{"id":"https://openalex.org/W3003635595","doi":"https://doi.org/10.1109/tc.2020.2970411","title":"Algorithms for Inversion Mod $p^k$","display_name":"Algorithms for Inversion Mod $p^k$","publication_year":2020,"publication_date":"2020-01-30","ids":{"openalex":"https://openalex.org/W3003635595","doi":"https://doi.org/10.1109/tc.2020.2970411","mag":"3003635595"},"language":"en","primary_location":{"id":"doi:10.1109/tc.2020.2970411","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tc.2020.2970411","pdf_url":null,"source":{"id":"https://openalex.org/S157670870","display_name":"IEEE Transactions on Computers","issn_l":"0018-9340","issn":["0018-9340","0016-9340","1557-9956","2326-3814"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Computers","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/A5018100877","display_name":"\u00c7etin Kaya Ko\u00e7","orcid":"https://orcid.org/0000-0002-2572-9565"},"institutions":[{"id":"https://openalex.org/I154570441","display_name":"University of California, Santa Barbara","ror":"https://ror.org/02t274463","country_code":"US","type":"education","lineage":["https://openalex.org/I154570441"]},{"id":"https://openalex.org/I4210121333","display_name":"Istinye University","ror":"https://ror.org/03081nz23","country_code":"TR","type":"education","lineage":["https://openalex.org/I4210121333"]},{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN","TR","US"],"is_corresponding":true,"raw_author_name":"Cetin Kaya Koc","raw_affiliation_strings":["Nanjing University of Aeronautics and Astronautics, Nanjing, China","University of California Santa Barbara, Santa Barbara, USA","\u0130stinye University, \u0130stanbul, Turkey"],"raw_orcid":"https://orcid.org/0000-0002-2572-9565","affiliations":[{"raw_affiliation_string":"Nanjing University of Aeronautics and Astronautics, Nanjing, China","institution_ids":["https://openalex.org/I9842412"]},{"raw_affiliation_string":"University of California Santa Barbara, Santa Barbara, USA","institution_ids":["https://openalex.org/I154570441"]},{"raw_affiliation_string":"\u0130stinye University, \u0130stanbul, Turkey","institution_ids":["https://openalex.org/I4210121333"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5018100877"],"corresponding_institution_ids":["https://openalex.org/I154570441","https://openalex.org/I4210121333","https://openalex.org/I9842412"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01244214,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"69","issue":"6","first_page":"907","last_page":"913"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9998000264167786,"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":0.9998000264167786,"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/T11130","display_name":"Coding theory and cryptography","score":0.9926999807357788,"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/T10237","display_name":"Cryptography and Data Security","score":0.9884999990463257,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.7274917364120483},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.5255172252655029},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.48759549856185913},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4569436013698578},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.40272170305252075},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36542704701423645},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.34110671281814575}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.7274917364120483},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.5255172252655029},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.48759549856185913},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4569436013698578},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.40272170305252075},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36542704701423645},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34110671281814575},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tc.2020.2970411","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tc.2020.2970411","pdf_url":null,"source":{"id":"https://openalex.org/S157670870","display_name":"IEEE Transactions on Computers","issn_l":"0018-9340","issn":["0018-9340","0016-9340","1557-9956","2326-3814"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Computers","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W1550170577","https://openalex.org/W1880212920","https://openalex.org/W2004814164","https://openalex.org/W2015464967","https://openalex.org/W2015855792","https://openalex.org/W2114496760","https://openalex.org/W2133708535","https://openalex.org/W2151163104","https://openalex.org/W2156476575","https://openalex.org/W2157672296","https://openalex.org/W3123845258","https://openalex.org/W3191099091","https://openalex.org/W4237773356","https://openalex.org/W6789511898"],"related_works":["https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2544423928","https://openalex.org/W2053286651","https://openalex.org/W2181743346","https://openalex.org/W2187401768","https://openalex.org/W2980963558","https://openalex.org/W2181413294","https://openalex.org/W2052122378","https://openalex.org/W2062023542"],"abstract_inverted_index":{"This":[0],"article":[1],"describes":[2],"and":[3,26,58,115],"analyzes":[4],"all":[5,91,142],"existing":[6,92,143],"algorithms":[7,93],"for":[8,22,94,110],"computing":[9],"x":[10,65],"=":[11,51,66],"a":[12,23,29,52,67,134],"<sup":[13,18,53,68,73],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[14,19,54,69,74],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">-1</sup>":[15,55,70],"(mod":[16,56,71],"p":[17,72],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">k</sup>":[20,75],")":[21,76],"prime":[24],"p,":[25,112],"also":[27],"introduces":[28],"new":[30,119],"algorithm":[31,44,86,120],"based":[32],"on":[33],"the":[34,47,60,63,85,105,118,127],"exact":[35],"solution":[36],"of":[37,62,84,97,130],"linear":[38],"equations":[39],"using":[40],"p-adic":[41],"expansions.":[42],"The":[43,80],"starts":[45],"with":[46],"initial":[48],"value":[49],"c":[50],"p)":[57],"iterativelycomputes":[59],"digits":[61],"inverse":[64],"in":[77],"base":[78],"p.":[79],"mod":[81],"2":[82],"version":[83],"is":[87,121],"more":[88],"efficient":[89],"than":[90],"small":[95],"values":[96],"k.":[98],"Moreover,":[99],"it":[100,125],"stands":[101],"out":[102],"as":[103,139],"being":[104],"only":[106],"one":[107],"that":[108],"works":[109],"any":[111,113],"k,":[114],"digit-by-digit.":[116],"While":[117],"asymptotically":[122],"worse":[123],"off,":[124],"requires":[126],"minimal":[128],"number":[129],"arithmetic":[131],"operations":[132],"(just":[133],"single":[135],"addition)":[136],"per":[137],"step,":[138],"compared":[140],"to":[141],"algorithms.":[144]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}