{"id":"https://openalex.org/W7128441521","doi":"https://doi.org/10.1109/tit.2026.3662335","title":"A Generalized \u03c7 <i> <sub>n</sub> </i> -Function","display_name":"A Generalized \u03c7 <i> <sub>n</sub> </i> -Function","publication_year":2026,"publication_date":"2026-02-09","ids":{"openalex":"https://openalex.org/W7128441521","doi":"https://doi.org/10.1109/tit.2026.3662335"},"language":null,"primary_location":{"id":"doi:10.1109/tit.2026.3662335","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2026.3662335","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","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/A5125467680","display_name":"Cheng Lyu","orcid":null},"institutions":[{"id":"https://openalex.org/I75900474","display_name":"Hubei University","ror":"https://ror.org/03a60m280","country_code":"CN","type":"education","lineage":["https://openalex.org/I75900474"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Cheng Lyu","raw_affiliation_strings":["Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China"],"raw_orcid":"https://orcid.org/0009-0009-6189-3846","affiliations":[{"raw_affiliation_string":"Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China","institution_ids":["https://openalex.org/I75900474"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031136617","display_name":"Mu Yuan","orcid":"https://orcid.org/0000-0002-1300-8170"},"institutions":[{"id":"https://openalex.org/I75900474","display_name":"Hubei University","ror":"https://ror.org/03a60m280","country_code":"CN","type":"education","lineage":["https://openalex.org/I75900474"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Mu Yuan","raw_affiliation_strings":["Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China"],"raw_orcid":"https://orcid.org/0000-0002-1300-8170","affiliations":[{"raw_affiliation_string":"Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China","institution_ids":["https://openalex.org/I75900474"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072537287","display_name":"Dabin Zheng","orcid":"https://orcid.org/0000-0003-3947-1590"},"institutions":[{"id":"https://openalex.org/I75900474","display_name":"Hubei University","ror":"https://ror.org/03a60m280","country_code":"CN","type":"education","lineage":["https://openalex.org/I75900474"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Dabin Zheng","raw_affiliation_strings":["Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China"],"raw_orcid":"https://orcid.org/0000-0003-3947-1590","affiliations":[{"raw_affiliation_string":"Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China","institution_ids":["https://openalex.org/I75900474"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5091454244","display_name":"Siwei Sun","orcid":"https://orcid.org/0000-0002-6568-6707"},"institutions":[{"id":"https://openalex.org/I4210132990","display_name":"State Key Laboratory of Cryptology","ror":"https://ror.org/02pn5rj08","country_code":"CN","type":"government","lineage":["https://openalex.org/I4210132990"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Siwei Sun","raw_affiliation_strings":["School of Cryptology, University of Chinese Academy of Sciences, Beijing, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Cryptology, University of Chinese Academy of Sciences, Beijing, China","institution_ids":["https://openalex.org/I4210132990","https://openalex.org/I4210165038"]}]},{"author_position":"last","author":{"id":null,"display_name":"Shun Li","orcid":"https://orcid.org/0000-0002-7663-8321"},"institutions":[{"id":"https://openalex.org/I4210132990","display_name":"State Key Laboratory of Cryptology","ror":"https://ror.org/02pn5rj08","country_code":"CN","type":"government","lineage":["https://openalex.org/I4210132990"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shun Li","raw_affiliation_strings":["School of Cryptology, University of Chinese Academy of Sciences, Beijing, China"],"raw_orcid":"https://orcid.org/0000-0002-7663-8321","affiliations":[{"raw_affiliation_string":"School of Cryptology, University of Chinese Academy of Sciences, Beijing, China","institution_ids":["https://openalex.org/I4210132990","https://openalex.org/I4210165038"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5125467680"],"corresponding_institution_ids":["https://openalex.org/I75900474"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.32760725,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"72","issue":"4","first_page":"2543","last_page":"2556"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12812","display_name":"Meromorphic and Entire Functions","score":0.23180000483989716,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12812","display_name":"Meromorphic and Entire Functions","score":0.23180000483989716,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11822","display_name":"Analytic and geometric function theory","score":0.226500004529953,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T14500","display_name":"Advanced Mathematical Theories","score":0.14480000734329224,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/modulo","display_name":"Modulo","score":0.6658999919891357},{"id":"https://openalex.org/keywords/abelian-group","display_name":"Abelian group","score":0.6427000164985657},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.6410999894142151},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.593500018119812},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.5271000266075134},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.516700029373169},{"id":"https://openalex.org/keywords/construct","display_name":"Construct (python library)","score":0.44609999656677246},{"id":"https://openalex.org/keywords/vector-space","display_name":"Vector space","score":0.39430001378059387}],"concepts":[{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6658999919891357},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6590999960899353},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.6427000164985657},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.6410999894142151},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.593500018119812},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.5271000266075134},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5175999999046326},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.516700029373169},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.44609999656677246},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4101000130176544},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.39430001378059387},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.35670000314712524},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.34040001034736633},{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.313400000333786},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.3093000054359436},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.2721000015735626},{"id":"https://openalex.org/C3020028006","wikidata":"https://www.wikidata.org/wiki/Q9158","display_name":"Electronic mail","level":2,"score":0.262800008058548},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2565999925136566},{"id":"https://openalex.org/C2779483109","wikidata":"https://www.wikidata.org/wiki/Q4681347","display_name":"Additive