{"id":"https://openalex.org/W4413827934","doi":"https://doi.org/10.3390/cryptography9030056","title":"Novel Approach to Degree, Balancedness, and Affine Equivalence of Boolean Functions and Construction of a Special Class of Non-Quadratic Balanced Boolean Functions","display_name":"Novel Approach to Degree, Balancedness, and Affine Equivalence of Boolean Functions and Construction of a Special Class of Non-Quadratic Balanced Boolean Functions","publication_year":2025,"publication_date":"2025-08-29","ids":{"openalex":"https://openalex.org/W4413827934","doi":"https://doi.org/10.3390/cryptography9030056"},"language":"en","primary_location":{"id":"doi:10.3390/cryptography9030056","is_oa":true,"landing_page_url":"https://doi.org/10.3390/cryptography9030056","pdf_url":"https://www.mdpi.com/2410-387X/9/3/56/pdf?version=1756458922","source":{"id":"https://openalex.org/S4210223320","display_name":"Cryptography","issn_l":"2410-387X","issn":["2410-387X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cryptography","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2410-387X/9/3/56/pdf?version=1756458922","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100751057","display_name":"Sunil Kumar","orcid":"https://orcid.org/0000-0002-1953-6273"},"institutions":[{"id":"https://openalex.org/I1340206300","display_name":"Defence Research and Development Organisation","ror":"https://ror.org/05k37v296","country_code":"IN","type":"government","lineage":["https://openalex.org/I1340206300","https://openalex.org/I4210150591"]},{"id":"https://openalex.org/I4210088856","display_name":"Laser Science & Technology Centre","ror":"https://ror.org/004tnhr10","country_code":"IN","type":"facility","lineage":["https://openalex.org/I1340206300","https://openalex.org/I4210088856","https://openalex.org/I4210150591"]},{"id":"https://openalex.org/I4210120887","display_name":"Scientific Analysis Group","ror":"https://ror.org/02rpzt041","country_code":"IN","type":"facility","lineage":["https://openalex.org/I1340206300","https://openalex.org/I4210120887","https://openalex.org/I4210150591"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Sunil Kumar","raw_affiliation_strings":["Defence Research and Development Organisation (DRDO), Near Metcalfe House, Delhi 110054, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Defence Research and Development Organisation (DRDO), Near Metcalfe House, Delhi 110054, India","institution_ids":["https://openalex.org/I1340206300","https://openalex.org/I4210088856","https://openalex.org/I4210120887"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5114109225","display_name":"Dharminder Chaudhary","orcid":null},"institutions":[{"id":"https://openalex.org/I81556334","display_name":"Amrita Vishwa Vidyapeetham","ror":"https://ror.org/03am10p12","country_code":"IN","type":"education","lineage":["https://openalex.org/I81556334"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Dharminder Chaudhary","raw_affiliation_strings":["Department of Artificial Intelligence, Amrita School of Artificial Intelligence, Amrita Vishwa Vidyapeetham, Faridabad 121002, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Artificial Intelligence, Amrita School of Artificial Intelligence, Amrita Vishwa Vidyapeetham, Faridabad 121002, India","institution_ids":["https://openalex.org/I81556334"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5112896564","display_name":"S A Lakshmanan","orcid":"https://orcid.org/0000-0002-4705-8088"},"institutions":[{"id":"https://openalex.org/I81556334","display_name":"Amrita Vishwa Vidyapeetham","ror":"https://ror.org/03am10p12","country_code":"IN","type":"education","lineage":["https://openalex.org/I81556334"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"S. A. Lakshmanan","raw_affiliation_strings":["Department of Electronics and Communication Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Chennai 601103, India"],"raw_orcid":"https://orcid.org/0000-0002-4705-8088","affiliations":[{"raw_affiliation_string":"Department of Electronics and Communication Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Chennai 601103, India","institution_ids":["https://openalex.org/I81556334"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047477860","display_name":"Cheng\u2010Chi Lee","orcid":"https://orcid.org/0000-0002-8918-1703"},"institutions":[{"id":"https://openalex.org/I114150738","display_name":"Fu Jen Catholic University","ror":"https://ror.org/04je98850","country_code":"TW","type":"education","lineage":["https://openalex.org/I114150738"]},{"id":"https://openalex.org/I143578492","display_name":"Asia University","ror":"https://ror.org/038a1tp19","country_code":"TW","type":"education","lineage":["https://openalex.org/I143578492"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Cheng-Chi Lee","raw_affiliation_strings":["Department of Computer Science and Information Engineering, Fintech and Blockchain Research Center, Asia University, Taichung City 41354, Taiwan","Department of Library and Information Science, Fu Jen Catholic University, New Taipei City 24205, Taiwan"],"raw_orcid":"https://orcid.org/0000-0002-8918-1703","affiliations":[{"raw_affiliation_string":"Department of Computer Science and Information Engineering, Fintech and Blockchain Research Center, Asia University, Taichung City 41354, Taiwan","institution_ids":["https://openalex.org/I143578492"]},{"raw_affiliation_string":"Department of Library and Information Science, Fu Jen Catholic University, New Taipei City 24205, Taiwan","institution_ids":["https://openalex.org/I114150738"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5047477860","https://openalex.org/A5112896564"],"corresponding_institution_ids":["https://openalex.org/I114150738","https://openalex.org/I143578492","https://openalex.org/I81556334"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":1.2699,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.85358505,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"9","issue":"3","first_page":"56","last_page":"56"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9983999729156494,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9932000041007996,"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/mathematics","display_name":"Mathematics","score":0.6607683897018433},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.6484054327011108},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.6093527674674988},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.5901950597763062},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.567259669303894},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.5306265354156494},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5127216577529907},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4485536515712738},{"id":"https://openalex.org/keywords/equivalence-class","display_name":"Equivalence