{"id":"https://openalex.org/W2988963440","doi":"https://doi.org/10.1515/jmc-2019-0032","title":"A framework for cryptographic problems from linear algebra","display_name":"A framework for cryptographic problems from linear algebra","publication_year":2019,"publication_date":"2019-11-08","ids":{"openalex":"https://openalex.org/W2988963440","doi":"https://doi.org/10.1515/jmc-2019-0032","mag":"2988963440"},"language":"en","primary_location":{"id":"doi:10.1515/jmc-2019-0032","is_oa":true,"landing_page_url":"https://doi.org/10.1515/jmc-2019-0032","pdf_url":"https://www.degruyter.com/document/doi/10.1515/jmc-2019-0032/pdf","source":{"id":"https://openalex.org/S100611479","display_name":"Journal of Mathematical Cryptology","issn_l":"1862-2976","issn":["1862-2976","1862-2984"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Mathematical Cryptology","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.degruyter.com/document/doi/10.1515/jmc-2019-0032/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082815475","display_name":"Carl Bootland","orcid":"https://orcid.org/0000-0002-8390-3410"},"institutions":[{"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"],"is_corresponding":true,"raw_author_name":"Carl Bootland","raw_affiliation_strings":["ESAT/COSIC, KU Leuven , Kasteelpark Arenberg 10, 3000 Leuven , Belgium","ESAT/COSIC, KU Leuven, Kasteelpark Arenberg 10, 3000Leuven, Belgium"],"raw_orcid":"https://orcid.org/0000-0002-8390-3410","affiliations":[{"raw_affiliation_string":"ESAT/COSIC, KU Leuven , Kasteelpark Arenberg 10, 3000 Leuven , Belgium","institution_ids":["https://openalex.org/I99464096"]},{"raw_affiliation_string":"ESAT/COSIC, KU Leuven, Kasteelpark Arenberg 10, 3000Leuven, Belgium","institution_ids":["https://openalex.org/I99464096"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041607172","display_name":"Wouter Castryck","orcid":"https://orcid.org/0000-0002-0191-5216"},"institutions":[{"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"],"is_corresponding":false,"raw_author_name":"Wouter Castryck","raw_affiliation_strings":["Department of Mathematics, KU Leuven , Celestijnenlaan 200B, 3000 Leuven , Belgium","Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3000Leuven, Belgium"],"raw_orcid":"https://orcid.org/0000-0002-0191-5216","affiliations":[{"raw_affiliation_string":"Department of Mathematics, KU Leuven , Celestijnenlaan 200B, 3000 Leuven , Belgium","institution_ids":["https://openalex.org/I99464096"]},{"raw_affiliation_string":"Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3000Leuven, Belgium","institution_ids":["https://openalex.org/I99464096"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044767875","display_name":"Alan Szepieniec","orcid":null},"institutions":[{"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"],"is_corresponding":false,"raw_author_name":"Alan Szepieniec","raw_affiliation_strings":["ESAT/COSIC, KU Leuven , Kasteelpark Arenberg 10, 3000 Leuven , Belgium","ESAT/COSIC, KU Leuven, Kasteelpark Arenberg 10, 3000Leuven, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ESAT/COSIC, KU Leuven , Kasteelpark Arenberg 10, 3000 Leuven , Belgium","institution_ids":["https://openalex.org/I99464096"]},{"raw_affiliation_string":"ESAT/COSIC, KU Leuven, Kasteelpark Arenberg 10, 3000Leuven, Belgium","institution_ids":["https://openalex.org/I99464096"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032574536","display_name":"Fr\u00e9derik Vercauteren","orcid":"https://orcid.org/0000-0002-7208-9599"},"institutions":[{"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"],"is_corresponding":false,"raw_author_name":"Frederik Vercauteren","raw_affiliation_strings":["ESAT/COSIC, KU Leuven , Kasteelpark Arenberg 10, 3000 Leuven , Belgium","ESAT/COSIC, KU Leuven, Kasteelpark Arenberg 10, 3000Leuven, Belgium"],"raw_orcid":"https://orcid.org/0000-0002-7208-9599","affiliations":[{"raw_affiliation_string":"ESAT/COSIC, KU Leuven , Kasteelpark Arenberg 10, 3000 Leuven , Belgium","institution_ids":["https://openalex.org/I99464096"]},{"raw_affiliation_string":"ESAT/COSIC, KU Leuven, Kasteelpark Arenberg 10, 3000Leuven, Belgium","institution_ids":["https://openalex.org/I99464096"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5082815475"],"corresponding_institution_ids":["https://openalex.org/I99464096"],"apc_list":{"value":1000,"currency":"EUR","value_usd":1078},"apc_paid":{"value":1000,"currency":"EUR","value_usd":1078},"fwci":0.4353,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.72945515,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"14","issue":"1","first_page":"202","last_page":"217"},"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/T11130","display_name":"Coding theory and