group","level":3,"score":0.2549000084400177},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2522999942302704}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2026.3662335","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2026.3662335","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2327870032","display_name":null,"funder_award_id":"62502494","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3790463292","display_name":null,"funder_award_id":"62532013","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4439847637","display_name":null,"funder_award_id":"2024M763221","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G5142255869","display_name":null,"funder_award_id":"2021YFA1000600","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5317161968","display_name":null,"funder_award_id":"12301671","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5497752695","display_name":null,"funder_award_id":"2021YFA1000600","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"},{"id":"https://openalex.org/G7735407926","display_name":null,"funder_award_id":"62032014","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8996357861","display_name":null,"funder_award_id":"62272148","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0],"mapping":[1,97],"<italic":[2,15,19,25,29,33,37,58,71,98,103,107,113,117,121,141,149,153,163,177,181,187,198,208,260,296,303,307,321,324,327,355,361,365],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[3,7,9,16,20,22,26,30,34,38,59,65,67,72,85,99,104,108,110,114,118,122,124,128,136,142,150,154,164,168,172,178,182,185,188,192,199,209,230,232,234,236,252,254,261,297,304,308,322,325,328,356,362,366],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7<sub>n</sub></i>":[4,60,363],"from":[5],"F<italic":[6,64,251],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><sup>n</sup></i><sub":[8,66,253],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[10,68,255],"to":[11,79,166,223,243,352,360],"itself":[12],"defined":[13,101],"by":[14,102,176],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">y</i>":[17,105],"=":[18,28,106,116,180],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7<sub>n</sub></i>(<italic":[21],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x</i>)":[23,111],"with":[24,112,148,264],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">y<sub>i</sub></i>":[27,115,179],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i</sub></i>":[31,119],"+":[32,36,120,126,130,138,194],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i</sub></i>+2(1":[35],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i</sub></i>+1),":[39],"where":[40,140,170],"the":[41,95,224,290],"indices":[42],"are":[43],"computed":[44],"modulo":[45],"n,":[46],"has":[47],"been":[48],"widely":[49],"studied":[50],"for":[51,197,256,293,299],"its":[52,77],"applications":[53],"in":[54,228,341],"lightweight":[55],"cryptography.":[56],"However,":[57],"is":[61,74,144,174],"bijective":[62],"on":[63],"only":[69],"when":[70],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</i>":[73,210,305],"odd,":[75],"restricting":[76],"use":[78],"odd-dimensional":[80],"vector":[81],"spaces":[82],"over":[83,250],"F<sub":[84,229],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>.":[86],"To":[87,156],"address":[88],"this":[89],"limitation,":[90],"we":[91,160,284,311],"introduce":[92],"and":[93,280,306,316,333,336,364],"analyze":[94,269],"generalized":[96],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7<sub>n,m</sub></i>":[100,165,298],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7<sub>n,m</sub></i>(<italic":[109],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i+m</sub></i>(<italic":[123],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i+m\u22121</sub></i>":[125],"1)(<italic":[127],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i+m\u22122</sub></i>":[129],"1)":[131,195],"\u00b7":[132,133,134],"(<italic":[135,191],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i+1</sub></i>":[137],"1),":[139],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">m</i>":[143,151],"a":[145,245,286,313,344],"fixed":[146,278],"integer":[147,259],"\u2224":[152],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</i>.":[155],"investigate":[157],"such":[158],"mappings,":[159,274],"further":[161],"generalize":[162],"\u03b8<italic":[167,171],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><sub>m,k</sub></i>,":[169],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><sub>m,k</sub></i>":[173],"given":[175],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i+mk</sub></i>":[183],"\u041f<italic":[184],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><sup>mk\u22121</sup></i>":[186],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><sub>j=1,":[189],"m\u2224j</sub></i>":[190],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x<sub>i+j</sub></i>":[193],",":[196,207],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">i</i>":[200],"\u2208":[201],"{0,":[202],"1,":[203],".":[204,205,206],"\u2212":[211],"1}.":[212],"We":[213,267],"prove":[214,337],"that":[215],"these":[216,273],"mappings":[217],"generate":[218],"an":[219],"abelian":[220],"group":[221,225],"isomorphic":[222],"of":[226,248,272,289,295,302,346,354],"units":[227],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>[<italic":[231],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">z</i>]/(<italic":[233],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">z</i><sup":[235],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u230a<i>n/m</i>\u230b+1</sup>).":[237],"This":[238],"structural":[239],"insight":[240],"enables":[241],"us":[242],"construct":[244],"broad":[246],"class":[247],"permutations":[249],"any":[257],"positive":[258],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</i>,":[262],"along":[263],"their":[265,276,334],"inverses.":[266],"rigorously":[268],"algebraic":[270],"properties":[271,292],"including":[275],"iterations,":[277],"points,":[279],"cycle":[281],"structures.":[282],"Additionally,":[283],"provide":[285],"comprehensive":[287],"database":[288],"cryptographic":[291],"iterates":[294],"small":[300],"values":[301],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">m</i>.":[309],"Finally,":[310],"conduct":[312],"comparative":[314],"security":[315],"implementation":[317],"cost":[318],"analysis":[319],"among":[320],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7<sub>n,m</sub></i>,":[323],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7<sub>n</sub></i>,":[326,357],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7\u03c7<sub>n</sub></i>":[329],"(EUROCRYPT":[330],"2025":[331],"[3])":[332],"variants,":[335],"Conjecture":[338],"1":[339],"proposed":[340],"[3]":[342],"as":[343],"by-product":[345],"our":[347],"study.":[348],"Our":[349],"results":[350],"lead":[351],"generalizations":[353],"providing":[358],"alternatives":[359],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03c7\u03c7<sub>n</sub></i>.":[367]},"counts_by_year":[],"updated_date":"2026-03-20T20:47:17.329874","created_date":"2026-02-10T00:00:00"}