{"id":"https://openalex.org/W3214473018","doi":"https://doi.org/10.1109/tit.2021.3128618","title":"Structural and Statistical Analysis of Multidimensional Linear Approximations of Random Functions and Permutations","display_name":"Structural and Statistical Analysis of Multidimensional Linear Approximations of Random Functions and Permutations","publication_year":2021,"publication_date":"2021-11-16","ids":{"openalex":"https://openalex.org/W3214473018","doi":"https://doi.org/10.1109/tit.2021.3128618","mag":"3214473018"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2021.3128618","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2021.3128618","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1109/tit.2021.3128618","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069940192","display_name":"Tomer Ashur","orcid":"https://orcid.org/0000-0001-6091-4857"},"institutions":[{"id":"https://openalex.org/I196972281","display_name":"Imec the Netherlands","ror":"https://ror.org/01ezq2j76","country_code":"NL","type":"facility","lineage":["https://openalex.org/I196972281"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Tomer Ashur","raw_affiliation_strings":["Imec-COSIC, KU Leuven, Leuven 3000, Belgium and TU Eindhoven, Eindhoven 5612, The Netherlands"],"raw_orcid":"https://orcid.org/0000-0001-6091-4857","affiliations":[{"raw_affiliation_string":"Imec-COSIC, KU Leuven, Leuven 3000, Belgium and TU Eindhoven, Eindhoven 5612, The Netherlands","institution_ids":["https://openalex.org/I196972281"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101506842","display_name":"Mohsin Khan","orcid":"https://orcid.org/0000-0002-8064-3166"},"institutions":[{"id":"https://openalex.org/I1306339040","display_name":"Ericsson (Sweden)","ror":"https://ror.org/05a7rhx54","country_code":"SE","type":"company","lineage":["https://openalex.org/I1306339040"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"Mohsin Khan","raw_affiliation_strings":["Ericsson Research, Ericsson AB, 16480 Stockholm, Sweden"],"raw_orcid":"https://orcid.org/0000-0002-8064-3166","affiliations":[{"raw_affiliation_string":"Ericsson Research, Ericsson AB, 16480 Stockholm, Sweden","institution_ids":["https://openalex.org/I1306339040"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5061268898","display_name":"Kaisa Nyberg","orcid":"https://orcid.org/0000-0003-2885-6364"},"institutions":[{"id":"https://openalex.org/I1306339040","display_name":"Ericsson (Sweden)","ror":"https://ror.org/05a7rhx54","country_code":"SE","type":"company","lineage":["https://openalex.org/I1306339040"]},{"id":"https://openalex.org/I133731052","display_name":"University of Helsinki","ror":"https://ror.org/040af2s02","country_code":"FI","type":"education","lineage":["https://openalex.org/I133731052"]},{"id":"https://openalex.org/I4210114974","display_name":"IMEC","ror":"https://ror.org/02kcbn207","country_code":"BE","type":"nonprofit","lineage":["https://openalex.org/I4210114974"]},{"id":"https://openalex.org/I83019370","display_name":"Eindhoven University of Technology","ror":"https://ror.org/02c2kyt77","country_code":"NL","type":"education","lineage":["https://openalex.org/I83019370"]},{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]},{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]}],"countries":["BE","FI","NL","SE"],"is_corresponding":false,"raw_author_name":"Kaisa Nyberg","raw_affiliation_strings":["Department of Computer Science, Aalto University, 00076 Aalto, Finland. (e-mail: kaisa.nyberg@aalto.fi)","imec-COSIC, KU Leuven, 3000 Leuven, Belgium,","Department of Computer Science, Univer-sity of Helsinki, 00560 Helsinki, Finland. He","the Ericsson Research, Ericsson AB, 16480 Stockholm, Sweden","Department of Mathematics and Computer Sci-ence, TU Eindhoven, 5612 AZ Eindhoven, The Netherlands","Department of Computer Science, Aalto University, 00076 Espoo, Finland. She resides in 00100 Helsinki, Finland"],"raw_orcid":"https://orcid.org/0000-0003-2885-6364","affiliations":[{"raw_affiliation_string":"Department of Computer Science, Aalto University, 00076 Aalto, Finland. (e-mail: kaisa.nyberg@aalto.fi)","institution_ids":["https://openalex.org/I9927081"]},{"raw_affiliation_string":"imec-COSIC, KU Leuven, 3000 Leuven, Belgium,","institution_ids":["https://openalex.org/I4210114974","https://openalex.org/I99464096"]},{"raw_affiliation_string":"Department of Computer Science, Univer-sity of Helsinki, 00560 Helsinki, Finland. He","institution_ids":["https://openalex.org/I133731052"]},{"raw_affiliation_string":"the Ericsson Research, Ericsson AB, 16480 Stockholm, Sweden","institution_ids":["https://openalex.org/I1306339040"]},{"raw_affiliation_string":"Department of Mathematics and Computer Sci-ence, TU Eindhoven, 5612 AZ Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]},{"raw_affiliation_string":"Department of Computer Science, Aalto University, 00076 Espoo, Finland. She resides in 00100 Helsinki, Finland","institution_ids":["https://openalex.org/I9927081"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5069940192"],"corresponding_institution_ids":["https://openalex.org/I196972281"],"apc_list":null,"apc_paid":null,"fwci":0.5597,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.74268294,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"68","issue":"2","first_page":"1296","last_page":"1315"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10951","display_name":"Cryptographic