{"id":"https://openalex.org/W2514005182","doi":"https://doi.org/10.1112/s1461157016000371","title":"An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero","display_name":"An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W2514005182","doi":"https://doi.org/10.1112/s1461157016000371","mag":"2514005182"},"language":"en","primary_location":{"id":"doi:10.1112/s1461157016000371","is_oa":true,"landing_page_url":"https://doi.org/10.1112/s1461157016000371","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/230ECFEEE6AF4D8027FF3E13998D560C/S1461157016000371a.pdf/div-class-title-an-algorithm-for-ntru-problems-and-cryptanalysis-of-the-ggh-multilinear-map-without-a-low-level-encoding-of-zero-div.pdf","source":{"id":"https://openalex.org/S118668765","display_name":"LMS Journal of Computation and Mathematics","issn_l":"1461-1570","issn":["1461-1570"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311874","host_organization_name":"London Mathematical Society","host_organization_lineage":["https://openalex.org/P4310311874"],"host_organization_lineage_names":["London Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"LMS Journal of Computation and Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/230ECFEEE6AF4D8027FF3E13998D560C/S1461157016000371a.pdf/div-class-title-an-algorithm-for-ntru-problems-and-cryptanalysis-of-the-ggh-multilinear-map-without-a-low-level-encoding-of-zero-div.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044011121","display_name":"Jung Hee Cheon","orcid":"https://orcid.org/0000-0002-7085-2220"},"institutions":[{"id":"https://openalex.org/I139264467","display_name":"Seoul National University","ror":"https://ror.org/04h9pn542","country_code":"KR","type":"education","lineage":["https://openalex.org/I139264467"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Jung Hee Cheon","raw_affiliation_strings":["Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Korea email"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Korea email","institution_ids":["https://openalex.org/I139264467"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057980212","display_name":"Jinhyuck Jeong","orcid":"https://orcid.org/0000-0002-6785-9835"},"institutions":[{"id":"https://openalex.org/I139264467","display_name":"Seoul National University","ror":"https://ror.org/04h9pn542","country_code":"KR","type":"education","lineage":["https://openalex.org/I139264467"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Jinhyuck Jeong","raw_affiliation_strings":["Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-42, Republic of Korea email"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-42, Republic of Korea email","institution_ids":["https://openalex.org/I139264467"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100609703","display_name":"Changmin Lee","orcid":"https://orcid.org/0000-0002-9870-3956"},"institutions":[{"id":"https://openalex.org/I139264467","display_name":"Seoul National University","ror":"https://ror.org/04h9pn542","country_code":"KR","type":"education","lineage":["https://openalex.org/I139264467"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Changmin Lee","raw_affiliation_strings":["Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Korea email"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Korea email","institution_ids":["https://openalex.org/I139264467"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I139264467"],"apc_list":null,"apc_paid":null,"fwci":22.5211,"has_fulltext":true,"cited_by_count":105,"citation_normalized_percentile":{"value":0.99412668,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"19","issue":"A","first_page":"255","last_page":"266"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.9994999766349792,"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":0.9994999766349792,"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/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"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/T10951","display_name":"Cryptographic Implementations and Security","score":0.9962000250816345,"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/ntru","display_name":"NTRU","score":0.7579735517501831},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7382867336273193},{"id":"https://openalex.org/keywords/multilinear-map","display_name":"Multilinear