{"id":"https://openalex.org/W2071825329","doi":"https://doi.org/10.1145/2535925","title":"On Ideal Lattices and Learning with Errors over Rings","display_name":"On Ideal Lattices and Learning with Errors over Rings","publication_year":2013,"publication_date":"2013-11-01","ids":{"openalex":"https://openalex.org/W2071825329","doi":"https://doi.org/10.1145/2535925","mag":"2071825329"},"language":"en","primary_location":{"id":"doi:10.1145/2535925","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2535925","pdf_url":null,"source":{"id":"https://openalex.org/S118992489","display_name":"Journal of the ACM","issn_l":"0004-5411","issn":["0004-5411","1557-735X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the ACM","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/A5074391920","display_name":"Vadim Lyubashevsky","orcid":"https://orcid.org/0009-0003-5149-264X"},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en informatique et en automatique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1326498283"]},{"id":"https://openalex.org/I4210103330","display_name":"\u00c9cole Normale Sup\u00e9rieure","ror":"https://ror.org/01dp7jr64","country_code":"BI","type":"education","lineage":["https://openalex.org/I4210103330"]}],"countries":["BI","FR"],"is_corresponding":true,"raw_author_name":"Vadim Lyubashevsky","raw_affiliation_strings":["INRIA and \u00c9cole Normale Sup\u00e9rieure, Paris"],"affiliations":[{"raw_affiliation_string":"INRIA and \u00c9cole Normale Sup\u00e9rieure, Paris","institution_ids":["https://openalex.org/I4210103330","https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086535374","display_name":"Chris Peikert","orcid":"https://orcid.org/0000-0003-0419-7501"},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Chris Peikert","raw_affiliation_strings":["Georgia Institute of Technology","[Georgia Institute of Technology.]"],"affiliations":[{"raw_affiliation_string":"Georgia Institute of Technology","institution_ids":["https://openalex.org/I130701444"]},{"raw_affiliation_string":"[Georgia Institute of Technology.]","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049700941","display_name":"Oded Regev","orcid":"https://orcid.org/0000-0002-8616-3163"},"institutions":[{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]},{"id":"https://openalex.org/I36672615","display_name":"Courant Institute of Mathematical Sciences","ror":"https://ror.org/037tm7f56","country_code":"US","type":"education","lineage":["https://openalex.org/I36672615","https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Oded Regev","raw_affiliation_strings":["Courant Institute, New York University"],"affiliations":[{"raw_affiliation_string":"Courant Institute, New York University","institution_ids":["https://openalex.org/I36672615","https://openalex.org/I57206974"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5074391920"],"corresponding_institution_ids":["https://openalex.org/I1326498283","https://openalex.org/I4210103330"],"apc_list":null,"apc_paid":null,"fwci":70.6755,"has_fulltext":false,"cited_by_count":805,"citation_normalized_percentile":{"value":0.99929254,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"60","issue":"6","first_page":"1","last_page":"35"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","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/T10237","display_name":"Cryptography and Data Security","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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9944000244140625,"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/T10764","display_name":"Privacy-Preserving Technologies in Data","score":0.9847000241279602,"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/learning-with-errors","display_name":"Learning with errors","score":0.9366142749786377},{"id":"https://openalex.org/keywords/cryptosystem","display_name":"Cryptosystem","score":0.657132625579834},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.6099038124084473},{"id":"https://openalex.org/keywords/lattice-problem","display_name":"Lattice problem","score":0.5930604338645935},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5101892948150635},{"id":"https://openalex.org/keywords/lattice-based-cryptography","display_name":"Lattice-based cryptography","score":0.4831051826477051},{"id":"https://openalex.org/keywords/pseudorandom-number-generator","display_name":"Pseudorandom number generator","score":0.4820869266986847},{"id":"https://openalex.org/keywords/post-quantum-cryptography","display_name":"Post-quantum cryptography","score":0.44252756237983704},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.43401357531547546},{"id":"https://openalex.org/keywords/oblivious-transfer","display_name":"Oblivious transfer","score":0.42584142088890076},{"id":"https://openalex.org/keywords/polynomial-ring","display_name":"Polynomial ring","score":0.4125877022743225},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4098566174507141},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38482069969177246},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.3799354135990143},{"id":"https://openalex.org/keywords/encryption","display_name":"Encryption","score":0.3613799512386322},{"id":"https://openalex.org/keywords/public-key-cryptography","display_name":"Public-key