{"id":"https://openalex.org/W3086270023","doi":"https://doi.org/10.4086/toc.2020.v016a002","title":"","display_name":"","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3086270023","doi":"https://doi.org/10.4086/toc.2020.v016a002","mag":"3086270023"},"language":"en","primary_location":{"id":"doi:10.4086/toc.2020.v016a002","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2020.v016a002","pdf_url":"http://www.theoryofcomputing.org/articles/v016a002/v016a002.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"http://www.theoryofcomputing.org/articles/v016a002/v016a002.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091529166","display_name":"Andrej Bogdanov","orcid":"https://orcid.org/0000-0002-0338-6151"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Andrej Bogdanov","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103160306","display_name":"Siyao Guo","orcid":"https://orcid.org/0000-0002-8402-1866"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Siyao Guo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5032483797","display_name":"Ilan Komargodski","orcid":"https://orcid.org/0000-0002-1647-2112"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ilan Komargodski","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":0.268,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.66612113,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"16","issue":"1","first_page":"1","last_page":"18"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","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/T10237","display_name":"Cryptography and Data Security","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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.991599977016449,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9653000235557556,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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/secret-sharing","display_name":"Secret sharing","score":0.8116251230239868},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6483961343765259},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5649312138557434},{"id":"https://openalex.org/keywords/verifiable-secret-sharing","display_name":"Verifiable secret sharing","score":0.5617539286613464},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5467355251312256},{"id":"https://openalex.org/keywords/homomorphic-secret-sharing","display_name":"Homomorphic secret sharing","score":0.49322670698165894},{"id":"https://openalex.org/keywords/secure-multi-party-computation","display_name":"Secure multi-party computation","score":0.47846007347106934},{"id":"https://openalex.org/keywords/shamirs-secret-sharing","display_name":"Shamir's Secret Sharing","score":0.47654229402542114},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.45261719822883606},{"id":"https://openalex.org/keywords/secrecy","display_name":"Secrecy","score":0.44931429624557495},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.43042275309562683},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.419354647397995},{"id":"https://openalex.org/keywords/prime-power","display_name":"Prime power","score":0.41140180826187134},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.39499151706695557},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.3921513259410858},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.25839346647262573},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.18046844005584717}],"concepts":[{"id":"https://openalex.org/C3087436","wikidata":"https://www.wikidata.org/wiki/Q1386603","display_name":"Secret sharing","level":3,"score":0.8116251230239868},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6483961343765259},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5649312138557434},{"id":"https://openalex.org/C85847156","wikidata":"https://www.wikidata.org/wiki/Q59015987","display_name":"Verifiable secret sharing","level":3,"score":0.5617539286613464},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5467355251312256},{"id":"https://openalex.org/C132161212","wikidata":"https://www.wikidata.org/wiki/Q5891458","display_name":"Homomorphic secret sharing","level":4,"score":0.49322670698165894},{"id":"https://openalex.org/C18396474","wikidata":"https://www.wikidata.org/wiki/Q2465888","display_name":"Secure multi-party computation","level":3,"score":0.47846007347106934},{"id":"https://openalex.org/C131755552","wikidata":"https://www.wikidata.org/wiki/Q935125","display_name":"Shamir's Secret Sharing","level":5,"score":0.47654229402542114},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.45261719822883606},{"id":"https://openalex.org/C2776452267","wikidata":"https://www.wikidata.org/wiki/Q1503443","display_name":"Secrecy","level":2,"score":0.44931429624557495},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.43042275309562683},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.419354647397995},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.41140180826187134},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.39499151706695557},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.3921513259410858},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25839346647262573},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.18046844005584717},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.4086/toc.2020.v016a002","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2020.v016a002","pdf_url":"http://www.theoryofcomputing.org/articles/v016a002/v016a002.