{"id":"https://openalex.org/W3045916595","doi":"https://doi.org/10.1145/3457341.3457342","title":"Computing one billion roots using the tangent Graeffe method","display_name":"Computing one billion roots using the tangent Graeffe method","publication_year":2020,"publication_date":"2020-09-01","ids":{"openalex":"https://openalex.org/W3045916595","doi":"https://doi.org/10.1145/3457341.3457342","mag":"3045916595"},"language":"en","primary_location":{"id":"doi:10.1145/3457341.3457342","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3457341.3457342","pdf_url":null,"source":{"id":"https://openalex.org/S176185422","display_name":"ACM communications in computer algebra","issn_l":"1932-2232","issn":["1932-2232","1932-2240"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320725","host_organization_name":"SIGSAM","host_organization_lineage":["https://openalex.org/P4310320725"],"host_organization_lineage_names":["SIGSAM"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Communications in Computer Algebra","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://hal.science/hal-02525408","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5001033788","display_name":"Joris van der Hoeven","orcid":"https://orcid.org/0000-0003-2244-1897"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA","FR"],"is_corresponding":false,"raw_author_name":"Joris van der Hoeven","raw_affiliation_strings":["CNRS, \u00c9cole polytechnique, Palaiseau, France","CNRS - Centre National de la Recherche Scientifique (France)","SFU - Department of Mathematics [Burnaby] (Simon Fraser University, 8888 University Drive, \r\nBurnaby, British Columbia  V5A 1S6 - Canada)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CNRS, \u00c9cole polytechnique, Palaiseau, France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I142476485"]},{"raw_affiliation_string":"CNRS - Centre National de la Recherche Scientifique (France)","institution_ids":["https://openalex.org/I1294671590"]},{"raw_affiliation_string":"SFU - Department of Mathematics [Burnaby] (Simon Fraser University, 8888 University Drive, \r\nBurnaby, British Columbia  V5A 1S6 - Canada)","institution_ids":["https://openalex.org/I18014758"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019266770","display_name":"Michael Monagan","orcid":"https://orcid.org/0000-0002-4652-2889"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Michael Monagan","raw_affiliation_strings":["Simon Fraser University, Burnaby, British Columbia, Canada","SFU - Department of Mathematics [Burnaby] (Simon Fraser University, 8888 University Drive, \r\nBurnaby, British Columbia V5A 1S6 - Canada)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Simon Fraser University, Burnaby, British Columbia, Canada","institution_ids":["https://openalex.org/I18014758"]},{"raw_affiliation_string":"SFU - Department of Mathematics [Burnaby] (Simon Fraser University, 8888 University Drive, \r\nBurnaby, British Columbia V5A 1S6 - Canada)","institution_ids":["https://openalex.org/I18014758"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.4021,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.707569,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"54","issue":"3","first_page":"65","last_page":"85"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9991000294685364,"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":0.9979000091552734,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5538995265960693},{"id":"https://openalex.org/keywords/tangent","display_name":"Tangent","score":0.5532978773117065},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5513305068016052},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.5460150837898254},{"id":"https://openalex.org/keywords/prime-factor","display_name":"Prime factor","score":0.5368445515632629},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5338372588157654},{"id":"https://openalex.org/keywords/fast-fourier-transform","display_name":"Fast Fourier transform","score":0.4824562668800354},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.4694252610206604},{"id":"https://openalex.org/keywords/tangent-vector","display_name":"Tangent vector","score":0.41944846510887146},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3973689675331116},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3810974359512329},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3412456512451172},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1329401433467865},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1048700213432312}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5538995265960693},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.5532978773117065},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5513305068016052},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5460150837898254},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.5368445515632629},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5338372588157654},{"id":"https://openalex.org/C75172450","wikidata":"https://www.wikidata.org/wiki/Q623950","display_name":"Fast Fourier transform","level":2,"score":0.4824562668800354},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4694252610206604},{"id":"https://openalex.org/C47890412","wikidata":"https://www.wikidata.org/wiki/Q1179296","display_name":"Tangent vector","level":3,"score":0.41944846510887146},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3973689675331116},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3810974359512329},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3412456512451172},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1329401433467865},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1048700213432312},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/3457341.3457342","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3457341.3457342","pdf_url":null,"source":{"id":"https://openalex.org/S176185422","display_name":"ACM