cryptography","score":0.35970214009284973},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.27482056617736816},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.21583297848701477},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.2053791582584381},{"id":"https://openalex.org/keywords/quantum-information","display_name":"Quantum information","score":0.1469743251800537},{"id":"https://openalex.org/keywords/quantum-cryptography","display_name":"Quantum cryptography","score":0.14467832446098328}],"concepts":[{"id":"https://openalex.org/C2779014939","wikidata":"https://www.wikidata.org/wiki/Q6510239","display_name":"Learning with errors","level":3,"score":0.9366142749786377},{"id":"https://openalex.org/C6295992","wikidata":"https://www.wikidata.org/wiki/Q976521","display_name":"Cryptosystem","level":3,"score":0.657132625579834},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.6099038124084473},{"id":"https://openalex.org/C168482242","wikidata":"https://www.wikidata.org/wiki/Q6497128","display_name":"Lattice problem","level":3,"score":0.5930604338645935},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5101892948150635},{"id":"https://openalex.org/C137660015","wikidata":"https://www.wikidata.org/wiki/Q6497083","display_name":"Lattice-based cryptography","level":5,"score":0.4831051826477051},{"id":"https://openalex.org/C140642157","wikidata":"https://www.wikidata.org/wiki/Q1623338","display_name":"Pseudorandom number generator","level":2,"score":0.4820869266986847},{"id":"https://openalex.org/C108277079","wikidata":"https://www.wikidata.org/wiki/Q7233576","display_name":"Post-quantum cryptography","level":4,"score":0.44252756237983704},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.43401357531547546},{"id":"https://openalex.org/C53076038","wikidata":"https://www.wikidata.org/wiki/Q2915383","display_name":"Oblivious transfer","level":3,"score":0.42584142088890076},{"id":"https://openalex.org/C9485509","wikidata":"https://www.wikidata.org/wiki/Q1455652","display_name":"Polynomial ring","level":3,"score":0.4125877022743225},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4098566174507141},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38482069969177246},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3799354135990143},{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.3613799512386322},{"id":"https://openalex.org/C203062551","wikidata":"https://www.wikidata.org/wiki/Q201339","display_name":"Public-key cryptography","level":3,"score":0.35970214009284973},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.27482056617736816},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.21583297848701477},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.2053791582584381},{"id":"https://openalex.org/C169699857","wikidata":"https://www.wikidata.org/wiki/Q2122243","display_name":"Quantum information","level":3,"score":0.1469743251800537},{"id":"https://openalex.org/C144901912","wikidata":"https://www.wikidata.org/wiki/Q471906","display_name":"Quantum cryptography","level":4,"score":0.14467832446098328},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/2535925","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2535925","pdf_url":null,"source":{"id":"https://openalex.org/S118992489","display_name":"Journal of the ACM","issn_l":"0004-5411","issn":["0004-5411","1557-735X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the ACM","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-00921792v1","is_oa":false,"landing_page_url":"https://inria.hal.science/hal-00921792","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of the ACM (JACM), 2013, 60 (6), ⟨10.1145/2535925⟩","raw_type":"Journal articles"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","score":0.5899999737739563,"id":"https://metadata.un.org/sdg/16"}],"awards":[{"id":"https://openalex.org/G2866203636","display_name":null,"funder_award_id":"CCF-1054495","funder_id":"https://openalex.org/F4320337387","funder_display_name":"Division of Computing and Communication Foundations"},{"id":"https://openalex.org/G7968489690","display_name":null,"funder_award_id":"CNS-0716786","funder_id":"https://openalex.org/F4320337388","funder_display_name":"Division of Computer and Network Systems"}],"funders":[{"id":"https://openalex.org/F4320306151","display_name":"Alfred P. Sloan Foundation","ror":"https://ror.org/052csg198"},{"id":"https://openalex.org/F4320308960","display_name":"United States - Israel Binational Science Foundation","ror":"https://ror.org/00j8z2m73"},{"id":"https://openalex.org/F4320320670","display_name":"Wolfson Foundation","ror":"https://ror.org/0333xzh65"},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"},{"id":"https://openalex.org/F4320334678","display_name":"European Research Council","ror":"https://ror.org/0472cxd90"},{"id":"https://openalex.org/F4320334962","display_name":"Sixth Framework Programme","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320337387","display_name":"Division of Computing and Communication Foundations","ror":"https://ror.org/01mng8331"},{"id":"https://openalex.org/F4320337388","display_name":"Division of Computer and Network