{"id":"https://openalex.org/W4390975310","doi":"https://doi.org/10.1007/s10623-023-01347-w","title":"Average case error estimates of the strong Lucas test","display_name":"Average case error estimates of the strong Lucas test","publication_year":2024,"publication_date":"2024-01-18","ids":{"openalex":"https://openalex.org/W4390975310","doi":"https://doi.org/10.1007/s10623-023-01347-w"},"language":"en","primary_location":{"id":"doi:10.1007/s10623-023-01347-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-023-01347-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-023-01347-w.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10623-023-01347-w.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5092206139","display_name":"Semira Einsele","orcid":"https://orcid.org/0000-0002-5525-736X"},"institutions":[{"id":"https://openalex.org/I75951250","display_name":"Freie Universit\u00e4t Berlin","ror":"https://ror.org/046ak2485","country_code":"DE","type":"education","lineage":["https://openalex.org/I75951250"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Semira Einsele","raw_affiliation_strings":["Department of Mathematics and Computer Science, FU Berlin, Takustr. 9, Berlin, Berlin, Germany"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, FU Berlin, Takustr. 9, Berlin, Berlin, Germany","institution_ids":["https://openalex.org/I75951250"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072987600","display_name":"Kenneth G. Paterson","orcid":"https://orcid.org/0000-0002-5145-4489"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Kenneth Paterson","raw_affiliation_strings":["Department of Computer Science, ETH Z\u00fcrich, Universit\u00e4tstrasse 6, 8092, Z\u00fcrich, Z\u00fcrich, Switzerland"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, ETH Z\u00fcrich, Universit\u00e4tstrasse 6, 8092, Z\u00fcrich, Z\u00fcrich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5092206139"],"corresponding_institution_ids":["https://openalex.org/I75951250"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":2.5167,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.84189723,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":96},"biblio":{"volume":"92","issue":"5","first_page":"1341","last_page":"1378"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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":0.9922999739646912,"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/primality-test","display_name":"Primality test","score":0.9611010551452637},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.644439697265625},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.5136606097221375},{"id":"https://openalex.org/keywords/cryptanalysis","display_name":"Cryptanalysis","score":0.45710644125938416},{"id":"https://openalex.org/keywords/test","display_name":"Test (biology)","score":0.4482293128967285},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.44186562299728394},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.41171616315841675},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40310463309288025},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36780720949172974},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.32067960500717163},{"id":"https://openalex.org/keywords/prime-number","display_name":"Prime number","score":0.23909002542495728}],"concepts":[{"id":"https://openalex.org/C191421660","wikidata":"https://www.wikidata.org/wiki/Q829546","display_name":"Primality test","level":3,"score":0.9611010551452637},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.644439697265625},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.5136606097221375},{"id":"https://openalex.org/C181149355","wikidata":"https://www.wikidata.org/wiki/Q897511","display_name":"Cryptanalysis","level":3,"score":0.45710644125938416},{"id":"https://openalex.org/C2777267654","wikidata":"https://www.wikidata.org/wiki/Q3519023","display_name":"Test (biology)","level":2,"score":0.4482293128967285},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.44186562299728394},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.41171616315841675},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40310463309288025},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36780720949172974},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.32067960500717163},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.23909002542495728},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10623-023-01347-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-023-01347-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-023-01347-w.