{"id":"https://openalex.org/W4395008407","doi":"https://doi.org/10.1007/s10623-024-01407-9","title":"Using $$P_\\tau $$ property for designing bent functions provably outside the completed Maiorana\u2013McFarland class","display_name":"Using $$P_\\tau $$ property for designing bent functions provably outside the completed Maiorana\u2013McFarland class","publication_year":2024,"publication_date":"2024-04-22","ids":{"openalex":"https://openalex.org/W4395008407","doi":"https://doi.org/10.1007/s10623-024-01407-9"},"language":"en","primary_location":{"id":"doi:10.1007/s10623-024-01407-9","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s10623-024-01407-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-024-01407-9.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10623-024-01407-9.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039271966","display_name":"Enes Pa\u0161ali\u0107","orcid":"https://orcid.org/0000-0001-6343-8796"},"institutions":[{"id":"https://openalex.org/I118905719","display_name":"University of Primorska","ror":"https://ror.org/05xefg082","country_code":"SI","type":"education","lineage":["https://openalex.org/I118905719"]},{"id":"https://openalex.org/I5343935","display_name":"Guilin University of Electronic Technology","ror":"https://ror.org/05arjae42","country_code":"CN","type":"education","lineage":["https://openalex.org/I5343935"]}],"countries":["CN","SI"],"is_corresponding":true,"raw_author_name":"Enes Pasalic","raw_affiliation_strings":["Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, People\u2019s Republic of China","University of Primorska, FAMNIT & IAM, Koper, Slovenia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, People\u2019s Republic of China","institution_ids":["https://openalex.org/I5343935"]},{"raw_affiliation_string":"University of Primorska, FAMNIT & IAM, Koper, Slovenia","institution_ids":["https://openalex.org/I118905719"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028148607","display_name":"Amar Bapi\u0107","orcid":"https://orcid.org/0000-0002-3568-4321"},"institutions":[{"id":"https://openalex.org/I118905719","display_name":"University of Primorska","ror":"https://ror.org/05xefg082","country_code":"SI","type":"education","lineage":["https://openalex.org/I118905719"]}],"countries":["SI"],"is_corresponding":false,"raw_author_name":"Amar Bapi\u0107","raw_affiliation_strings":["University of Primorska, FAMNIT & IAM, Koper, Slovenia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Primorska, FAMNIT & IAM, Koper, Slovenia","institution_ids":["https://openalex.org/I118905719"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056616258","display_name":"Fengrong Zhang","orcid":"https://orcid.org/0000-0002-6804-6975"},"institutions":[{"id":"https://openalex.org/I149594827","display_name":"Xidian University","ror":"https://ror.org/05s92vm98","country_code":"CN","type":"education","lineage":["https://openalex.org/I149594827"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fengrong Zhang","raw_affiliation_strings":["School of Cyber Engineering, Xidian University, Xi\u2019an, 710071, People\u2019s Republic of China","School of Cyber Engineering, Xidian University, Xi'an, 710071, People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Cyber Engineering, Xidian University, Xi\u2019an, 710071, People\u2019s Republic of China","institution_ids":["https://openalex.org/I149594827"]},{"raw_affiliation_string":"School of Cyber Engineering, Xidian University, Xi'an, 710071, People's Republic of China","institution_ids":["https://openalex.org/I149594827"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5027674774","display_name":"Yongzhuang Wei","orcid":"https://orcid.org/0000-0002-3371-4865"},"institutions":[{"id":"https://openalex.org/I5343935","display_name":"Guilin University of Electronic Technology","ror":"https://ror.org/05arjae42","country_code":"CN","type":"education","lineage":["https://openalex.org/I5343935"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yongzhuang Wei","raw_affiliation_strings":["Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, People\u2019s Republic of China","Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, People\u2019s Republic of China","institution_ids":["https://openalex.org/I5343935"]},{"raw_affiliation_string":"Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, People's Republic of China","institution_ids":["https://openalex.org/I5343935"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5039271966"],"corresponding_institution_ids":["https://openalex.org/I118905719","https://openalex.org/I5343935"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":{"value":2890,"currency":"USD","value_usd":2890},"fwci":0.486,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.69382901,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"92","issue":"9","first_page":"2639","last_page":"2654"},"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/T10951","display_name":"Cryptographic Implementations and