cryptography","score":0.9976000189781189,"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":0.9909999966621399,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mersenne-prime","display_name":"Mersenne prime","score":0.7697204947471619},{"id":"https://openalex.org/keywords/ntru","display_name":"NTRU","score":0.7486951947212219},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.6150137782096863},{"id":"https://openalex.org/keywords/cryptosystem","display_name":"Cryptosystem","score":0.5637530088424683},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5332911014556885},{"id":"https://openalex.org/keywords/ciphertext","display_name":"Ciphertext","score":0.5109211802482605},{"id":"https://openalex.org/keywords/plaintext","display_name":"Plaintext","score":0.5014917850494385},{"id":"https://openalex.org/keywords/encryption","display_name":"Encryption","score":0.4098171591758728},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4017412066459656},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3447904586791992},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3353559672832489},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.17067721486091614},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.15293103456497192}],"concepts":[{"id":"https://openalex.org/C29641680","wikidata":"https://www.wikidata.org/wiki/Q186875","display_name":"Mersenne prime","level":2,"score":0.7697204947471619},{"id":"https://openalex.org/C122368425","wikidata":"https://www.wikidata.org/wiki/Q6955493","display_name":"NTRU","level":4,"score":0.7486951947212219},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.6150137782096863},{"id":"https://openalex.org/C6295992","wikidata":"https://www.wikidata.org/wiki/Q976521","display_name":"Cryptosystem","level":3,"score":0.5637530088424683},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5332911014556885},{"id":"https://openalex.org/C93974786","wikidata":"https://www.wikidata.org/wiki/Q1589480","display_name":"Ciphertext","level":3,"score":0.5109211802482605},{"id":"https://openalex.org/C92717368","wikidata":"https://www.wikidata.org/wiki/Q1162538","display_name":"Plaintext","level":3,"score":0.5014917850494385},{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.4098171591758728},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4017412066459656},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3447904586791992},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3353559672832489},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.17067721486091614},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.15293103456497192},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1515/jmc-2019-0032","is_oa":true,"landing_page_url":"https://doi.org/10.1515/jmc-2019-0032","pdf_url":"https://www.degruyter.com/document/doi/10.1515/jmc-2019-0032/pdf","source":{"id":"https://openalex.org/S100611479","display_name":"Journal of Mathematical Cryptology","issn_l":"1862-2976","issn":["1862-2976","1862-2984"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Mathematical Cryptology","raw_type":"journal-article"},{"id":"pmh:oai:lirias2repo.kuleuven.be:123456789/646822","is_oa":false,"landing_page_url":"https://lirias.kuleuven.be/handle/123456789/646822","pdf_url":null,"source":{"id":"https://openalex.org/S4306401954","display_name":"Lirias (KU Leuven)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I99464096","host_organization_name":"KU Leuven","host_organization_lineage":["https://openalex.org/I99464096"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:lirias2repo.kuleuven.be:123456789/649924","is_oa":true,"landing_page_url":"https://lirias.kuleuven.be/handle/123456789/649924","pdf_url":null,"source":{"id":"https://openalex.org/S4306401954","display_name":"Lirias (KU Leuven)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I99464096","host_organization_name":"KU Leuven","host_organization_lineage":["https://openalex.org/I99464096"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"Journal of Mathematical Cryptology, vol. 14 (1), (202-217)","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:archive.ugent.be:8665510","is_oa":true,"landing_page_url":"https://biblio.ugent.be/publication/8665510","pdf_url":null,"source":{"id":"https://openalex.org/S4306400478","display_name":"Ghent University Academic Bibliography (Ghent University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I32597200","host_organization_name":"Ghent University","host_organization_lineage":["https://openalex.org/I32597200"],"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":"ISSN: 1862-2984","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:doaj.org/article:86d340c212bc4362bc87186406ac5df7","is_oa":true,"landing_page_url":"https://doaj.org/article/86d340c212bc4362bc87186406ac5df7","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Mathematical