Implementations and Security","score":0.9998000264167786,"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/T10951","display_name":"Cryptographic Implementations and Security","score":0.9998000264167786,"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":0.9995999932289124,"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"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9977999925613403,"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":0.5807194113731384},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.5717203617095947},{"id":"https://openalex.org/keywords/independence","display_name":"Independence (probability theory)","score":0.5461746454238892},{"id":"https://openalex.org/keywords/linear-approximation","display_name":"Linear approximation","score":0.45278307795524597},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.44230854511260986},{"id":"https://openalex.org/keywords/linear-cryptanalysis","display_name":"Linear cryptanalysis","score":0.439549058675766},{"id":"https://openalex.org/keywords/cryptanalysis","display_name":"Cryptanalysis","score":0.31424471735954285},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3017326593399048},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.2921028435230255},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.13886898756027222},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1318073868751526}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5807194113731384},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.5717203617095947},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.5461746454238892},{"id":"https://openalex.org/C160824197","wikidata":"https://www.wikidata.org/wiki/Q2071054","display_name":"Linear approximation","level":3,"score":0.45278307795524597},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.44230854511260986},{"id":"https://openalex.org/C82424418","wikidata":"https://www.wikidata.org/wiki/Q1826463","display_name":"Linear cryptanalysis","level":4,"score":0.439549058675766},{"id":"https://openalex.org/C181149355","wikidata":"https://www.wikidata.org/wiki/Q897511","display_name":"Cryptanalysis","level":3,"score":0.31424471735954285},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3017326593399048},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.2921028435230255},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.13886898756027222},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1318073868751526},{"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/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","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":6,"locations":[{"id":"doi:10.1109/tit.2021.3128618","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2021.3128618","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:pure.tue.nl:openaire/32c0d9c0-9910-4fba-9c97-5c1897c8b6e5","is_oa":true,"landing_page_url":"https://research.tue.nl/en/publications/32c0d9c0-9910-4fba-9c97-5c1897c8b6e5","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","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","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Ashur, T, Khan, M & Nyberg, K 2022, 'Structural and Statistical Analysis of Multidimensional Linear Approximations of Random Functions and Permutations', IEEE Transactions on Information Theory, vol. 68, no. 2, pp. 1296-1315. https://doi.org/10.1109/TIT.2021.3128618","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:lirias2repo.kuleuven.be:20.500.12942/691367","is_oa":true,"landing_page_url":"https://lirias.kuleuven.be/handle/20.500.12942/691367","pdf_url":"https://lirias.kuleuven.be/retrieve/036865ff-6d72-4702-9da6-1887ae7a7208","source":{"id":"https://openalex.org/S7407055369","display_name":"Lirias","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","license_id":"https://openalex.org/licenses/cc-by","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"Ieee Transactions On Information Theory, vol. 68 (2), (1296-1315)","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:aaltodoc.aalto.fi:123456789/113031","is_oa":true,"landing_page_url":"https://research.aalto.fi/en/publications/73a35631-2909-43cf-9759-7cd3798605d7","pdf_url":"https://research.aalto.fi/files/79268463/Structural_and_Statistical_Analysis_of_Multidimensional_Linear_Approximations_of_Random_Functions_and_Permutations.pdf","source":{"id":"https://openalex.org/S4306401663","display_name":"Aaltodoc (Aalto University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I9927081","host_organization_name":"Aalto University","host_organization_lineage":["https://openalex.org/I9927081"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},{"id":"pmh:oai:helda.helsinki.fi:10138/341560","is_oa":true,"landing_page_url":"http://hdl.handle.net/10138/341560","pdf_url":null,"source":{"id":"https://openalex.org/S4210213322","display_name":"Ty\u00f6v\u00e4entutkimus