class (music)","score":0.41753146052360535},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.39277756214141846},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.29279571771621704},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2177950143814087}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6607683897018433},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.6484054327011108},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.6093527674674988},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5901950597763062},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.567259669303894},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.5306265354156494},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5127216577529907},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4485536515712738},{"id":"https://openalex.org/C2777044963","wikidata":"https://www.wikidata.org/wiki/Q5384707","display_name":"Equivalence class (music)","level":2,"score":0.41753146052360535},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.39277756214141846},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29279571771621704},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2177950143814087},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/cryptography9030056","is_oa":true,"landing_page_url":"https://doi.org/10.3390/cryptography9030056","pdf_url":"https://www.mdpi.com/2410-387X/9/3/56/pdf?version=1756458922","source":{"id":"https://openalex.org/S4210223320","display_name":"Cryptography","issn_l":"2410-387X","issn":["2410-387X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cryptography","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:0631cc06381d4e66a7265a321f1a8c5c","is_oa":true,"landing_page_url":"https://doaj.org/article/0631cc06381d4e66a7265a321f1a8c5c","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Cryptography, Vol 9, Iss 3, p 56 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/cryptography9030056","is_oa":true,"landing_page_url":"https://doi.org/10.3390/cryptography9030056","pdf_url":"https://www.mdpi.com/2410-387X/9/3/56/pdf?version=1756458922","source":{"id":"https://openalex.org/S4210223320","display_name":"Cryptography","issn_l":"2410-387X","issn":["2410-387X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cryptography","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4413827934.pdf","grobid_xml":"https://content.openalex.org/works/W4413827934.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W140633939","https://openalex.org/W606334457","https://openalex.org/W1588183317","https://openalex.org/W2004229705","https://openalex.org/W2465207115","https://openalex.org/W2469100438","https://openalex.org/W2981485902","https://openalex.org/W3120067464","https://openalex.org/W3134842914","https://openalex.org/W3173152432","https://openalex.org/W4205482204","https://openalex.org/W4244821898","https://openalex.org/W4281493197","https://openalex.org/W4385251533","https://openalex.org/W4385704355","https://openalex.org/W4390636105","https://openalex.org/W4391756456","https://openalex.org/W4402532449","https://openalex.org/W4410577940","https://openalex.org/W4411224589"],"related_works":["https://openalex.org/W2382232876","https://openalex.org/W2404823920","https://openalex.org/W1571968879","https://openalex.org/W2182642161","https://openalex.org/W3029349442","https://openalex.org/W4255196547","https://openalex.org/W2358659170","https://openalex.org/W146502352","https://openalex.org/W2322917093","https://openalex.org/W2543212557"],"abstract_inverted_index":{"In":[0,51],"several":[1],"stream":[2,31],"cipher":[3],"designs,":[4],"Boolean":[5,90,104,128,160,202],"functions":[6,19,129,161,203],"(BFs)":[7],"play":[8,66],"a":[9,79,89,119,123,136,183],"crucial":[10],"role":[11,69],"as":[12,17],"non-linear":[13],"components,":[14],"either":[15],"serving":[16],"filtering":[18],"or":[20],"being":[21],"used":[22],"within":[23],"the":[24,60,71,85,150,154,176,187,199],"combining":[25],"process.":[26],"The":[27],"overall":[28],"strength":[29],"of":[30,39,63,73,88,125,178,201,208,212],"ciphers":[32],"mainly":[33],"depends":[34],"on":[35],"certain":[36],"cryptographic":[37,75,133],"properties":[38,134],"BFs,":[40,64],"including":[41],"their":[42,114,169],"balancedness,":[43],"non-linearity,":[44],"resistance":[45],"to":[46,59,82,156,205],"correlation,":[47],"and":[48,77,130,139,181],"algebraic":[49,61,86],"degrees.":[50],"this":[52,143,179,191],"paper,":[53],"we":[54,117,148,174],"present":[55,182],"novel":[56,80],"findings":[57],"related":[58],"degrees":[62],"which":[65],"an":[67],"important":[68],"in":[70,135,153,196],"design":[72],"symmetric":[74],"systems,":[76],"propose":[78,118,149],"algorithm":[81,152,180],"directly":[83],"deduce":[84],"degree":[87],"function":[91],"(BF)":[92],"from":[93],"its":[94,194],"truth":[95],"table.":[96],"We":[97],"also":[98],"explore":[99],"new":[100,111,120],"results":[101,112],"concerning":[102],"balanced":[103,159],"functions,":[105],"specifically":[106],"characterizing":[107],"them":[108],"by":[109,190],"establishing":[110],"regarding":[113],"support.":[115],"Additionally,":[116],"approach":[121],"for":[122,198],"subclass":[124],"affine":[126],"equivalent":[127],"discuss":[131,175],"well-known":[132],"very":[137],"simple":[138],"lucid":[140],"manner":[141],"using":[142],"newly":[144],"introduced":[145],"approach.":[146],"Moreover,":[147],"first":[151],"literature":[155],"construct":[157],"non-quadratic":[158],"(NQBBFs)":[162],"that":[163,185],"possess":[164],"no":[165],"linear":[166],"structure":[167],"where":[168],"derivative":[170],"equals":[171],"1.":[172],"Finally,":[173],"complexity":[177],"table":[184],"shows":[186],"time":[188],"taken":[189],"algorithm,":[192],"after":[193],"implementation":[195],"SageMath,":[197],"generation":[200],"corresponding":[204],"different":[206],"values":[207],"n":[209],"(i.e.,":[210],"number":[211],"variables).":[213]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}