","publication_year":2009,"publication_date":"2009-04-27","ids":{"openalex":"https://openalex.org/W2007615168","doi":"https://doi.org/10.1090/s0025-5718-09-02199-1","mag":"2007615168"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-09-02199-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02199-1","pdf_url":"https://www.ams.org/mcom/2009-78-267/S0025-5718-09-02199-1/S0025-5718-09-02199-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2009-78-267/S0025-5718-09-02199-1/S0025-5718-09-02199-1.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007318055","display_name":"Andreas Enge","orcid":null},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en informatique et en automatique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Andreas Enge","raw_affiliation_strings":["INRIA Saclay\u00e2\u0080\u0093\u00c3\u008ele-de-France & Laboratoire d\u00e2\u0080\u0099Informatique (CNRS/UMR 7161), \u00c3\u0089cole polytechnique, 91128 Palaiseau Cedex, France","Lithe and fast algorithmic number theory"],"affiliations":[{"raw_affiliation_string":"INRIA Saclay\u00e2\u0080\u0093\u00c3\u008ele-de-France & Laboratoire d\u00e2\u0080\u0099Informatique (CNRS/UMR 7161), \u00c3\u0089cole polytechnique, 91128 Palaiseau Cedex, France","institution_ids":["https://openalex.org/I1326498283","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"Lithe and fast algorithmic number theory","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5007318055"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I1326498283"],"apc_list":null,"apc_paid":null,"fwci":11.6445,"has_fulltext":true,"cited_by_count":46,"citation_normalized_percentile":{"value":0.98218304,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"78","issue":"267","first_page":"1809","last_page":"1824"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9958999752998352,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8112525939941406},{"id":"https://openalex.org/keywords/modular-form","display_name":"Modular form","score":0.5848126411437988},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.526153028011322},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4897427558898926},{"id":"https://openalex.org/keywords/modular-design","display_name":"Modular design","score":0.4818771779537201},{"id":"https://openalex.org/keywords/elliptic-curve","display_name":"Elliptic curve","score":0.4706662893295288},{"id":"https://openalex.org/keywords/correctness","display_name":"Correctness","score":0.43883639574050903},{"id":"https://openalex.org/keywords/modular-curve","display_name":"Modular curve","score":0.4347783625125885},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.427834153175354},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.41678714752197266},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41606035828590393},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3671413064002991},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2466646432876587},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2425483763217926},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11208224296569824},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09607243537902832}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8112525939941406},{"id":"https://openalex.org/C75764964","wikidata":"https://www.wikidata.org/wiki/Q870797","display_name":"Modular form","level":2,"score":0.5848126411437988},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.526153028011322},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4897427558898926},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.4818771779537201},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.4706662893295288},{"id":"https://openalex.org/C55439883","wikidata":"https://www.wikidata.org/wiki/Q360812","display_name":"Correctness","level":2,"score":0.43883639574050903},{"id":"https://openalex.org/C59766168","wikidata":"https://www.wikidata.org/wiki/Q3001220","display_name":"Modular curve","level":3,"score":0.4347783625125885},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.427834153175354},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.41678714752197266},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41606035828590393},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3671413064002991},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2466646432876587},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2425483763217926},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11208224296569824},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09607243537902832},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1090/s0025-5718-09-02199-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02199-1","pdf_url":"https://www.ams.org/mcom/2009-78-267/S0025-5718-09-02199-1/S0025-5718-09-02199-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.192.4597","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.4597","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/journals/mcom/2009-78-267/S0025-5718-09-02199-1/S0025-5718-09-02199-1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.311.1014","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.311.1014","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0704.3177v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.311.4237","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.311.4237","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0704.3177v2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.405.4483","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.405.4483","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://hal.archives-ouvertes.fr/docs/00/30/51/86/PDF/modcomp.