{"id":"https://openalex.org/W4296020464","doi":"https://doi.org/10.3390/sym14091897","title":"Small Private Exponent Attacks on RSA Using Continued Fractions and Multicore Systems","display_name":"Small Private Exponent Attacks on RSA Using Continued Fractions and Multicore Systems","publication_year":2022,"publication_date":"2022-09-10","ids":{"openalex":"https://openalex.org/W4296020464","doi":"https://doi.org/10.3390/sym14091897"},"language":"en","primary_location":{"id":"doi:10.3390/sym14091897","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091897","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1897/pdf?version=1663815683","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/9/1897/pdf?version=1663815683","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037170867","display_name":"Hatem M. Bahig","orcid":"https://orcid.org/0000-0002-8137-7939"},"institutions":[{"id":"https://openalex.org/I107720978","display_name":"Ain Shams University","ror":"https://ror.org/00cb9w016","country_code":"EG","type":"education","lineage":["https://openalex.org/I107720978"]},{"id":"https://openalex.org/I4210088963","display_name":"University of Ha'il","ror":"https://ror.org/013w98a82","country_code":"SA","type":"education","lineage":["https://openalex.org/I4210088963"]}],"countries":["EG","SA"],"is_corresponding":false,"raw_author_name":"Hatem M. Bahig","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt","Information and Computer Science Department, College of Computer Science and Engineering, University of Ha\u2019il, Hail, Ha\u2019il 81481, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-8137-7939","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt","institution_ids":["https://openalex.org/I107720978"]},{"raw_affiliation_string":"Information and Computer Science Department, College of Computer Science and Engineering, University of Ha\u2019il, Hail, Ha\u2019il 81481, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006723199","display_name":"Dieaa I. Nassr","orcid":"https://orcid.org/0000-0002-5550-3372"},"institutions":[{"id":"https://openalex.org/I107720978","display_name":"Ain Shams University","ror":"https://ror.org/00cb9w016","country_code":"EG","type":"education","lineage":["https://openalex.org/I107720978"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Dieaa I. Nassr","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt"],"raw_orcid":"https://orcid.org/0000-0002-5550-3372","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt","institution_ids":["https://openalex.org/I107720978"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062403617","display_name":"Mohammed A. Mahdi","orcid":"https://orcid.org/0000-0002-0703-5826"},"institutions":[{"id":"https://openalex.org/I4210088963","display_name":"University of Ha'il","ror":"https://ror.org/013w98a82","country_code":"SA","type":"education","lineage":["https://openalex.org/I4210088963"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Mohammed A. Mahdi","raw_affiliation_strings":["Information and Computer Science Department, College of Computer Science and Engineering, University of Ha\u2019il, Hail, Ha\u2019il 81481, Saudi Arabia","Information and Computer Science Department, College of Computer Science and Engineering, University of Ha'il, Hail, Ha'il 81481, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-0703-5826","affiliations":[{"raw_affiliation_string":"Information and Computer Science Department, College of Computer Science and Engineering, University of Ha\u2019il, Hail, Ha\u2019il 81481, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]},{"raw_affiliation_string":"Information and Computer Science Department, College of Computer Science and Engineering, University of Ha'il, Hail, Ha'il 81481, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024925431","display_name":"Hazem M. Bahig","orcid":"https://orcid.org/0000-0001-9448-6168"},"institutions":[{"id":"https://openalex.org/I107720978","display_name":"Ain Shams University","ror":"https://ror.org/00cb9w016","country_code":"EG","type":"education","lineage":["https://openalex.org/I107720978"]},{"id":"https://openalex.org/I4210088963","display_name":"University of Ha'il","ror":"https://ror.org/013w98a82","country_code":"SA","type":"education","lineage":["https://openalex.org/I4210088963"]}],"countries":["EG","SA"],"is_corresponding":true,"raw_author_name":"Hazem M. Bahig","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt","Information and Computer Science Department, College of Computer Science and Engineering, University of Ha\u2019il, Hail, Ha\u2019il 81481, Saudi Arabia","Information and Computer Science Department, College of Computer Science and Engineering, University of Ha'il, Hail, Ha'il 