communications in computer algebra","issn_l":"1932-2232","issn":["1932-2232","1932-2240"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320725","host_organization_name":"SIGSAM","host_organization_lineage":["https://openalex.org/P4310320725"],"host_organization_lineage_names":["SIGSAM"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Communications in Computer Algebra","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-02525408v4","is_oa":true,"landing_page_url":"https://hal.science/hal-02525408","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":"ACM Communications in Computer Algebra, 2021, 54 (3), pp.65-85. &#x27E8;10.1145/3457341.3457342&#x27E9;","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:HAL:hal-02525408v4","is_oa":true,"landing_page_url":"https://hal.science/hal-02525408","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":"ACM Communications in Computer Algebra, 2021, 54 (3), pp.65-85. &#x27E8;10.1145/3457341.3457342&#x27E9;","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W1974321539","https://openalex.org/W1978784845","https://openalex.org/W1979243540","https://openalex.org/W2001224529","https://openalex.org/W2002137405","https://openalex.org/W2010953531","https://openalex.org/W2022028092","https://openalex.org/W2025561788","https://openalex.org/W2028186291","https://openalex.org/W2045706641","https://openalex.org/W2052965490","https://openalex.org/W2065660785","https://openalex.org/W2067507719","https://openalex.org/W2069448237","https://openalex.org/W2070279173","https://openalex.org/W2072518300","https://openalex.org/W2073842331","https://openalex.org/W2074053034","https://openalex.org/W2078861427","https://openalex.org/W2088142312","https://openalex.org/W2096280747","https://openalex.org/W2102182691","https://openalex.org/W2104070031","https://openalex.org/W2108872752","https://openalex.org/W2136952590","https://openalex.org/W2143563649","https://openalex.org/W2168662780","https://openalex.org/W2223609897","https://openalex.org/W2274238134","https://openalex.org/W2400405359","https://openalex.org/W2504258127","https://openalex.org/W2737933154","https://openalex.org/W2818570966","https://openalex.org/W3018597471","https://openalex.org/W3040773772","https://openalex.org/W3102065507","https://openalex.org/W4234858060","https://openalex.org/W4297824757","https://openalex.org/W6786022642"],"related_works":["https://openalex.org/W2054570290","https://openalex.org/W2348092648","https://openalex.org/W4249328994","https://openalex.org/W4243792188","https://openalex.org/W2021174839","https://openalex.org/W4242263702","https://openalex.org/W2361271799","https://openalex.org/W4243493531","https://openalex.org/W2060675723","https://openalex.org/W2785123783"],"abstract_inverted_index":{"Let":[0,28],"p":[1,8,20,26,41,51,54],"be":[2,33],"a":[3,34,90,137,211,221,248,254,285],"prime":[4],"of":[5,36,66,69,84,106,122,139,161,187,213,226,242,250,305],"the":[6,22,64,78,126,146,151,172,185,188,192,205,227,232,240,243,310],"form":[7],"=":[9],"\u03c32":[10],"k":[11],"+":[12,59],"1":[13,60],"with":[14,25,45,93,231,272],"\u03c3":[15],"small":[16],"and":[17,103,118,174,201,223],"let":[18],"F":[19,40,50],"denote":[21],"finite":[23,282],"field":[24],"elements.":[27],"P":[29],"(":[30],"z":[31,43],")":[32,143],"polynomial":[35,168,212,274,292],"degree":[37,70,214],"d":[38,46,142],"in":[39,49,155],"[":[42],"]":[44],"distinct":[47],"roots":[48,65],".":[52,73,217],"For":[53],"=5":[55],"\u00b7":[56],"2":[57],"55":[58],"we":[61,76,183,195,198,203,208],"can":[62],"compute":[63],"such":[67,82],"polynomials":[68,83],"10":[71,95,116,215],"9":[72,216],"We":[74,88,124,149,218,238,252,265],"believe":[75],"are":[77,295],"first":[79],"to":[80],"factor":[81,138,210,249],"size":[85],"one":[86,304],"billion.":[87],"used":[89,125],"multi-core":[91],"computer":[92],"two":[94],"core":[96],"Intel":[97],"Xeon":[98],"E5":[99],"2680":[100],"v2":[101],"CPUs":[102],"128":[104],"gigabytes":[105,121],"RAM.":[107,123],"The":[108],"factorization":[109,275],"takes":[110],"just":[111],"under":[112],"4,000":[113],"seconds":[114],"on":[115,297],"cores":[117],"uses":[119],"121":[120],"tangent":[127,152,189,228,244],"Graeffe":[128,153,190,229,245],"root":[129,236,278],"finding":[130,279],"algorithm":[131,154,230,234,246,260],"from":[132],"[27,":[133],"19]":[134],"which":[135],"is":[136,262,284],"O":[140],"(log":[141],"faster":[144],"than":[145],"Cantor-Zassenhaus":[147,233],"algorithm.":[148],"implemented":[150],"C":[156],"using":[157,177],"our":[158,270,306],"own":[159],"library":[160],"64":[162],"bit":[163],"integer":[164],"FFT":[165,173],"based":[166,296],"in-place":[167,256],"algorithms":[169,289],"then":[170],"parallelized":[171,199],"main":[175,307],"steps":[176,186],"Cilk":[178],"C.":[179],"In":[180],"this":[181],"article":[182],"discuss":[184],"algorithm,":[191],"sub-algorithms":[193],"that":[194,261,294],"used,":[196],"how":[197,202],"them,":[200],"organized":[204],"memory":[206],"so":[207],"could":[209],"give":[219],"both":[220],"theoretical":[222],"practical":[224],"comparison":[225],"for":[235,288,290,309],"finding.":[237],"improve":[239],"complexity":[241],"by":[247],"2.":[251],"present":[253,266,311],"new":[255],"product":[257],"tree":[258],"multiplication":[259],"fully":[263],"parallelizable.":[264],"some":[267],"timings":[268],"comparing":[269],"software":[271],"Magma's":[273],"command.":[276],"Polynomial":[277],"over":[280],"smooth":[281],"fields":[283],"key":[286],"ingredient":[287],"sparse":[291],"interpolation":[293],"geometric":[298],"sequences.":[299],"This":[300],"application":[301],"was":[302],"also":[303],"motivations":[308],"work.":[312]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2025-10-10T00:00:00"}