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:inria-00143084v2","is_oa":true,"landing_page_url":"https://inria.hal.science/inria-00143084","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics of Computation, 2009, 78 (267), pp.1809-1824","raw_type":"Journal articles"},{"id":"pmh:oai:oskar-bordeaux.fr:20.500.12278/116500","is_oa":false,"landing_page_url":"https://oskar-bordeaux.fr/handle/20.500.12278/116500","pdf_url":null,"source":{"id":"https://openalex.org/S4306402569","display_name":"Oskar-Bordeaux (Universite de Bordeaux)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article de revue"},{"id":"pmh:oai:oskar-bordeaux.fr:20.500.12278/166012","is_oa":false,"landing_page_url":"https://oskar-bordeaux.fr/handle/20.500.12278/166012","pdf_url":null,"source":{"id":"https://openalex.org/S4306402569","display_name":"Oskar-Bordeaux (Universite de Bordeaux)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article de revue"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-09-02199-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-09-02199-1","pdf_url":"https://www.ams.org/mcom/2009-78-267/S0025-5718-09-02199-1/S0025-5718-09-02199-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2007615168.pdf","grobid_xml":"https://content.openalex.org/works/W2007615168.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W628698491","https://openalex.org/W1495655187","https://openalex.org/W1511091366","https://openalex.org/W1546559129","https://openalex.org/W1562183207","https://openalex.org/W1846216834","https://openalex.org/W2018927995","https://openalex.org/W2019135503","https://openalex.org/W2026644646","https://openalex.org/W2034285964","https://openalex.org/W2054592337","https://openalex.org/W2060520271","https://openalex.org/W2067361453","https://openalex.org/W2083368929","https://openalex.org/W2107012927","https://openalex.org/W2113672208","https://openalex.org/W2146323046","https://openalex.org/W2151843313","https://openalex.org/W2156704072","https://openalex.org/W2170835539","https://openalex.org/W2298873378","https://openalex.org/W4390506026"],"related_works":["https://openalex.org/W1994841333","https://openalex.org/W4286912068","https://openalex.org/W2963841585","https://openalex.org/W3102323471","https://openalex.org/W4302317481","https://openalex.org/W4300573607","https://openalex.org/W1919889031","https://openalex.org/W3169715737","https://openalex.org/W1559870539","https://openalex.org/W4221167579"],"abstract_inverted_index":{"We":[0],"analyse":[1],"and":[2,47,172,183],"compare":[3],"the":[4,14,22,25,61,64,71,114,145,180,190,193],"complexity":[5,52],"of":[6,24,44,63,77,142,175,189,200],"several":[7],"algorithms":[8,138],"for":[9,126],"computing":[10,197],"modular":[11,45,73,124,170,198],"polynomials.":[12,66],"Under":[13],"assumption":[15],"that":[16,36,53],"rounding":[17],"errors":[18],"do":[19],"not":[20],"influence":[21],"correctness":[23],"result,":[26],"which":[27,131],"appears":[28],"to":[29,56,117,150],"be":[30],"satisfied":[31],"in":[32,60,81,136,205],"practice,":[33],"we":[34],"show":[35],"an":[37,133],"algorithm":[38,146,191],"relying":[39],"on":[40,48,210],"floating":[41],"point":[42],"evaluation":[43],"functions":[46],"interpolation":[49],"has":[50],"a":[51,169],"is":[54,113,147],"up":[55],"logarithmic":[57],"factors":[58],"linear":[59],"size":[62],"computed":[65],"In":[67],"particular,":[68],"it":[69],"obtains":[70],"classical":[72],"polynomial":[74],"$\\Phi":[75],"_\\ell$":[76],"prime":[78,162,176],"level":[79,202],"$\\ell$":[80],"time":[82,115],"\\[":[83],"O":[84,97],"\\left":[85,98],"(\\ell":[86,92,129],"^2":[87],"\\log":[88,102],"^3":[89,101],"\\ell":[90,100],"M":[91],")":[93,95],"\\right":[94,107],"\\subseteq":[96],"(":[99],"^{4":[103],"+":[104],"\\varepsilon":[105],"}\\ell":[106],"),":[108],"\\]":[109],"where":[110],"$M(\\ell":[111],")$":[112],"needed":[116],"multiply":[118],"two":[119,208],"$\\ell$-bit":[120],"numbers.":[121],"Besides":[122],"treating":[123],"polynomials":[125,167,182],"$\\Gamma":[127],"^0":[128],")$,":[130],"are":[132,156],"important":[134],"ingredient":[135],"many":[137],"dealing":[139],"with":[140],"isogenies":[141],"elliptic":[143],"curves,":[144],"easily":[148,160],"adapted":[149],"more":[151],"general":[152],"situations.":[153],"Composite":[154],"levels":[155],"handled":[157],"just":[158],"as":[159,161,164,166,179],"levels,":[163],"well":[165],"between":[168],"function":[171],"its":[173],"transform":[174],"level,":[177],"such":[178],"Schl\u00c3\u00a4fli":[181],"their":[184],"generalisations.":[185],"Our":[186],"distributed":[187],"implementation":[188],"confirms":[192],"theoretical":[194],"analysis":[195],"by":[196],"equations":[199],"record":[201],"around":[203],"$10000$":[204],"less":[206],"than":[207],"weeks":[209],"ten":[211],"processors.":[212]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":4},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}