{"id":"https://openalex.org/W2749626892","doi":"https://doi.org/10.1080/01611194.2017.1325788","title":"On non-linearity and affine equivalence of permutations over an arbitrary finite commutative ring with unity","display_name":"On non-linearity and affine equivalence of permutations over an arbitrary finite commutative ring with unity","publication_year":2017,"publication_date":"2017-08-14","ids":{"openalex":"https://openalex.org/W2749626892","doi":"https://doi.org/10.1080/01611194.2017.1325788","mag":"2749626892"},"language":"en","primary_location":{"id":"doi:10.1080/01611194.2017.1325788","is_oa":false,"landing_page_url":"https://doi.org/10.1080/01611194.2017.1325788","pdf_url":null,"source":{"id":"https://openalex.org/S185196701","display_name":"Cryptologia","issn_l":"0161-1194","issn":["0161-1194","1558-1586"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cryptologia","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/A5009290876","display_name":"P. R. Mishra","orcid":"https://orcid.org/0000-0003-0473-6616"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"P. R. Mishra","raw_affiliation_strings":["View further author information"],"affiliations":[{"raw_affiliation_string":"View further author information","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101868886","display_name":"Yogesh Kumar","orcid":"https://orcid.org/0000-0003-4203-8958"},"institutions":[{"id":"https://openalex.org/I1325784139","display_name":"Yahoo (United Kingdom)","ror":"https://ror.org/038p3gq39","country_code":"GB","type":"company","lineage":["https://openalex.org/I1325784139","https://openalex.org/I4210134091"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Yogesh Kumar","raw_affiliation_strings":["Correspondenceyogesh_lather@yahoo.co.in"],"affiliations":[{"raw_affiliation_string":"Correspondenceyogesh_lather@yahoo.co.in","institution_ids":["https://openalex.org/I1325784139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009805746","display_name":"N. Rajesh Pillai","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"N. R. Pillai","raw_affiliation_strings":["View further author information"],"affiliations":[{"raw_affiliation_string":"View further author information","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031205881","display_name":"Rajat Sharma","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"R. K. Sharma","raw_affiliation_strings":["View further author information"],"affiliations":[{"raw_affiliation_string":"View further author information","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5101868886"],"corresponding_institution_ids":["https://openalex.org/I1325784139"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.11007076,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"42","issue":"1","first_page":"81","last_page":"94"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":1.0,"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":1.0,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/bijection","display_name":"Bijection","score":0.8535664081573486},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.7813138365745544},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.7686935067176819},{"id":"https://openalex.org/keywords/bijection-injection-and-surjection","display_name":"Bijection, injection and surjection","score":0.7444524765014648},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6839468479156494},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.6625837087631226},{"id":"https://openalex.org/keywords/commutative-ring","display_name":"Commutative ring","score":0.6138005256652832},{"id":"https://openalex.org/keywords/linearity","display_name":"Linearity","score":0.5564016699790955},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.508592963218689},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.500025749206543},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.49449893832206726},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.4788820147514343},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39047425985336304},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.35532650351524353},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.19852402806282043},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.06167420744895935}],"concepts":[{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.8535664081573486},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.7813138365745544},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.7686935067176819},{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.7444524765014648},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6839468479156494},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6625837087631226},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.6138005256652832},{"id":"https://openalex.org/C77170095","wikidata":"https://www.wikidata.org/wiki/Q1753188","display_name":"Linearity","level":2,"score":0.5564016699790955},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.508592963218689},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.500025749206543},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.49449893832206726},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.4788820147514343},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39047425985336304},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35532650351524353},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.19852402806282043},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.06167420744895935},{"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},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","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:10.1080/01611194.2017.1325788","is_oa":false,"landing_page_url":"https://doi.org/10.1080/01611194.2017.1325788","pdf_url":null,"source":{"id":"https://openalex.org/S185196701","display_name":"Cryptologia","issn_l":"0161-1194","issn":["0161-1194","1558-1586"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cryptologia","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":25,"referenced_works":["https://openalex.org/W567095041","https://openalex.org/W584171051","https://openalex.org/W606334457","https://openalex.org/W1505761511","https://openalex.org/W1513703559","https://openalex.org/W1531334709","https://openalex.org/W1556443961","https://openalex.org/W1587687098","https://openalex.org/W1606480398","https://openalex.org/W1660562555","https://openalex.org/W2021094554","https://openalex.org/W2024384464","https://openalex.org/W2032685643","https://openalex.org/W2043694411","https://openalex.org/W2082143349","https://openalex.org/W2390070928","https://openalex.org/W2401365507","https://openalex.org/W2504274129","https://openalex.org/W2798943694","https://openalex.org/W4211127378","https://openalex.org/W4213095169","https://openalex.org/W4234295943","https://openalex.org/W4243205024","https://openalex.org/W4289127653","https://openalex.org/W4298109338"],"related_works":["https://openalex.org/W2952259524","https://openalex.org/W2160566290","https://openalex.org/W2806276129","https://openalex.org/W4299611798","https://openalex.org/W2963449060","https://openalex.org/W1893226401","https://openalex.org/W1582142388","https://openalex.org/W2952981421","https://openalex.org/W2949947309","https://openalex.org/W4213409680"],"abstract_inverted_index":{"Cryptographic":[0],"properties":[1],"of":[2,48],"permutations":[3],"viz":[4],"non-linearity,":[5],"affine":[6],"equivalence,":[7],"and":[8],"mode":[9],"transform":[10],"have":[11],"been":[12],"studied":[13],"in":[14,62,74],"the":[15,26,49,71],"literature,":[16],"treating":[17],"them":[18,29,41],"as":[19,30],"bijections":[20],"on":[21,32],"\u2124n.":[22],"In":[23],"this":[24,42],"article,":[25],"authors":[27,54],"consider":[28],"bijection":[31],"an":[33,57],"arbitrary":[34],"finite":[35],"commutative":[36],"ring":[37],"with":[38],"unity.":[39],"Treating":[40],"way,":[43],"they":[44],"achieve":[45],"a":[46],"generalization":[47],"above":[50],"mentioned":[51],"properties.":[52],"The":[53],"also":[55],"propose":[56],"algorithm":[58],"for":[59],"computing":[60],"non-linearity":[61],"their":[63],"generalized":[64],"scenario,":[65],"which":[66],"is":[67],"faster":[68],"compared":[69],"to":[70],"direct":[72],"approach":[73],"many":[75],"cases.":[76]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}