81481, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-9448-6168","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt","institution_ids":["https://openalex.org/I107720978"]},{"raw_affiliation_string":"Information and Computer Science Department, College of Computer Science and Engineering, University of Ha\u2019il, Hail, Ha\u2019il 81481, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]},{"raw_affiliation_string":"Information and Computer Science Department, College of Computer Science and Engineering, University of Ha'il, Hail, Ha'il 81481, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5024925431"],"corresponding_institution_ids":["https://openalex.org/I107720978","https://openalex.org/I4210088963"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":2.4921,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.9125001,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"14","issue":"9","first_page":"1897","last_page":"1897"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9998999834060669,"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"}},{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.9998000264167786,"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":0.9998000264167786,"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/eulers-totient-function","display_name":"Euler's totient function","score":0.7728196382522583},{"id":"https://openalex.org/keywords/cryptosystem","display_name":"Cryptosystem","score":0.7144831418991089},{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.6543136239051819},{"id":"https://openalex.org/keywords/public-key-cryptography","display_name":"Public-key cryptography","score":0.644931435585022},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.6341787576675415},{"id":"https://openalex.org/keywords/public-key-cryptosystem","display_name":"Public key cryptosystem","score":0.515159010887146},{"id":"https://openalex.org/keywords/fraction","display_name":"Fraction (chemistry)","score":0.5034162402153015},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.46309447288513184},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4617367684841156},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.4343809485435486},{"id":"https://openalex.org/keywords/multi-core-processor","display_name":"Multi-core processor","score":0.4215162396430969},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4198755919933319},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4031675159931183},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.3806307911872864},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.3790530860424042},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.29454702138900757},{"id":"https://openalex.org/keywords/computer-security","display_name":"Computer security","score":0.27896350622177124},{"id":"https://openalex.org/keywords/encryption","display_name":"Encryption","score":0.24797981977462769},{"id":"https://openalex.org/keywords/parallel-computing","display_name":"Parallel computing","score":0.175360769033432}],"concepts":[{"id":"https://openalex.org/C198944804","wikidata":"https://www.wikidata.org/wiki/Q190026","display_name":"Euler's totient function","level":3,"score":0.7728196382522583},{"id":"https://openalex.org/C6295992","wikidata":"https://www.wikidata.org/wiki/Q976521","display_name":"Cryptosystem","level":3,"score":0.7144831418991089},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6543136239051819},{"id":"https://openalex.org/C203062551","wikidata":"https://www.wikidata.org/wiki/Q201339","display_name":"Public-key cryptography","level":3,"score":0.644931435585022},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.6341787576675415},{"id":"https://openalex.org/C2984195096","wikidata":"https://www.wikidata.org/wiki/Q201339","display_name":"Public key cryptosystem","level":4,"score":0.515159010887146},{"id":"https://openalex.org/C149629883","wikidata":"https://www.wikidata.org/wiki/Q660926","display_name":"Fraction (chemistry)","level":2,"score":0.5034162402153015},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.46309447288513184},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4617367684841156},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4343809485435486},{"id":"https://openalex.org/C78766204","wikidata":"https://www.wikidata.org/wiki/Q555032","display_name":"Multi-core