{"id":"https://openalex.org/W2020296150","doi":"https://doi.org/10.1109/tsp.2011.2113176","title":"Closed-Form Orthogonal Number Theoretic Transform Eigenvectors and the Fast Fractional NTT","display_name":"Closed-Form Orthogonal Number Theoretic Transform Eigenvectors and the Fast Fractional NTT","publication_year":2011,"publication_date":"2011-02-11","ids":{"openalex":"https://openalex.org/W2020296150","doi":"https://doi.org/10.1109/tsp.2011.2113176","mag":"2020296150"},"language":"en","primary_location":{"id":"doi:10.1109/tsp.2011.2113176","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2011.2113176","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"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 Signal Processing","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/A5085279585","display_name":"Soo\u2010Chang Pei","orcid":"https://orcid.org/0000-0003-2448-4196"},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Soo-Chang Pei","raw_affiliation_strings":["Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan","Dept. of Electr. Eng.. Nat. Taiwan Univ., Taipei, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan","institution_ids":["https://openalex.org/I16733864"]},{"raw_affiliation_string":"Dept. of Electr. Eng.. Nat. Taiwan Univ., Taipei, Taiwan","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048690899","display_name":"Chia-Chang Wen","orcid":null},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Chia-Chang Wen","raw_affiliation_strings":["Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan","Dept. of Electr. Eng.. Nat. Taiwan Univ., Taipei, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan","institution_ids":["https://openalex.org/I16733864"]},{"raw_affiliation_string":"Dept. of Electr. Eng.. Nat. Taiwan Univ., Taipei, Taiwan","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5022371410","display_name":"Jian\u2013Jiun Ding","orcid":"https://orcid.org/0000-0003-4510-2273"},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Jian-Jiun Ding","raw_affiliation_strings":["Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan","Dept. of Electr. Eng.. Nat. Taiwan Univ., Taipei, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan","institution_ids":["https://openalex.org/I16733864"]},{"raw_affiliation_string":"Dept. of Electr. Eng.. Nat. Taiwan Univ., Taipei, Taiwan","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5085279585"],"corresponding_institution_ids":["https://openalex.org/I16733864"],"apc_list":null,"apc_paid":null,"fwci":1.8374,"has_fulltext":false,"cited_by_count":19,"citation_normalized_percentile":{"value":0.85414865,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"59","issue":"5","first_page":"2124","last_page":"2135"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11034","display_name":"Digital Filter Design and Implementation","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T11034","display_name":"Digital Filter Design and Implementation","score":0.9994999766349792,"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"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994000196456909,"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":0.9976000189781189,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mersenne-prime","display_name":"Mersenne prime","score":0.8683505058288574},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6378870606422424},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6271636486053467},{"id":"https://openalex.org/keywords/fermats-last-theorem","display_name":"Fermat's Last Theorem","score":0.4564627707004547},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.44878271222114563},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.43481892347335815},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3780641257762909},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1918753981590271}],"concepts":[{"id":"https://openalex.org/C29641680","wikidata":"https://www.wikidata.org/wiki/Q186875","display_name":"Mersenne prime","level":2,"score":0.8683505058288574},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6378870606422424},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6271636486053467},{"id":"https://openalex.org/C87670640","wikidata":"https://www.wikidata.org/wiki/Q132469","display_name":"Fermat's Last Theorem","level":2,"score":0.4564627707004547},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.44878271222114563},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.43481892347335815},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3780641257762909},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1918753981590271},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tsp.2011.2113176","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2011.2113176","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"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 Signal Processing","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":28,"referenced_works":["https://openalex.org/W615268498","https://openalex.org/W1567766094","https://openalex.org/W1593683169","https://openalex.org/W1983343479","https://openalex.org/W2013244555","https://openalex.org/W2018076255","https://openalex.org/W2022652042","https://openalex.org/W2073962970","https://openalex.org/W2090697566","https://openalex.org/W2090778219","https://openalex.org/W2096305981","https://openalex.org/W2098083269","https://openalex.org/W2102337953","https://openalex.org/W2116983948","https://openalex.org/W2123759036","https://openalex.org/W2132538465","https://openalex.org/W2133355175","https://openalex.org/W2135675826","https://openalex.org/W2137539933","https://openalex.org/W2138410653","https://openalex.org/W2140029228","https://openalex.org/W2147357069","https://openalex.org/W2149510369","https://openalex.org/W2153470366","https://openalex.org/W2154123414","https://openalex.org/W2155295891","https://openalex.org/W2157280639","https://openalex.org/W4235903219"],"related_works":["https://openalex.org/W1486266213","https://openalex.org/W4200236666","https://openalex.org/W4379791326","https://openalex.org/W2943053837","https://openalex.org/W2073069373","https://openalex.org/W117931972","https://openalex.org/W1729285580","https://openalex.org/W4388508657","https://openalex.org/W4388508656","https://openalex.org/W2950403881"],"abstract_inverted_index":{"In":[0,85],"this":[1],"paper,":[2],"we":[3,92],"propose":[4],"a":[5,71],"new":[6,118],"method":[7],"to":[8,34],"find":[9],"the":[10,21,27,36,42,48,51,57,95,102,108,116,125,128,134],"closed-form":[11],"solution":[12],"of":[13,127,136],"Number":[14],"Theoretic":[15],"Transform":[16],"(NTT)":[17],"eigenvectors.":[18],"We":[19,40],"construct":[20,94],"complete":[22],"generalized":[23],"Legendre":[24],"sequence":[25],"over":[26,56],"finite":[28],"field":[29,53],"(CGLSF)":[30],"and":[31,59,69,115,131],"use":[32],"it":[33],"solve":[35],"NTT":[37,44,64,83],"eigenvector":[38,65],"problem.":[39],"derive":[41],"CGLSF-like":[43],"eigenvectors":[45],"successfully,":[46],"including":[47,101],"case":[49],"where":[50],"operation":[52],"is":[54,67,75],"defined":[55],"Fermat":[58,104],"Mersenne":[60,111,119],"numbers.":[61],"The":[62],"derived":[63],"set":[66],"orthogonal":[68],"has":[70],"closed":[72],"form.":[73],"It":[74],"suitable":[76],"for":[77,82],"constructing":[78],"sub-NTT":[79],"building":[80],"blocks":[81],"implementation.":[84],"addition,":[86],"with":[87],"different":[88],"eigenvalue":[89],"assignment":[90],"rule,":[91],"can":[93],"fractional":[96,103,109,117],"number":[97,105,112,120],"theoretic":[98],"transform":[99,106,113,121],"(FNTT),":[100],"(FFNT),":[107],"complex":[110],"(FCMNT),":[114],"(FNMNT).":[122],"They":[123],"are":[124],"generalizations":[126],"original":[129],"transforms":[130],"all":[132],"have":[133],"complexities":[135],"<i":[137,141,148],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[138,142,146,149],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">O</i>":[139],"(":[140],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">N</i>":[143,150],"log":[144],"<sub":[145],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[147],").":[151]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}