{"id":"https://openalex.org/W1999474022","doi":"https://doi.org/10.1137/050640941","title":"Pseudorandom Bits for Constant\u2010Depth Circuits with Few Arbitrary Symmetric Gates","display_name":"Pseudorandom Bits for Constant\u2010Depth Circuits with Few Arbitrary Symmetric Gates","publication_year":2007,"publication_date":"2007-01-01","ids":{"openalex":"https://openalex.org/W1999474022","doi":"https://doi.org/10.1137/050640941","mag":"1999474022"},"language":"en","primary_location":{"id":"doi:10.1137/050640941","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050640941","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5072846764","display_name":"Emanuele Viola","orcid":"https://orcid.org/0000-0001-6091-1824"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Emanuele Viola","raw_affiliation_strings":["Division of Engineering and Applied Sciences Harvard University Cambridge, MA 02138"],"affiliations":[{"raw_affiliation_string":"Division of Engineering and Applied Sciences Harvard University Cambridge, MA 02138","institution_ids":["https://openalex.org/I136199984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5072846764"],"corresponding_institution_ids":["https://openalex.org/I136199984"],"apc_list":null,"apc_paid":null,"fwci":9.732,"has_fulltext":false,"cited_by_count":64,"citation_normalized_percentile":{"value":0.98422574,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"36","issue":"5","first_page":"1387","last_page":"1403"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10237","display_name":"Cryptography and Data Security","score":0.9994000196456909,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9919000267982483,"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/pseudorandom-number-generator","display_name":"Pseudorandom number generator","score":0.834213137626648},{"id":"https://openalex.org/keywords/pseudorandom-generator","display_name":"Pseudorandom generator","score":0.7858560085296631},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.6792346239089966},{"id":"https://openalex.org/keywords/generator","display_name":"Generator (circuit theory)","score":0.6739060878753662},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.6152346134185791},{"id":"https://openalex.org/keywords/pseudorandomness","display_name":"Pseudorandomness","score":0.5788336992263794},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5763880610466003},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.5492950081825256},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5376039743423462},{"id":"https://openalex.org/keywords/circuit-complexity","display_name":"Circuit complexity","score":0.5185642838478088},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5097419619560242},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5079465508460999},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.48616573214530945},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.455483615398407},{"id":"https://openalex.org/keywords/parity","display_name":"Parity (physics)","score":0.43589848279953003},{"id":"https://openalex.org/keywords/random-number-generation","display_name":"Random number generation","score":0.4193544089794159},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.241943359375},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.22516363859176636},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.19021111726760864},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17024824023246765},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11265507340431213}],"concepts":[{"id":"https://openalex.org/C140642157","wikidata":"https://www.wikidata.org/wiki/Q1623338","display_name":"Pseudorandom number generator","level":2,"score":0.834213137626648},{"id":"https://openalex.org/C92913381","wikidata":"https://www.wikidata.org/wiki/Q7255474","display_name":"Pseudorandom generator","level":3,"score":0.7858560085296631},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.6792346239089966},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.6739060878753662},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.6152346134185791},{"id":"https://openalex.org/C19086982","wikidata":"https://www.wikidata.org/wiki/Q2115856","display_name":"Pseudorandomness","level":3,"score":0.5788336992263794},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5763880610466003},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.5492950081825256},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5376039743423462},{"id":"https://openalex.org/C90702460","wikidata":"https://www.wikidata.org/wiki/Q1055112","display_name":"Circuit complexity","level":3,"score":0.5185642838478088},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5097419619560242},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5079465508460999},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.48616573214530945},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.455483615398407},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.43589848279953003},{"id":"https://openalex.org/C201866948","wikidata":"https://www.wikidata.org/wiki/Q228206","display_name":"Random number