map","score":0.6666197180747986},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6611731648445129},{"id":"https://openalex.org/keywords/ciphertext","display_name":"Ciphertext","score":0.6136820316314697},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.46156802773475647},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.43259960412979126},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42709848284721375},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.42293402552604675},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36278751492500305},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.1543205976486206},{"id":"https://openalex.org/keywords/encryption","display_name":"Encryption","score":0.15283459424972534},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10805198550224304},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.07480508089065552}],"concepts":[{"id":"https://openalex.org/C122368425","wikidata":"https://www.wikidata.org/wiki/Q6955493","display_name":"NTRU","level":4,"score":0.7579735517501831},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7382867336273193},{"id":"https://openalex.org/C84392682","wikidata":"https://www.wikidata.org/wiki/Q1952404","display_name":"Multilinear map","level":2,"score":0.6666197180747986},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6611731648445129},{"id":"https://openalex.org/C93974786","wikidata":"https://www.wikidata.org/wiki/Q1589480","display_name":"Ciphertext","level":3,"score":0.6136820316314697},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.46156802773475647},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.43259960412979126},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42709848284721375},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.42293402552604675},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36278751492500305},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.1543205976486206},{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.15283459424972534},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10805198550224304},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.07480508089065552},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"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/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C6295992","wikidata":"https://www.wikidata.org/wiki/Q976521","display_name":"Cryptosystem","level":3,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1112/s1461157016000371","is_oa":true,"landing_page_url":"https://doi.org/10.1112/s1461157016000371","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/230ECFEEE6AF4D8027FF3E13998D560C/S1461157016000371a.pdf/div-class-title-an-algorithm-for-ntru-problems-and-cryptanalysis-of-the-ggh-multilinear-map-without-a-low-level-encoding-of-zero-div.pdf","source":{"id":"https://openalex.org/S118668765","display_name":"LMS Journal of Computation and Mathematics","issn_l":"1461-1570","issn":["1461-1570"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311874","host_organization_name":"London Mathematical Society","host_organization_lineage":["https://openalex.org/P4310311874"],"host_organization_lineage_names":["London Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"LMS Journal of Computation and Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1112/s1461157016000371","is_oa":true,"landing_page_url":"https://doi.org/10.1112/s1461157016000371","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/230ECFEEE6AF4D8027FF3E13998D560C/S1461157016000371a.pdf/div-class-title-an-algorithm-for-ntru-problems-and-cryptanalysis-of-the-ggh-multilinear-map-without-a-low-level-encoding-of-zero-div.pdf","source":{"id":"https://openalex.org/S118668765","display_name":"LMS Journal of Computation and Mathematics","issn_l":"1461-1570","issn":["1461-1570"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311874","host_organization_name":"London Mathematical Society","host_organization_lineage":["https://openalex.org/P4310311874"],"host_organization_lineage_names":["London Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"LMS Journal of Computation and Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320671","display_name":"National Research Foundation","ror":"https://ror.org/05s0g1g46"},{"id":"https://openalex.org/F4320322030","display_name":"Ministry of Science, ICT and Future Planning","ror":"https://ror.org/032e49973"},{"id":"https://openalex.org/F4320322120","display_name":"National Research Foundation of Korea","ror":"https://ror.org/013aysd81"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2514005182.pdf","grobid_xml":"https://content.openalex.org/works/W2514005182.