Security","score":0.9988999962806702,"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/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/bent-molecular-geometry","display_name":"Bent molecular geometry","score":0.8011575937271118},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.7106285095214844},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.7031245827674866},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4540191888809204},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.41775763034820557},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3793826699256897},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3717277944087982},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.12572187185287476},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.11673781275749207},{"id":"https://openalex.org/keywords/structural-engineering","display_name":"Structural engineering","score":0.07398733496665955},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.06319156289100647}],"concepts":[{"id":"https://openalex.org/C138211643","wikidata":"https://www.wikidata.org/wiki/Q675211","display_name":"Bent molecular geometry","level":2,"score":0.8011575937271118},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.7106285095214844},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.7031245827674866},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4540191888809204},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.41775763034820557},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3793826699256897},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3717277944087982},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.12572187185287476},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.11673781275749207},{"id":"https://openalex.org/C66938386","wikidata":"https://www.wikidata.org/wiki/Q633538","display_name":"Structural engineering","level":1,"score":0.07398733496665955},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.06319156289100647},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10623-024-01407-9","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s10623-024-01407-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-024-01407-9.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10623-024-01407-9","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s10623-024-01407-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-024-01407-9.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2407458217","display_name":null,"funder_award_id":"J1-4084","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2439551750","display_name":"Mathematical modelling and encryption: from theoretical concepts to real-life applications","funder_award_id":"P1-0404","funder_id":"https://openalex.org/F4320322554","funder_display_name":"Javna Agencija za Raziskovalno Dejavnost RS"},{"id":"https://openalex.org/G3967027307","display_name":"Symmetries in graphs via rigid cells","funder_award_id":"J1-2451","funder_id":"https://openalex.org/F4320322554","funder_display_name":"Javna Agencija za Raziskovalno Dejavnost RS"},{"id":"https://openalex.org/G4292803698","display_name":null,"funder_award_id":"2019GXNSFGA245004","funder_id":"https://openalex.org/F4320322768","funder_display_name":"Natural Science Foundation of Guangxi Province"},{"id":"https://openalex.org/G4556023968","display_name":null,"funder_award_id":"2019GXNSFGA245004","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5150125490","display_name":"Designing certain perfect discrete combinatorial objects in spectral domain","funder_award_id":"J1-1694","funder_id":"https://openalex.org/F4320322554","funder_display_name":"Javna Agencija za Raziskovalno Dejavnost RS"},{"id":"https://openalex.org/G5675997893","display_name":"Designing certain discrete mathematical objects in spectral domain","funder_award_id":"N1-0159","funder_id":"https://openalex.org/F4320322554","funder_display_name":"Javna Agencija za Raziskovalno Dejavnost RS"},{"id":"https://openalex.org/G6179361739","display_name":null,"funder_award_id":"62372346","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6940565730","display_name":null,"funder_award_id":"J1-2451","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7563951053","display_name":null,"funder_award_id":"N1-0159","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7793498958","display_name":"Discrete combinatorial objects in the spectral domain - intersection analysis","funder_award_id":"J1-4084","funder_id":"https://openalex.org/F4320322554","funder_display_name":"Javna Agencija za Raziskovalno Dejavnost RS"},{"id":"https://openalex.org/G780546674","display_name":"Bent\u51fd\u6570\u548cplateaued\u51fd\u6570\u7684\u6784\u9020\u548c\u5206\u7c7b\u7814\u7a76","funder_award_id":"61972400","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/F4320322554","display_name":"Javna Agencija za Raziskovalno Dejavnost RS","ror":"https://ror.org/059bp8k51"},{"id":"https://openalex.org/F4320322768","display_name":"Natural Science Foundation of Guangxi