{"id":"https://openalex.org/W7214700164","doi":"https://doi.org/10.48550/arxiv.2609.35689","title":"Linearity bounds for APN functions","display_name":"Linearity bounds for APN functions","publication_year":2026,"publication_date":"2026-09-28","ids":{"openalex":"https://openalex.org/W7214700164","doi":"https://doi.org/10.48550/arxiv.2609.35689"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.35689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35689","pdf_url":"https://arxiv.org/pdf/2609.35689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.35689","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037504840","display_name":"Christof Beierle","orcid":"https://orcid.org/0000-0002-5558-0722","observed_orcids":["https://orcid.org/0000-0002-5558-0722"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Beierle, Christof","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5103192760","display_name":"Lukas K\u00f6lsch","orcid":"https://orcid.org/0000-0002-2966-0710","observed_orcids":["https://orcid.org/0000-0002-2966-0710"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"K\u00f6lsch, Lukas","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9945999979972839,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9945999979972839,"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"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.0005000000237487257,"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"}},{"id":"https://openalex.org/T10951","display_name":"Cryptographic Implementations and Security","score":0.0003000000142492354,"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/almost-perfect-nonlinear-functions","display_name":"almost perfect nonlinear functions","score":0.9539999961853027},{"id":"https://openalex.org/keywords/linearity","display_name":"linearity","score":0.4869999885559082},{"id":"https://openalex.org/keywords/walsh-coefficients","display_name":"Walsh coefficients","score":0.9589999914169312},{"id":"https://openalex.org/keywords/boolean-functions","display_name":"Boolean functions","score":0.4880000054836273},{"id":"https://openalex.org/keywords/finite-fields","display_name":"finite fields","score":0.5099999904632568},{"id":"https://openalex.org/keywords/upper-bounds","display_name":"upper bounds","score":0.5249999761581421},{"id":"https://openalex.org/keywords/nonlinearity","display_name":"nonlinearity","score":0.46299999952316284},{"id":"https://openalex.org/keywords/even-and-odd-dimensions","display_name":"even and odd dimensions","score":0.6100000143051147}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.853600025177002},{"id":"https://openalex.org/C77170095","wikidata":"https://www.wikidata.org/wiki/Q1753188","display_name":"Linearity","level":2,"score":0.7842000126838684},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.7251999974250793},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.588100016117096},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.45399999618530273},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42100000381469727},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.3898000121116638},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35199999809265137},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.32499998807907104},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.2946000099182129},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.25519999861717224}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.35689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35689","pdf_url":"https://arxiv.org/pdf/2609.35689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.35689","doi":"https://doi.org/10.48550/arxiv.2609.35689","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.35689","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.35689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35689","pdf_url":"https://arxiv.org/pdf/2609.35689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214700164.pdf","grobid_xml":"https://content.openalex.org/works/W7214700164.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W1490691825","https://openalex.org/W1498892963","https://openalex.org/W1537327665","https://openalex.org/W1575149347","https://openalex.org/W1587687098","https://openalex.org/W1659081910","https://openalex.org/W1935580934","https://openalex.org/W2002865028","https://openalex.org/W2010961669","https://openalex.org/W2016744051","https://openalex.org/W2020301329","https://openalex.org/W2024729203","https://openalex.org/W2090432705","https://openalex.org/W2149881492","https://openalex.org/W2173356534","https://openalex.org/W2759739548","https://openalex.org/W2761574822","https://openalex.org/W2795322485","https://openalex.org/W2912209775","https://openalex.org/W2984903117","https://openalex.org/W3014299119","https://openalex.org/W3086858932","https://openalex.org/W3093954335","https://openalex.org/W3114045166","https://openalex.org/W4387184589","https://openalex.org/W4403820664","https://openalex.org/W4415257479","https://openalex.org/W4415309877","https://openalex.org/W4416442132","https://openalex.org/W7124766530","https://openalex.org/W7204260943","https://openalex.org/W7214168333"],"related_works":["https://openalex.org/W4409798403","https://openalex.org/W7062897178","https://openalex.org/W4405723619","https://openalex.org/W7101110048","https://openalex.org/W2016744051","https://openalex.org/W2035311227","https://openalex.org/W3082441468","https://openalex.org/W2087896741","https://openalex.org/W1485694296","https://openalex.org/W2051373917"],"abstract_inverted_index":{"For":[0],"$n\\ge5$,":[1],"let":[2],"$F\\colon\\mathbb{F}_2^n\\to":[3],"\\mathbb{F}_2^n$":[4],"be":[5],"almost":[6],"perfect":[7],"nonlinear":[8],"and":[9,59,96,105],"write":[10],"$N=2^n$.":[11],"It":[12,67],"is":[13,31,37,40,68,88,124],"proven":[14,42,70],"that":[15,43,136],"the":[16,22,50,76,112,117,147],"linearity":[17,113],"$\\mathcal{L}(F)$":[18],"of":[19,27,54,116,120],"$F$,":[20],"i.e.,":[21],"largest":[23,78,98],"absolute":[24,79],"Walsh":[25,80],"coefficient":[26,81],"a":[28,109],"nonzero":[29,83,122],"component,":[30],"at":[32,89,101],"most":[33,90,102],"$N-6$.":[34],"If":[35],"$n$":[36],"even,":[38],"it":[39],"further":[41,69],"$\\mathcal{L}(F)\\le":[44,151,157],"N-\\sqrt":[45],"N/4-2$.":[46],"These":[47],"bounds":[48],"improve":[49],"general":[51],"upper":[52],"bound":[53,110],"$N-4$":[55],"in":[56,61,114],"odd":[57,127],"dimension":[58],"$N-6$":[60],"even":[62,141],"dimensions":[63],"$n":[64],"\\geq":[65],"10$.":[66],"that,":[71],"for":[72,129,143],"each":[73],"fixed":[74],"$k$,":[75],"$k$-th":[77],"among":[82],"components,":[84],"counted":[85],"with":[86],"multiplicity,":[87],"$(1+O_k(2^{-n}))N/\\sqrt{k}$":[91],"as":[92],"$n\\to\\infty$.":[93],"The":[94],"second":[95],"fourth":[97],"coefficients":[99],"are":[100],"$2\\lfloor":[103],"N/3\\rfloor$":[104],"$N/2$,":[106],"respectively.":[107],"Finally,":[108],"on":[111],"terms":[115],"number":[118],"$q$":[119],"nonplateaued":[121],"components":[123],"derived.":[125],"In":[126,140,153],"dimension,":[128,142],"$q>0$,":[130],"we":[131],"have":[132],"$\\mathcal{L}(F)^2\\le":[133],"N(1+\\sqrt{q(N-1)})$,":[134],"so":[135],"$\\mathcal{L}(F)/N\\to1$":[137],"implies":[138,150,156],"$q/N\\to1$.":[139],"every":[144],"$1/2\\le":[145],"C<1$,":[146],"condition":[148],"$q\\le(4C-C^2-1)N/4+1$":[149],"CN$.":[152],"particular,":[154],"$q\\le3N/16+2$":[155],"N/2$.":[158]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-30T00:00:00"}