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.4086/toc.2020.v016a002","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2020.v016a002","pdf_url":"http://www.theoryofcomputing.org/articles/v016a002/v016a002.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1523888516","display_name":null,"funder_award_id":"FA9550-","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G8525538178","display_name":null,"funder_award_id":"FA9550-15-1-0262","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"}],"funders":[{"id":"https://openalex.org/F4320320946","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306"},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"},{"id":"https://openalex.org/F4320322942","display_name":"Chinese University of Hong Kong","ror":"https://ror.org/00t33hh48"},{"id":"https://openalex.org/F4320335016","display_name":"Israeli Centers for Research Excellence","ror":"https://ror.org/03w2v9593"},{"id":"https://openalex.org/F4320335375","display_name":"Planning and Budgeting Committee of the Council for Higher Education of Israel","ror":null},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3086270023.pdf","grobid_xml":"https://content.openalex.org/works/W3086270023.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W51265332","https://openalex.org/W324663928","https://openalex.org/W1534695542","https://openalex.org/W1550421702","https://openalex.org/W1574535057","https://openalex.org/W1968182724","https://openalex.org/W1984242286","https://openalex.org/W1987772082","https://openalex.org/W1987975907","https://openalex.org/W2002568678","https://openalex.org/W2015875314","https://openalex.org/W2068101455","https://openalex.org/W2076645411","https://openalex.org/W2080975816","https://openalex.org/W2099000090","https://openalex.org/W2099950221","https://openalex.org/W2102302508","https://openalex.org/W2116360492","https://openalex.org/W2120551550","https://openalex.org/W2133474031","https://openalex.org/W2141420453","https://openalex.org/W2143656893","https://openalex.org/W2151877312","https://openalex.org/W2169672310","https://openalex.org/W2293606446","https://openalex.org/W2307125543","https://openalex.org/W2397763371","https://openalex.org/W2397857137","https://openalex.org/W2571456248","https://openalex.org/W2741846989","https://openalex.org/W2809318426","https://openalex.org/W2809402609","https://openalex.org/W3022074522","https://openalex.org/W3030169579","https://openalex.org/W3128663643","https://openalex.org/W3195059766","https://openalex.org/W4226347946"],"related_works":["https://openalex.org/W2083084382","https://openalex.org/W2006088653","https://openalex.org/W2348427614","https://openalex.org/W1557783852","https://openalex.org/W2768332800","https://openalex.org/W2129756153","https://openalex.org/W2357909631","https://openalex.org/W3125135494","https://openalex.org/W1552579947","https://openalex.org/W2032053269"],"abstract_inverted_index":{"$":[0,1],"We":[2,152],"prove":[3,153],"that":[4,65,103],"for":[5,20,85,88,104,148,159],"every":[6],"$n$":[7,40],"and":[8,39,48,122,168],"$1":[9,94,106],"<":[10,12,95,97,107,109,111],"t":[11,56,96],"n$":[13],"any":[14,113],"$t$-out-of-$n$":[15],"threshold":[16,120,124,160],"secret":[17,71,115,146,161],"sharing":[18,72,116,147,162],"scheme":[19,73,117],"one-bit":[21,86],"secrets":[22,87],"requires":[23,126],"share":[24,67,127],"size":[25,68,128],"$\\log(t":[26],"+":[27,57,130],"1)$.":[28],"Our":[29],"bound":[30,53],"is":[31,41,77],"tight":[32],"when":[33],"$t":[34],"=":[35],"n":[36],"-":[37,55,132],"1$":[38],"a":[42,141,171],"prime":[43],"power.":[44],"In":[45],"1990":[46],"Kilian":[47],"Nisan":[49],"proved":[50],"the":[51,61,66,89,154],"incomparable":[52],"$\\log(n":[54],"2)$.":[58],"Taken":[59],"together,":[60],"two":[62],"bounds":[63],"imply":[64],"of":[69,92,136,145,156],"Shamir's":[70],"(Comm.":[74],"ACM":[75],"1979)":[76],"optimal":[78],"up":[79],"to":[80,165],"an":[81],"additive":[82],"constant":[83],"even":[84],"whole":[90],"range":[91],"parameters":[93],"n$.":[98],"More":[99],"generally,":[100],"we":[101,139],"show":[102],"all":[105],"s":[108],"r":[110],"n$,":[112],"ramp":[114],"with":[118,163],"secrecy":[119],"$s$":[121],"reconstruction":[123],"$r$":[125],"$\\log((r":[129],"1)/(r":[131],"s))$.":[133],"As":[134],"part":[135],"our":[137,157],"analysis":[138,158],"formulate":[140],"simple":[142],"game-theoretic":[143],"relaxation":[144],"arbitrary":[149],"access":[150],"structures.":[151],"optimality":[155],"respect":[164],"this":[166],"method":[167],"point":[169],"out":[170],"general":[172],"limitation.":[173]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2020-09-21T00:00:00"}