{"id":"https://openalex.org/W2043657477","doi":"https://doi.org/10.1145/1731022.1731026","title":"Efficient and formally proven reduction of large integers by small moduli","display_name":"Efficient and formally proven reduction of large integers by small moduli","publication_year":2010,"publication_date":"2010-04-01","ids":{"openalex":"https://openalex.org/W2043657477","doi":"https://doi.org/10.1145/1731022.1731026","mag":"2043657477"},"language":"en","primary_location":{"id":"doi:10.1145/1731022.1731026","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1731022.1731026","pdf_url":null,"source":{"id":"https://openalex.org/S163326656","display_name":"ACM Transactions on Mathematical Software","issn_l":"0098-3500","issn":["0098-3500","1557-7295"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Mathematical Software","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013720399","display_name":"L.M.W.J. Rutten","orcid":null},"institutions":[{"id":"https://openalex.org/I4210148065","display_name":"IBM (Netherlands)","ror":"https://ror.org/055yb9v95","country_code":"NL","type":"company","lineage":["https://openalex.org/I1341412227","https://openalex.org/I4210148065"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Luc Rutten","raw_affiliation_strings":["IBM, Delft, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"IBM, Delft, The Netherlands","institution_ids":["https://openalex.org/I4210148065"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5112510889","display_name":"Marko van Eekelen","orcid":null},"institutions":[{"id":"https://openalex.org/I145872427","display_name":"Radboud University Nijmegen","ror":"https://ror.org/016xsfp80","country_code":"NL","type":"education","lineage":["https://openalex.org/I145872427"]},{"id":"https://openalex.org/I7876267","display_name":"Open University of the Netherlands","ror":"https://ror.org/018dfmf50","country_code":"NL","type":"education","lineage":["https://openalex.org/I7876267"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Marko Van Eekelen","raw_affiliation_strings":["Radboud University Nijmegen and Open University of the Netherlands, Nijmegen, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Radboud University Nijmegen and Open University of the Netherlands, Nijmegen, The Netherlands","institution_ids":["https://openalex.org/I7876267","https://openalex.org/I145872427"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5013720399"],"corresponding_institution_ids":["https://openalex.org/I4210148065"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.12325117,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"37","issue":"2","first_page":"1","last_page":"21"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9993000030517578,"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"}},"topics":[{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9993000030517578,"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/T10054","display_name":"Parallel Computing and Optimization Techniques","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"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/T11034","display_name":"Digital Filter Design and Implementation","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/modular-arithmetic","display_name":"Modular arithmetic","score":0.7527160048484802},{"id":"https://openalex.org/keywords/moduli","display_name":"Moduli","score":0.7300258874893188},{"id":"https://openalex.org/keywords/correctness","display_name":"Correctness","score":0.6564844846725464},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.6433796286582947},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.5097789168357849},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5027613639831543},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.45920899510383606},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.44142773747444153},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.43242591619491577},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38211488723754883},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.364755243062973},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.26762664318084717},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.14423426985740662}],"concepts":[{"id":"https://openalex.org/C32049820","wikidata":"https://www.wikidata.org/wiki/Q319400","display_name":"Modular arithmetic","level":3,"score":0.7527160048484802},{"id":"https://openalex.org/C121089165","wikidata":"https://www.wikidata.org/wiki/Q6889796","display_name":"Moduli","level":2,"score":0.7300258874893188},{"id":"https://openalex.org/C55439883","wikidata":"https://www.wikidata.org/wiki/Q360812","display_name":"Correctness","level":2,"score":0.6564844846725464},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.6433796286582947},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.5097789168357849},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5027613639831543},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.45920899510383606},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.44142773747444153},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.43242591619491577},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38211488723754883},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.364755243062973},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.26762664318084717},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.14423426985740662},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1145/1731022.1731026","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1731022.1731026","pdf_url":null,"source":{"id":"https://openalex.org/S163326656","display_name":"ACM