{"id":"https://openalex.org/W2794416297","doi":"https://doi.org/10.3934/amc.2018007","title":"Generalized nonlinearity of <inline-formula><tex-math id=\"M1\">$ S$</tex-math></inline-formula>-boxes","display_name":"Generalized nonlinearity of <inline-formula><tex-math id=\"M1\">$ S$</tex-math></inline-formula>-boxes","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2794416297","doi":"https://doi.org/10.3934/amc.2018007","mag":"2794416297"},"language":"en","primary_location":{"id":"doi:10.3934/amc.2018007","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2018007","pdf_url":"https://www.aimsciences.org/article/exportPdf?id=35ad105c-021a-4535-818a-cf5ac89f9135","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.aimsciences.org/article/exportPdf?id=35ad105c-021a-4535-818a-cf5ac89f9135","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013399016","display_name":"Sugata Gangopadhyay","orcid":"https://orcid.org/0000-0002-7329-7349"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Sugata Gangopadhyay","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026354959","display_name":"Goutam Paul","orcid":"https://orcid.org/0000-0002-5160-6032"},"institutions":[{"id":"https://openalex.org/I6498739","display_name":"Indian Statistical Institute","ror":"https://ror.org/00q2w1j53","country_code":"IN","type":"education","lineage":["https://openalex.org/I6498739"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Goutam Paul","raw_affiliation_strings":["Cryptology and Security Research Unit, R. C. Bose Centre for Cryptology and Security, Indian Statistical Institute, Kolkata 700108, India"],"affiliations":[{"raw_affiliation_string":"Cryptology and Security Research Unit, R. C. Bose Centre for Cryptology and Security, Indian Statistical Institute, Kolkata 700108, India","institution_ids":["https://openalex.org/I6498739"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102867969","display_name":"Nishant Sinha","orcid":"https://orcid.org/0000-0002-5393-9965"},"institutions":[{"id":"https://openalex.org/I154851008","display_name":"Indian Institute of Technology Roorkee","ror":"https://ror.org/00582g326","country_code":"IN","type":"education","lineage":["https://openalex.org/I154851008"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Nishant Sinha","raw_affiliation_strings":["Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India","institution_ids":["https://openalex.org/I154851008"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021290362","display_name":"Pantelimon St\u0103nic\u0103","orcid":"https://orcid.org/0000-0002-8622-7120"},"institutions":[{"id":"https://openalex.org/I35364215","display_name":"Naval Postgraduate School","ror":"https://ror.org/033yfkj90","country_code":"US","type":"education","lineage":["https://openalex.org/I1330347796","https://openalex.org/I3130687028","https://openalex.org/I35364215"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Pantelimon St\u01cenic\u01ce","raw_affiliation_strings":["Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216, USA"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216, USA","institution_ids":["https://openalex.org/I35364215"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5013399016"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5077,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.72784055,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"12","issue":"1","first_page":"115","last_page":"122"},"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/T10951","display_name":"Cryptographic Implementations and Security","score":0.9997000098228455,"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":0.8641672134399414},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.6503661870956421},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.6429250240325928},{"id":"https://openalex.org/keywords/hadamard-transform","display_name":"Hadamard transform","score":0.6152429580688477},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5495133399963379},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.5242079496383667},{"id":"https://openalex.org/keywords/cryptanalysis","display_name":"Cryptanalysis","score":0.4674108326435089},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41953182220458984},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.34133780002593994},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32261455059051514},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.318159818649292},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3057171702384949},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.29511743783950806},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13094913959503174}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8641672134399414},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6503661870956421},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.6429250240325928},{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.6152429580688477},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5495133399963379},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.5242079496383667},{"id":"https://openalex.org/C181149355","wikidata":"https://www.wikidata.org/wiki/Q897511","display_name":"Cryptanalysis","level":3,"score":0.4674108326435089},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41953182220458984},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.34133780002593994},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32261455059051514},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.318159818649292},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3057171702384949},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.29511743783950806},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13094913959503174},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3934/amc.2018007","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2018007","pdf_url":"https://www.aimsciences.org/article/exportPdf?id=35ad105c-021a-4535-818a-cf5ac89f9135","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3934/amc.2018007","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2018007","pdf_url":"https://www.aimsciences.org/article/exportPdf?id=35ad105c-021a-4535-818a-cf5ac89f9135","source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2794416297.pdf","grobid_xml":"https://content.openalex.org/works/W2794416297.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1498876551","https://openalex.org/W1529556207","https://openalex.org/W1891181203","https://openalex.org/W1964723977","https://openalex.org/W2269930284","https://openalex.org/W3143219376","https://openalex.org/W3204162628","https://openalex.org/W4244821898"],"related_works":["https://openalex.org/W1982241170","https://openalex.org/W2116282083","https://openalex.org/W2386549147","https://openalex.org/W1974664230","https://openalex.org/W4390188610","https://openalex.org/W3089223619","https://openalex.org/W4288849478","https://openalex.org/W2940752727","https://openalex.org/W1561797008","https://openalex.org/W2516549250"],"abstract_inverted_index":{"While":[0],"analyzing":[1],"$":[2,34,42,62,86],"S$-boxes,":[3],"or":[4],"vectorial":[5],"Boolean":[6],"functions,":[7],"it":[8,25],"is":[9,26,44],"of":[10,40,57,60,74,80,84],"interest":[11],"to":[12,32,47],"approximate":[13],"its":[14],"component":[15],"functions":[16],"by":[17,64],"affine":[18],"functions.":[19],"In":[20,50],"the":[21,55,81],"usual":[22],"attack":[23],"models,":[24],"assumed":[27],"that":[28],"all":[29],"input":[30],"vectors":[31],"an":[33,41,61,85],"S$-box":[35,43,63],"are":[36],"equiprobable.":[37],"The":[38],"nonlinearity":[39,75],"defined,":[45],"subject":[46],"this":[48,51],"assumption.":[49],"paper,":[52],"we":[53],"explore":[54],"possibility":[56],"linear":[58],"cryptanalysis":[59],"introducing":[65],"biased":[66],"inputs":[67],"and":[68],"thus":[69],"propose":[70],"a":[71,78],"generalized":[72],"notion":[73],"along":[76],"with":[77],"generalization":[79],"Walsh-Hadamard":[82],"spectrum":[83],"S$-box.":[87]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-03-25T13:04:00.132906","created_date":"2025-10-10T00:00:00"}