processor","level":2,"score":0.4215162396430969},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4198755919933319},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4031675159931183},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.3806307911872864},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.3790530860424042},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.29454702138900757},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.27896350622177124},{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.24797981977462769},{"id":"https://openalex.org/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"score":0.175360769033432},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14091897","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091897","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1897/pdf?version=1663815683","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:177ea26937e24098ad2d6508a0730db2","is_oa":true,"landing_page_url":"https://doaj.org/article/177ea26937e24098ad2d6508a0730db2","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":"Symmetry, Vol 14, Iss 9, p 1897 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/9/1897/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14091897","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14091897","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091897","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1897/pdf?version=1663815683","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Partnerships for the goals","score":0.41999998688697815,"id":"https://metadata.un.org/sdg/17"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4296020464.pdf","grobid_xml":"https://content.openalex.org/works/W4296020464.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W93179768","https://openalex.org/W94103598","https://openalex.org/W180417168","https://openalex.org/W1026355344","https://openalex.org/W1499461890","https://openalex.org/W1557608351","https://openalex.org/W1865276762","https://openalex.org/W1961022628","https://openalex.org/W2019689629","https://openalex.org/W2101040389","https://openalex.org/W2106729090","https://openalex.org/W2149180563","https://openalex.org/W2156666175","https://openalex.org/W2166780626","https://openalex.org/W2757648903","https://openalex.org/W3007149643","https://openalex.org/W3011125593","https://openalex.org/W3021302326","https://openalex.org/W3023478445","https://openalex.org/W3026505891","https://openalex.org/W3119277422","https://openalex.org/W3210063686","https://openalex.org/W4205359176","https://openalex.org/W4225725352","https://openalex.org/W4232836212","https://openalex.org/W4280634534","https://openalex.org/W4285105943","https://openalex.org/W4399612325","https://openalex.org/W6676051169","https://openalex.org/W6787995931","https://openalex.org/W6810794339"],"related_works":["https://openalex.org/W4385402977","https://openalex.org/W1806453146","https://openalex.org/W1977146020","https://openalex.org/W2120749142","https://openalex.org/W2077900604","https://openalex.org/W2157130609","https://openalex.org/W807354026","https://openalex.org/W2100244348","https://openalex.org/W2071811380","https://openalex.org/W2201391556"],"abstract_inverted_index":{"The":[0,73,120],"RSA":[1,17,35,66,130],"(Rivest\u2013Shamir\u2013Adleman)":[2],"asymmetric-key":[3],"cryptosystem":[4],"is":[5,36,48,79,116,143,150],"widely":[6],"used":[7,102,154],"for":[8,165],"encryptions":[9],"and":[10,20,70,87,106,169,181],"digital":[11],"signatures.":[12],"Let":[13],"(n,e)":[14],"be":[15,22,101],"the":[16,23,32,39,76,97,111,135,139,159],"public":[18],"key":[19,26,41,141],"d":[21,42,47,162],"corresponding":[24],"private":[25,28,40,62,112,126,140],"(or":[27],"exponent).":[29],"One":[30],"of":[31,75,123,138,161],"attacks":[33,128,173],"on":[34,65,129],"to":[37,58,80,92,109,157],"find":[38,81],"using":[43,67],"continued":[44,68,104,146],"fractions":[45,69],"when":[46],"small.":[49],"In":[50],"this":[51],"paper,":[52],"we":[53,133,153],"present":[54],"a":[55,60,90],"new":[56],"technique":[57,78],"improve":[59],"small":[61,125],"exponent":[63,127],"attack":[64],"multicore":[71,107],"systems.":[72],"idea":[74],"proposed":[77],"an":[82],"interval":[83,98],"that":[84,99,132,142],"contains":[85],"\u03d5(n),":[86],"then":[88],"propose":[89],"method":[91],"generate":[93],"different":[94],"points":[95],"in":[96,174],"can":[100],"by":[103,145,163],"fraction":[105],"systems":[108],"recover":[110],"key,":[113],"where":[114],"\u03d5":[115],"Euler\u2019s":[117],"totient":[118],"function.":[119],"practical":[121],"results":[122],"three":[124],"show":[131],"extended":[134],"previous":[136],"bound":[137,160],"discovered":[144],"fractions.":[147],"When":[148],"n":[149],"1024":[151],"bits,":[152],"20":[155],"cores":[156],"extend":[158],"0.016":[164],"de":[166],"Weger,":[167],"Maitra-Sarkar,":[168],"Nassr":[170],"et":[171],"al.":[172],"average":[175],"times":[176],"7.67":[177],"h,":[178,180],"2.7":[179],"44":[182],"min,":[183],"respectively.":[184]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}