Systems","ror":"https://ror.org/02rdzmk74"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":61,"referenced_works":["https://openalex.org/W31130112","https://openalex.org/W74973629","https://openalex.org/W121367636","https://openalex.org/W139987668","https://openalex.org/W149470655","https://openalex.org/W1491152510","https://openalex.org/W1491861813","https://openalex.org/W1496801598","https://openalex.org/W1499766499","https://openalex.org/W1506156414","https://openalex.org/W1525538795","https://openalex.org/W1537360597","https://openalex.org/W1546634390","https://openalex.org/W1548880861","https://openalex.org/W1560474949","https://openalex.org/W1576470039","https://openalex.org/W1606729219","https://openalex.org/W1675339804","https://openalex.org/W1761303171","https://openalex.org/W1818243338","https://openalex.org/W1856342626","https://openalex.org/W1936753088","https://openalex.org/W1985439922","https://openalex.org/W1992282993","https://openalex.org/W1994790157","https://openalex.org/W2000956176","https://openalex.org/W2001093624","https://openalex.org/W2007466965","https://openalex.org/W2013623332","https://openalex.org/W2031533839","https://openalex.org/W2038761522","https://openalex.org/W2048245339","https://openalex.org/W2048994663","https://openalex.org/W2050689197","https://openalex.org/W2058785578","https://openalex.org/W2089477764","https://openalex.org/W2106121141","https://openalex.org/W2106970188","https://openalex.org/W2108834246","https://openalex.org/W2113333997","https://openalex.org/W2120030642","https://openalex.org/W2122086332","https://openalex.org/W2122973102","https://openalex.org/W2127640229","https://openalex.org/W2130410178","https://openalex.org/W2140256428","https://openalex.org/W2146633075","https://openalex.org/W2151237818","https://openalex.org/W2152783950","https://openalex.org/W2152926062","https://openalex.org/W2156186849","https://openalex.org/W2163469693","https://openalex.org/W2167236842","https://openalex.org/W2169649500","https://openalex.org/W2212077392","https://openalex.org/W2233233025","https://openalex.org/W2403438130","https://openalex.org/W2979771948","https://openalex.org/W4285719527","https://openalex.org/W4406271161","https://openalex.org/W6605963903"],"related_works":["https://openalex.org/W3165155389","https://openalex.org/W4298094689","https://openalex.org/W2061949491","https://openalex.org/W2883034031","https://openalex.org/W2895855268","https://openalex.org/W3172324685","https://openalex.org/W3127629500","https://openalex.org/W2951286680","https://openalex.org/W2007466965","https://openalex.org/W2315723423"],"abstract_inverted_index":{"The":[0,26],"\u201clearning":[1],"with":[2,160],"errors\u201d":[3],"(LWE)":[4],"problem":[5,27],"is":[6,136],"to":[7,31,62],"distinguish":[8],"random":[9],"linear":[10],"equations,":[11],"which":[12],"have":[13],"been":[14,29],"perturbed":[15],"by":[16,87,109],"a":[17,50],"small":[18],"amount":[19],"of":[20,52,70,114,167,171,182],"noise,":[21],"from":[22],"truly":[23,85,155],"uniform":[24],"ones.":[25],"has":[28,44],"shown":[30],"be":[32,83,174],"as":[33,35,46,92],"hard":[34,146],"worst-case":[36,140],"lattice":[37],"problems,":[38],"and":[39,79,119],"in":[40,67,106],"recent":[41],"years":[42],"it":[43,122],"served":[45],"the":[47,68,107,133,153,168,180],"foundation":[48],"for":[49,95,147],"plethora":[51],"cryptographic":[53],"applications.":[54],"Unfortunately,":[55],"these":[56],"applications":[57,81,170],"are":[58,145],"rather":[59],"inefficient":[60],"due":[61],"an":[63,111,161],"inherent":[64],"quadratic":[65],"overhead":[66],"use":[69,181],"LWE.":[71],"A":[72],"main":[73],"open":[74],"question":[75,105],"was":[76,93],"whether":[77],"LWE":[78,115,172],"its":[80],"could":[82],"made":[84,175],"efficient":[86,162,178],"exploiting":[88],"extra":[89],"algebraic":[90,112],"structure,":[91],"done":[94],"lattice-based":[96,157],"hash":[97],"functions":[98],"(and":[99],"related":[100],"primitives).":[101],"We":[102],"resolve":[103],"this":[104],"affirmative":[108],"introducing":[110],"variant":[113],"called":[116],"ring-LWE":[117,134],",":[118],"proving":[120],"that":[121,132,139],"too":[123],"enjoys":[124],"very":[125],"strong":[126],"hardness":[127],"guarantees.":[128],"Specifically,":[129],"we":[130],"show":[131],"distribution":[135],"pseudorandom,":[137],"assuming":[138],"problems":[141],"on":[142],"ideal":[143],"lattices":[144],"polynomial-time":[148],"quantum":[149],"algorithms.":[150],"Applications":[151],"include":[152],"first":[154],"practical":[156],"public-key":[158],"cryptosystem":[159],"security":[163],"reduction;":[164],"moreover,":[165],"many":[166],"other":[169],"can":[173],"much":[176],"more":[177],"through":[179],"ring-LWE.":[183]},"counts_by_year":[{"year":2026,"cited_by_count":12},{"year":2025,"cited_by_count":97},{"year":2024,"cited_by_count":88},{"year":2023,"cited_by_count":99},{"year":2022,"cited_by_count":70},{"year":2021,"cited_by_count":62},{"year":2020,"cited_by_count":70},{"year":2019,"cited_by_count":50},{"year":2018,"cited_by_count":45},{"year":2017,"cited_by_count":46},{"year":2016,"cited_by_count":42},{"year":2015,"cited_by_count":53},{"year":2014,"cited_by_count":36},{"year":2013,"cited_by_count":12},{"year":2012,"cited_by_count":11}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}