Vuosikirja","issn_l":"0784-1272","issn":["0784-1272","1459-7780"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:tue:oai:pure.tue.nl:publications/32c0d9c0-9910-4fba-9c97-5c1897c8b6e5","is_oa":true,"landing_page_url":"https://research.tue.nl/nl/publications/32c0d9c0-9910-4fba-9c97-5c1897c8b6e5","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Transactions on Information Theory, 68(2), 1296 - 1315. Institute of Electrical and Electronics Engineers","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1109/tit.2021.3128618","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2021.3128618","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8445168529","display_name":null,"funder_award_id":"12ZH420N","funder_id":"https://openalex.org/F4320321730","funder_display_name":"Fonds Wetenschappelijk Onderzoek"}],"funders":[{"id":"https://openalex.org/F4320321730","display_name":"Fonds Wetenschappelijk Onderzoek","ror":"https://ror.org/03qtxy027"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W32121401","https://openalex.org/W57809933","https://openalex.org/W1519447501","https://openalex.org/W1537206908","https://openalex.org/W1553413085","https://openalex.org/W1892798954","https://openalex.org/W1964723977","https://openalex.org/W2045479946","https://openalex.org/W2067771147","https://openalex.org/W2083383916","https://openalex.org/W2087928947","https://openalex.org/W2133371372","https://openalex.org/W2150208140","https://openalex.org/W2167266508","https://openalex.org/W2185027341","https://openalex.org/W2397410996","https://openalex.org/W2404788957","https://openalex.org/W2515193397","https://openalex.org/W2516291978","https://openalex.org/W2888123155","https://openalex.org/W2977797264","https://openalex.org/W3005773786","https://openalex.org/W3028734200","https://openalex.org/W3029690034","https://openalex.org/W4250476278","https://openalex.org/W4253548338","https://openalex.org/W6713416448","https://openalex.org/W6768936726","https://openalex.org/W6777776007","https://openalex.org/W6778306023"],"related_works":["https://openalex.org/W1907974733","https://openalex.org/W2802560047","https://openalex.org/W2093538406","https://openalex.org/W2026284256","https://openalex.org/W2397489728","https://openalex.org/W2102023965","https://openalex.org/W1849484603","https://openalex.org/W1996683293","https://openalex.org/W1536590213","https://openalex.org/W2373276407"],"abstract_inverted_index":{"The":[0,217],"goal":[1],"of":[2,10,23,41,45,89,105,116,133,153,155,189,213],"this":[3,50,144,183,222],"paper":[4,145,223],"is":[5,17],"to":[6,97,125,167,229,250],"investigate":[7],"linear":[8,66,91,106,163,191,215],"approximations":[9,67,120,149,171],"random":[11],"functions":[12],"and":[13,58,81,135,200,209,227,244,248],"permutations.":[14],"Our":[15],"motivation":[16],"twofold.":[18],"First,":[19],"before":[20],"the":[21,34,42,46,78,90,109,114,126,140,147,151,156,161,174,207,210],"distinguishability":[22],"a":[24,131,178],"practical":[25,254],"cipher":[26],"from":[27],"an":[28,38],"ideal":[29,47],"one":[30],"can":[31],"be":[32],"analysed,":[33],"cryptanalyst":[35],"must":[36],"have":[37,74],"accurate":[39,226],"understanding":[40],"statistical":[43,103,180,203,237],"behaviour":[44,80],"cipher.":[48],"Secondly,":[49],"issue":[51],"has":[52,113,197],"been":[53,75],"neglected":[54],"both":[55,206],"in":[56,59,143,221],"old":[57],"more":[60],"recent":[61],"studies,":[62],"particularly":[63],"when":[64],"multiple":[65],"are":[68,224,232],"being":[69],"used":[70],"simultaneously.":[71],"Traditional":[72],"models":[73,204,219],"based":[76],"on":[77],"average":[79],"simplified":[82],"using":[83,253],"other":[84,110],"assumptions":[85,101],"such":[86,239],"as":[87,240],"independence":[88,104],"approximations.":[92,107,216],"Multidimensional":[93],"cryptanalysis":[94,164],"was":[95,165],"introduced":[96],"avoid":[98],"making":[99],"artificial":[100],"about":[102],"On":[108],"hand,":[111],"it":[112],"drawback":[115],"including":[117],"many":[118],"trivial":[119,148,170],"that":[121,146],"do":[122],"not":[123],"contribute":[124],"attack":[127],"but":[128],"just":[129],"cause":[130],"waste":[132],"time":[134,142],"memory.":[136],"We":[137],"show":[138],"for":[139,205],"first":[141],"reduce":[150],"degree":[152],"freedom":[154],"related":[157],"\u03c72":[158,242],"distribution.":[159],"Previously,":[160],"affine":[162,208],"proposed":[166],"allow":[168],"removing":[169],"and,":[172],"at":[173],"same":[175],"time,":[176],"admitting":[177],"solid":[179],"model.":[181],"In":[182],"paper,":[184],"we":[185],"identify":[186],"another":[187],"type":[188,212],"multidimensional":[190,214],"approximation,":[192,195],"called":[193],"Davies-Meyer":[194,211],"which":[196],"similar":[198],"advantages,":[199],"present":[201],"full":[202],"new":[218],"given":[220],"realistic,":[225],"easy":[228],"use.":[230],"They":[231],"backed":[233],"up":[234],"by":[235],"standard":[236],"tools":[238],"Pearson\u2019s":[241],"test":[243],"finite":[245],"population":[246],"correction":[247],"demonstrated":[249],"work":[251],"accurately":[252],"examples.":[255]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-01T08:36:08.643496","created_date":"2025-10-10T00:00:00"}