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10623-023-01347-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-023-01347-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-023-01347-w.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1306977718","display_name":null,"funder_award_id":"16KISK025","funder_id":"https://openalex.org/F4320321114","funder_display_name":"Bundesministerium f\u00fcr Bildung und Forschung"},{"id":"https://openalex.org/G2295054390","display_name":null,"funder_award_id":"6G-RIC","funder_id":"https://openalex.org/F4320321114","funder_display_name":"Bundesministerium f\u00fcr Bildung und Forschung"},{"id":"https://openalex.org/G3043571677","display_name":null,"funder_award_id":"16KISK020K","funder_id":"https://openalex.org/F4320321114","funder_display_name":"Bundesministerium f\u00fcr Bildung und Forschung"},{"id":"https://openalex.org/G352791218","display_name":null,"funder_award_id":"(BMBF)","funder_id":"https://openalex.org/F4320321114","funder_display_name":"Bundesministerium f\u00fcr Bildung und Forschung"}],"funders":[{"id":"https://openalex.org/F4320321114","display_name":"Bundesministerium f\u00fcr Bildung und Forschung","ror":"https://ror.org/04pz7b180"},{"id":"https://openalex.org/F4320324521","display_name":"Freie Universit\u00e4t Berlin","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4390975310.pdf"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1536546174","https://openalex.org/W1998777945","https://openalex.org/W1998884984","https://openalex.org/W2021376368","https://openalex.org/W2047316746","https://openalex.org/W2056939948","https://openalex.org/W2070029754","https://openalex.org/W2071056578","https://openalex.org/W2089613796","https://openalex.org/W2159904901","https://openalex.org/W2319770796","https://openalex.org/W2597011185","https://openalex.org/W2889633772","https://openalex.org/W2920106046","https://openalex.org/W4245090983","https://openalex.org/W4388288416"],"related_works":["https://openalex.org/W4385966166","https://openalex.org/W4300905728","https://openalex.org/W3150335004","https://openalex.org/W1604915031","https://openalex.org/W3089223619","https://openalex.org/W4288849478","https://openalex.org/W4288674537","https://openalex.org/W2940752727","https://openalex.org/W1561797008","https://openalex.org/W2516549250"],"abstract_inverted_index":{"Abstract":[0],"Reliable":[1],"probabilistic":[2],"primality":[3,23,75],"tests":[4],"are":[5,45,57,162],"fundamental":[6],"in":[7,40,60],"public-key":[8],"cryptography.":[9],"In":[10,28,64,171],"adversarial":[11],"scenarios,":[12,42],"a":[13,16,21,51,144,151],"composite":[14,145],"with":[15,94,120],"high":[17],"probability":[18,142],"of":[19,36,115,154],"passing":[20],"specific":[22],"test":[24,93,119],"could":[25],"be":[26],"chosen.":[27],"such":[29,69],"cases,":[30],"we":[31,67,149,173],"need":[32],"worst-case":[33],"error":[34,54,62,85],"estimates":[35],"the":[37,43,72,116,126,141,175],"test.":[38],"However,":[39],"many":[41],"numbers":[44,176],"randomly":[46,121],"chosen":[47,122],"and":[48,109,124],"thus":[49],"have":[50],"significantly":[52],"smaller":[53],"probability.":[55],"We":[56,135],"hence":[58],"interested":[59],"average-case":[61,84],"estimates.":[63],"this":[65,92,155],"paper":[66],"establish":[68],"bounds":[70,139],"for":[71],"strong":[73,117],"Lucas":[74,118],"test,":[76],"as":[77],"there":[78],"exist":[79],"only":[80,158],"worst-case,":[81],"but":[82],"no":[83],"bounds.":[86,170],"This":[87],"allows":[88],"us":[89,98],"to":[90,180],"use":[91],"more":[95],"confidence.":[96],"Let":[97],"examine":[99,150],"an":[100],"algorithm":[101,156],"that":[102,129,143,157,161,177],"draws":[103],"odd":[104],"k":[105],"-bit":[106],"integers":[107,160],"uniformly":[108],"independently,":[110],"runs":[111],"t":[112,132],"independent":[113],"iterations":[114],"parameters,":[123],"outputs":[125],"first":[127],"number":[128],"passes":[130],"all":[131],"consecutive":[133],"rounds.":[134],"attain":[136],"numerical":[137],"upper":[138],"on":[140],"is":[146],"returned.":[147],"Moreover,":[148],"slight":[152],"modification":[153],"considers":[159],"not":[163],"divisible":[164],"by":[165],"small":[166],"primes,":[167],"yielding":[168],"improved":[169],"addition,":[172],"classify":[174],"contribute":[178],"most":[179],"our":[181],"estimate.":[182]},"counts_by_year":[{"year":2025,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