generation","level":2,"score":0.4193544089794159},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.241943359375},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.22516363859176636},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.19021111726760864},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17024824023246765},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11265507340431213},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/050640941","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050640941","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G286214036","display_name":"CAREER: A Unified Theory of Pseudorandomness","funder_award_id":"0133096","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7425413337","display_name":null,"funder_award_id":"-0001","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1901622511","https://openalex.org/W1998918799","https://openalex.org/W2023915883","https://openalex.org/W2027328186","https://openalex.org/W2041210515","https://openalex.org/W2042194938","https://openalex.org/W2058859323","https://openalex.org/W2060270693","https://openalex.org/W2075848688","https://openalex.org/W2078387207","https://openalex.org/W2080210666","https://openalex.org/W2107998050","https://openalex.org/W2135453633","https://openalex.org/W2175992284","https://openalex.org/W2912080987","https://openalex.org/W2912133662","https://openalex.org/W2998808137","https://openalex.org/W3021570316","https://openalex.org/W4212956348"],"related_works":["https://openalex.org/W4241950541","https://openalex.org/W2077146536","https://openalex.org/W44435986","https://openalex.org/W2161451071","https://openalex.org/W2062045810","https://openalex.org/W2988803943","https://openalex.org/W1503711153","https://openalex.org/W3111658143","https://openalex.org/W2025294130","https://openalex.org/W2978186572"],"abstract_inverted_index":{"We":[0],"exhibit":[1],"an":[2,150],"explicitly":[3],"computable":[4,113],"pseudorandom":[5,185],"generator":[6,38,79,89,95,145,186],"stretching":[7],"l":[8],"bits":[9,15],"into":[10,182],"$m(l)":[11],"=":[12,170],"l^{\\Omega(\\log":[13],"l)}$":[14],"that":[16,58,110,158],"look":[17],"random":[18],"to":[19,131,139],"constant\u2010depth":[20,91,118,165],"circuits":[21,66,86,92,119,166],"of":[22,45,52,67,85,167,198,202,208,226,234,255],"size":[23,168],"$m(l)$":[24],"with":[25,70,120,173,263],"$\\log":[26,174],"m(l)$":[27],"arbitrary":[28,72,123,176],"symmetric":[29,73,124,177],"gates":[30,125],"(e.g.,":[31],"PARITY,":[32],"MAJORITY).":[33],"This":[34,129],"improves":[35],"on":[36,50,162,231],"a":[37,81,97,141,206,264],"by":[39,114,148,210,270],"Luby,":[40],"Velickovic,":[41],"and":[42,179,212,221,223,248,273],"Wigderson":[43,213],"[Proceedings":[44,225],"the":[46,60,76,133,183,199,227],"Second":[47],"Israel":[48],"Symposium":[49,230],"Theory":[51],"Computing":[53],"Systems,":[54],"1993,":[55],"pp.":[56,104,194,219,246,281],"18\u201324]":[57],"achieves":[59],"same":[61],"stretch":[62],"but":[63],"fools":[64,80],"only":[65],"depth":[68],"2":[69],"one":[71],"gate":[74],"at":[75],"top.":[77],"Our":[78,144],"strictly":[82],"richer":[83],"class":[84,137],"than":[87],"Nisan\u2019s":[88,94],"for":[90,164],"(but":[93],"has":[96],"much":[98],"bigger":[99],"stretch)":[100],"[Combinatorica,":[101],"11":[102],"(1991),":[103],"63\u201370].":[105],"In":[106],"particular,":[107],"we":[108],"conclude":[109],"every":[111],"function":[112,152,204],"uniform":[115],"$\\poly(n)$\u2010size":[116],"probabilistic":[117,135],"$O(\\log":[121],"n)$":[122],"is":[126,146,159,205],"in":[127,239],"$\\mathit{TIME}(2^{n^{o(1)}})$.":[128],"seems":[130],"be":[132],"richest":[134],"circuit":[136],"known":[138],"admit":[140],"subexponential":[142],"derandomization.":[143],"obtained":[147],"constructing":[149],"explicit":[151],"$f":[153],":":[154],"\\zo^n":[155],"\\to":[156],"\\zo$":[157],"very":[160],"hard":[161],"average":[163],"$s(n)":[169],"n^{\\Omega(\\log":[171],"n)}$":[172],"s(n)$":[175],"gates,":[178],"plugging":[180],"it":[181],"Nisan\u2013Wigderson":[184],"construction":[187],"[J.":[188,275],"Comput.":[189,240,276],"System":[190,277],"Sci.,":[191,278],"49":[192],"(1994),":[193],"149\u2013167].":[195],"The":[196],"proof":[197],"average\u2010case":[200],"hardness":[201],"this":[203],"modification":[207],"arguments":[209],"Razborov":[211],"[Inform.":[214],"Process.":[215],"Lett.,":[216],"45":[217,279],"(1993),":[218],"303\u2013307]":[220],"Hansen":[222],"Miltersen":[224],"29th":[228],"International":[229],"Mathematical":[232],"Foundations":[233],"Computer":[235],"Science,":[236],"Lecture":[237],"Notes":[238],"Sci.":[241],"3153,":[242],"Springer\u2010Verlag,":[243],"Berlin,":[244],"2004,":[245],"334\u2013345]":[247],"combines":[249],"H\u00e5stad\u2019s":[250],"switching":[251],"lemma":[252],"[Computational":[253],"Limitations":[254],"Small\u2010Depth":[256],"Circuits,":[257],"MIT":[258],"Press,":[259],"Cambridge,":[260],"MA,":[261],"1987]":[262],"multiparty":[265],"communication":[266],"complexity":[267],"lower":[268],"bound":[269],"Babai,":[271],"Nisan,":[272],"Szegedy":[274],"(1992),":[280],"204\u2013232].":[282]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}