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W7540341","https://openalex.org/W57039937","https://openalex.org/W130353434","https://openalex.org/W1503892142","https://openalex.org/W1603256047","https://openalex.org/W1675339804","https://openalex.org/W1893972099","https://openalex.org/W1915197037","https://openalex.org/W2048994663","https://openalex.org/W2168596303","https://openalex.org/W2233194383","https://openalex.org/W2295080346","https://openalex.org/W2295328551","https://openalex.org/W2397835572","https://openalex.org/W2398614482","https://openalex.org/W2399042670","https://openalex.org/W2470510195","https://openalex.org/W2477092523","https://openalex.org/W2495595500","https://openalex.org/W2500250645","https://openalex.org/W2610863760","https://openalex.org/W2900370870","https://openalex.org/W2950700143","https://openalex.org/W2951200405","https://openalex.org/W3121827341","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W4390824577","https://openalex.org/W3206398586","https://openalex.org/W1603256047","https://openalex.org/W4390511315","https://openalex.org/W4280589839","https://openalex.org/W1571236230","https://openalex.org/W2950440893","https://openalex.org/W2362903264","https://openalex.org/W3201042007","https://openalex.org/W2921056627"],"abstract_inverted_index":{"Let":[0],"$\\mathbf{f}$":[1],"and":[2,63,151,157,171,197],"$\\mathbf{g}$":[3],"be":[4,105,136],"polynomials":[5],"of":[6,73,79,96,114,131,149,155,169,179,186],"a":[7,34,80,83,88,152,165,176],"bounded":[8],"Euclidean":[9,36],"norm":[10,37],"in":[11,53,182,206],"the":[12,18,24,48,77,93,108,125,183,191],"ring":[13],"$\\mathbb{Z}[X]/\\langle":[14],"X^{n}+1\\rangle$":[15,22,32],".":[16,41],"Given":[17],"polynomial":[19],"$[\\mathbf{f}/\\mathbf{g}]_{q}\\in":[20],"\\mathbb{Z}_{q}[X]/\\langle":[21],",":[23,62],"NTRU":[25,49],"problem":[26,81],"is":[27,76],"to":[28,46,85,104,139,174],"find":[29],"$\\mathbf{a},\\mathbf{b}\\in":[30],"\\mathbb{Z}[X]/\\langle":[31],"with":[33],"small":[35],"such":[38],"that":[39],"$[\\mathbf{a}/\\mathbf{b}]_{q}=[\\mathbf{f}/\\mathbf{g}]_{q}$":[40],"We":[42],"propose":[43],"an":[44,97],"algorithm":[45,75,134],"solve":[47],"problem,":[50],"which":[51],"runs":[52],"$2^{O(\\log":[54],"^{2}\\unicode[STIX]{x1D706})}$":[55],"time":[56,185],"when":[57],"$\\Vert":[58,64],"\\mathbf{g}\\Vert":[59],",\\Vert":[60],"\\mathbf{f}\\Vert$":[61],"\\mathbf{g}^{-1}\\Vert$":[65],"are":[66],"within":[67],"some":[68,146],"range.":[69],"The":[70,90],"main":[71],"technique":[72],"our":[74,133,160],"reduction":[78],"on":[82,87],"field":[84],"one":[86],"subfield.":[89],"GGH":[91,126],"scheme,":[92],"first":[94],"candidate":[95],"(approximate)":[98],"multilinear":[99,200],"map,":[100],"was":[101,120],"recently":[102],"found":[103],"insecure":[106],"by":[107,194],"Hu\u2013Jia":[109],"attack":[110,119,140,175],"using":[111,190],"low-level":[112,129],"encodings":[113,130,148],"zero,":[115,132],"but":[116],"no":[117],"polynomial-time":[118],"known":[121,153],"without":[122,128],"them.":[123],"In":[124],"scheme":[127,142,181],"can":[135,163],"directly":[137],"applied":[138],"this":[141,180],"if":[143],"we":[144,162],"have":[145],"top-level":[147],"zero":[150,170],"pair":[154],"plaintext":[156],"ciphertext.":[158],"Using":[159],"algorithm,":[161],"construct":[164],"level-":[166],"$0$":[167],"encoding":[168],"utilize":[172],"it":[173],"security":[177,188],"ground":[178],"quasi-polynomial":[184],"its":[187],"parameter":[189],"parameters":[192],"suggested":[193],"Garg,":[195],"Gentry":[196],"Halevi":[198],"[\u2018Candidate":[199],"maps":[201],"from":[202],"ideal":[203],"lattices\u2019,":[204],"Advances":[205],"cryptology":[207],"\u2014":[208],"EUROCRYPT":[209],"2013":[210],"(Springer,":[211],"2013)":[212],"1\u201317].":[213]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":12},{"year":2021,"cited_by_count":12},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":14},{"year":2018,"cited_by_count":10},{"year":2017,"cited_by_count":19},{"year":2016,"cited_by_count":9},{"year":2015,"cited_by_count":1}],"updated_date":"2026-06-24T06:17:17.354583","created_date":"2025-10-10T00:00:00"}