Cryptology, Vol 14, Iss 1, Pp 202-217 (2020)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1515/jmc-2019-0032","is_oa":true,"landing_page_url":"https://doi.org/10.1515/jmc-2019-0032","pdf_url":"https://www.degruyter.com/document/doi/10.1515/jmc-2019-0032/pdf","source":{"id":"https://openalex.org/S100611479","display_name":"Journal of Mathematical Cryptology","issn_l":"1862-2976","issn":["1862-2976","1862-2984"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Mathematical Cryptology","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6263826853","display_name":null,"funder_award_id":"C14/18/067","funder_id":"https://openalex.org/F4320322681","funder_display_name":"Onderzoeksraad, KU Leuven"},{"id":"https://openalex.org/G6892013326","display_name":null,"funder_award_id":"STG/17/019","funder_id":"https://openalex.org/F4320322681","funder_display_name":"Onderzoeksraad, KU Leuven"}],"funders":[{"id":"https://openalex.org/F4320321730","display_name":"Fonds Wetenschappelijk Onderzoek","ror":"https://ror.org/03qtxy027"},{"id":"https://openalex.org/F4320322308","display_name":"KU Leuven","ror":"https://ror.org/05f950310"},{"id":"https://openalex.org/F4320322603","display_name":"Universiteit Gent","ror":"https://ror.org/00cv9y106"},{"id":"https://openalex.org/F4320322681","display_name":"Onderzoeksraad, KU Leuven","ror":"https://ror.org/05f950310"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2988963440.pdf","grobid_xml":"https://content.openalex.org/works/W2988963440.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W1985439922","https://openalex.org/W2103398070","https://openalex.org/W2122086332","https://openalex.org/W2152926062","https://openalex.org/W2398538644","https://openalex.org/W2512450221","https://openalex.org/W2514005182","https://openalex.org/W2514893051","https://openalex.org/W2729344185","https://openalex.org/W2741098187","https://openalex.org/W2795267660","https://openalex.org/W2807738027","https://openalex.org/W2888966703","https://openalex.org/W2894450396","https://openalex.org/W2950573224","https://openalex.org/W2986959318","https://openalex.org/W3028867652","https://openalex.org/W3172550433","https://openalex.org/W3217399103","https://openalex.org/W4254023826"],"related_works":["https://openalex.org/W3099266986","https://openalex.org/W4385711271","https://openalex.org/W2556463690","https://openalex.org/W2348168089","https://openalex.org/W2733141493","https://openalex.org/W4312347076","https://openalex.org/W2795067504","https://openalex.org/W2949316140","https://openalex.org/W1565128838","https://openalex.org/W2921056627"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"introduce":[2,56],"a":[3,51,101,111],"general":[4],"framework":[5,32,189],"encompassing":[6],"the":[7,18,62,91,125,147,163,170,188],"main":[8],"hard":[9,38,59,71],"problems":[10,26,39,60,72],"emerging":[11],"in":[12],"lattice-based":[13],"cryptography,":[14],"which":[15],"naturally":[16],"includes":[17],"recently":[19,164],"proposed":[20,165],"Mersenne":[21,166],"prime":[22,167],"cryptosystem,":[23],"but":[24],"also":[25],"coming":[27],"from":[28,48],"code-based":[29],"cryptography.":[30],"The":[31],"allows":[33],"to":[34,41,79,115,141],"easily":[35],"instantiate":[36],"new":[37,58],"and":[40,61,77,104,133,144,155,179,192],"automatically":[42],"construct":[43],"plausibly":[44],"post-quantum":[45],"secure":[46],"primitives":[47],"them.":[49],"As":[50],"first":[52],"basic":[53],"application,":[54],"we":[55,67],"two":[57],"corresponding":[63],"encryption":[64],"schemes.":[65],"Concretely,":[66],"study":[68],"generalisations":[69],"of":[70,84,90,122,176,190],"such":[73],"as":[74],"SIS,":[75],"LWE":[76,191],"NTRU":[78],"free":[80],"modules":[81,120,175],"over":[82],"quotients":[83],"\u2124[":[85,107],"X":[86,108,129,151,158],"]":[87,109],"by":[88],"ideals":[89],"form":[92],"(":[93],"f":[94,99,116,127,149,182],",":[95],"g":[96,105,134,156],"),":[97],"where":[98],"is":[100,110],"monic":[102],"polynomial":[103],"\u2208":[106,137],"ciphertext":[112],"modulus":[113],"coprime":[114],".":[117],"For":[118],"trivial":[119],"(i.e.":[121],"rank":[123,178],"one),":[124],"case":[126],"=":[128,135,150,157,184],"n":[130,152],"+":[131],"1":[132,154],"q":[136],"\u2124":[138],">1":[139],"corresponds":[140],"ring-LWE,":[142],"ring-SIS":[143],"NTRU,":[145],"while":[146],"choices":[148],"\u2013":[153,159],"2":[160],"essentially":[161],"cover":[162],"cryptosystems.":[168],"At":[169],"other":[171],"extreme,":[172],"when":[173],"considering":[174],"large":[177],"letting":[180],"deg(":[181],")":[183],"1,":[185],"one":[186],"recovers":[187],"SIS.":[193]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2019-11-22T00:00:00"}