Province","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4395008407.pdf"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W637485159","https://openalex.org/W1981357736","https://openalex.org/W1998858124","https://openalex.org/W2032865557","https://openalex.org/W2063388923","https://openalex.org/W2077642171","https://openalex.org/W2090432705","https://openalex.org/W2124818982","https://openalex.org/W2139312425","https://openalex.org/W2286164450","https://openalex.org/W2592639675","https://openalex.org/W2624131751","https://openalex.org/W2710562044","https://openalex.org/W2750816313","https://openalex.org/W2970864062","https://openalex.org/W3025611388","https://openalex.org/W3032045624","https://openalex.org/W3038044508","https://openalex.org/W3103624135","https://openalex.org/W3132449836","https://openalex.org/W3194066911","https://openalex.org/W4205482204","https://openalex.org/W4205675957","https://openalex.org/W4220861603","https://openalex.org/W4221077859","https://openalex.org/W4236140721","https://openalex.org/W4283800974","https://openalex.org/W4312539711","https://openalex.org/W4353091551"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2335716720","https://openalex.org/W2071445534","https://openalex.org/W2329706164","https://openalex.org/W2949764448","https://openalex.org/W4297068499","https://openalex.org/W4255845461","https://openalex.org/W2391219220","https://openalex.org/W4235916564","https://openalex.org/W2361221749"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2,69],"article,":[3],"we":[4],"identify":[5],"certain":[6],"instances":[7,86],"of":[8,87,96,161,177,183],"bent":[9,88,97,359],"functions,":[10,98],"constructed":[11],"using":[12],"the":[13,29,73,93,100,137,145,172,181,184],"so-called":[14],"$$P_\\tau":[15,74],"$$":[16,75],"<mml:math":[17,34,76,103,117,128,201,363],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[18,35,77,104,118,129,202,364],"<mml:msub>":[19,78,130,227,241,259,268,282,304,314,328,346],"<mml:mi>P</mml:mi>":[20,79],"<mml:mi>\u03c4</mml:mi>":[21,80,306],"</mml:msub>":[22,81,133,230,244,262,271,289,307,317,331,349],"</mml:math>":[23,43,82,112,124,134,356],"property,":[24],"that":[25,68,90,166],"are":[26,91],"provably":[27],"outside":[28,92,361],"completed":[30],"Maiorana\u2013McFarland":[31],"(":[32],"$${\\mathcal{M}\\mathcal{M}}^\\#$$":[33,102,362],"<mml:msup>":[36,105,365],"<mml:mrow>":[37,106,119,203,205,215,219,231,239,257,272,280,284,302,318,326,344],"<mml:mi>M</mml:mi>":[38,39,107,108],"</mml:mrow>":[40,109,123,211,223,225,247,265,267,288,292,310,312,334,352,354,355],"<mml:mo>#</mml:mo>":[41,110],"</mml:msup>":[42,111],")":[44],"class.":[45],"This":[46],"also":[47],"partially":[48],"answers":[49],"an":[50],"open":[51],"problem":[52],"in":[53,144],"posed":[54],"by":[55,142],"Kan":[56],"et":[57],"al.":[58],"(IEEE":[59],"Trans":[60],"Inf":[61],"Theory,":[62],"https://doi.org/10.1109/TIT.2022.3140180":[63],",":[64,113,135],"2022).":[65],"We":[66,148],"show":[67],"design":[70],"framework":[71],"(using":[72],"property),":[83],"can":[84],"provide":[85,149],"functions":[89,182],"known":[94],"classes":[95,101],"including":[99],"$${{\\mathcal":[114,126],"{C}}},{{\\mathcal":[115],"{D}}}$$":[116],"<mml:mi>C</mml:mi>":[120],"<mml:mo>,</mml:mo>":[121,208,248,250,293,295,335,337],"<mml:mi>D</mml:mi>":[122,131],"and":[125,360],"{D}}}_0$$":[127],"<mml:mn>0</mml:mn>":[132],"where":[136,157],"latter":[138],"three":[139],"were":[140],"introduced":[141],"Carlet":[143],"early":[146],"nineties.":[147],"two":[150],"generic":[151],"methods":[152,163],"for":[153,180],"identifying":[154],"such":[155],"instances,":[156],"most":[158],"notably":[159],"one":[160],"these":[162],"uses":[164],"permutations":[165],"may":[167],"admit":[168],"linear":[169],"structures.":[170],"For":[171],"first":[173],"time,":[174],"a":[175],"set":[176],"sufficient":[178],"conditions":[179],"form":[185],"$$h(y,z)=Tr(y\\pi":[186],"(z))":[187],"+":[188],"G_1(Tr_1^m(\\alpha":[189],"_1y),\\ldots":[190,198],",Tr_1^m(\\alpha":[191,199],"_ky))G_2(Tr_1^m(\\beta":[192],"_{k+1}z),\\ldots":[193],",Tr_1^m(\\beta":[194],"_{\\tau":[195],"}z))+":[196],"G_3(Tr_1^m(\\alpha":[197],"_ky))$$":[200],"<mml:mi>h</mml:mi>":[204],"<mml:mo>(</mml:mo>":[206,216,220,232,240,258,273,281,303,319,327,345],"<mml:mi>y</mml:mi>":[207,217,245,263,332,350],"<mml:mi>z</mml:mi>":[209,221,290,308],"<mml:mo>)</mml:mo>":[210,222,224,246,264,266,291,309,311,333,351,353],"<mml:mo>=</mml:mo>":[212],"<mml:mi>T</mml:mi>":[213,233,251,274,296,320,338],"<mml:mi>r</mml:mi>":[214,235,253,276,298,322,340],"<mml:mi>\u03c0</mml:mi>":[218],"<mml:mo>+</mml:mo>":[226,286,313],"<mml:mi>G</mml:mi>":[228,269,315],"<mml:mn>1</mml:mn>":[229,236,243,254,277,287,299,323,330,341],"<mml:msubsup>":[234,252,275,297,321,339],"<mml:mi>m</mml:mi>":[237,255,278,300,324,342],"</mml:msubsup>":[238,256,279,301,325,343],"<mml:mi>\u03b1</mml:mi>":[242,260,329,347],"<mml:mo>\u2026</mml:mo>":[249,294,336],"<mml:mi>k</mml:mi>":[261,285,348],"<mml:mn>2</mml:mn>":[270],"<mml:mi>\u03b2</mml:mi>":[283,305],"<mml:mn>3</mml:mn>":[316],"to":[357],"be":[358],"<mm":[366]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}