Transactions on Mathematical Software","issn_l":"0098-3500","issn":["0098-3500","1557-7295"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Mathematical Software","raw_type":"journal-article"},{"id":"pmh:oai:repository.ubn.ru.nl:2066/72253","is_oa":false,"landing_page_url":"http://hdl.handle.net/2066/72253","pdf_url":null,"source":{"id":"https://openalex.org/S4306401067","display_name":"Radboud Repository (Radboud University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I145872427","host_organization_name":"Radboud University Nijmegen","host_organization_lineage":["https://openalex.org/I145872427"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"External research report"},{"id":"pmh:oai:repository.ubn.ru.nl:2066/84108","is_oa":false,"landing_page_url":"http://hdl.handle.net/2066/84108","pdf_url":null,"source":{"id":"https://openalex.org/S4306401067","display_name":"Radboud Repository (Radboud University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I145872427","host_organization_name":"Radboud University Nijmegen","host_organization_lineage":["https://openalex.org/I145872427"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ACM Transactions on Mathematical Software, 37, 2, pp. 1-21","raw_type":"Article / Letter to editor"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W956810774","https://openalex.org/W1498685334","https://openalex.org/W1525685818","https://openalex.org/W1546768125","https://openalex.org/W1603543937","https://openalex.org/W1973779036","https://openalex.org/W1981663184","https://openalex.org/W2015918966","https://openalex.org/W2051277798","https://openalex.org/W2145428071","https://openalex.org/W2157018930","https://openalex.org/W2752853835","https://openalex.org/W4245800583","https://openalex.org/W4249577874","https://openalex.org/W4291439952"],"related_works":["https://openalex.org/W2405527256","https://openalex.org/W4386489432","https://openalex.org/W2163717605","https://openalex.org/W4311943969","https://openalex.org/W1976550211","https://openalex.org/W4363649707","https://openalex.org/W2589564280","https://openalex.org/W4313824220","https://openalex.org/W3159034445","https://openalex.org/W2954610826"],"abstract_inverted_index":{"On":[0],"w":[1,11,18,22,35,75,134,157],"-bit":[2,12,19,23],"processors":[3,165],"which":[4],"are":[5,37],"much":[6,200],"faster":[7,144],"at":[8,15],"multiplying":[9],"two":[10],"integers":[13,20,28,128],"than":[14,33,73,145],"dividing":[16],"2":[17,34,74,133,156],"by":[21,29,56,61,129,162,213],"integers,":[24],"reductions":[25],"of":[26,64,82,126,138,147,183,197,209],"large":[27,127],"moduli":[30,71,87,130,153],"M":[31],"smaller":[32,86],"-1":[36,76,135],"often":[38],"implemented":[39],"suboptimally,":[40],"leading":[41],"applications":[42],"to":[43,85,91,100,117,132,155,175,193],"take":[44],"excessive":[45],"processing":[46],"time.":[47],"We":[48,78],"present":[49],"a":[50,57,62,67,110,114,214],"modular":[51,94],"reduction":[52,125],"algorithm":[53,108,123,140],"implementing":[54],"division":[55],"modulus":[58],"through":[59],"multiplication":[60,169],"reciprocal":[63],"that":[65,80,150],"modulus,":[66],"well-known":[68],"method":[69,84],"for":[70,102,124,152],"larger":[72],".":[77,136,158],"show":[79],"application":[81],"this":[83,139],"makes":[88],"it":[89],"possible":[90],"express":[92],"certain":[93],"sums":[95],"and":[96,112],"differences":[97],"without":[98],"having":[99],"compensate":[101],"word":[103],"overflows.":[104],"By":[105],"embedding":[106],"the":[107,118,181,195],"in":[109,187],"loop":[111],"applying":[113],"few":[115],"transformations":[116],"loop,":[119],"we":[120,203],"obtain":[121],"an":[122,188],"up":[131,154],"Implementations":[137],"can":[141],"run":[142],"considerably":[143],"implementations":[146],"similar":[148],"algorithms":[149,179],"allow":[151],"This":[159],"is":[160,172],"substantiated":[161],"measurements":[163],"on":[164,180],"with":[166],"relatively":[167],"fast":[168],"instructions.":[170],"It":[171],"notoriously":[173],"hard":[174],"specify":[176],"efficient":[177],"mathematical":[178],"level":[182],"abstract":[184],"machine":[185],"instructions":[186],"error-free":[189],"manner.":[190],"In":[191],"order":[192],"eliminate":[194],"chance":[196],"errors":[198],"as":[199,201],"possible,":[202],"have":[204],"created":[205],"formal":[206],"correctness":[207],"proofs":[208],"our":[210],"algorithms,":[211],"checked":[212],"mechanized":[215],"proof":[216],"assistant.":[217]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2016-06-24T00:00:00"}