{"id":"https://openalex.org/W2198520256","doi":null,"title":"Discrete Logarithms on Elliptic Curves","display_name":"Discrete Logarithms on Elliptic Curves","publication_year":2011,"publication_date":"2011-01-01","ids":{"openalex":"https://openalex.org/W2198520256","mag":"2198520256"},"language":"en","primary_location":{"id":"pmh:oai:scholar.rose-hulman.edu:math_mstr-1024","doi":null,"is_oa":true,"landing_page_url":"https://scholar.rose-hulman.edu/math_mstr/25","pdf_url":"https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1024&context=math_mstr","source":{"id":"https://openalex.org/S4377196619","display_name":"Rose-Hulman Scholar (Rose\u2013Hulman Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I192578771","host_organization_name":"Rose\u2013Hulman Institute of Technology","host_organization_lineage":["https://openalex.org/I192578771"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematical Sciences Technical Reports (MSTR)","raw_type":"text"},"volume":null,"type":"article","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1024&context=math_mstr","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023484994","display_name":"Aaron Blumenfeld","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I5388228","display_name":"University of Rochester","ror":"https://ror.org/022kthw22","country_code":"US","type":"education","lineage":["https://openalex.org/I5388228"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Aaron Blumenfeld","raw_affiliation_strings":["University of Rochester"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Rochester","institution_ids":["https://openalex.org/I5388228"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5023484994"],"corresponding_institution_ids":["https://openalex.org/I5388228"],"apc_list":null,"apc_paid":null,"fwci":0.5457,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.75731252,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":87,"max":95},"biblio":{"volume":"12","issue":"1","first_page":"3","last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9810000061988831,"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":0.9810000061988831,"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/T11130","display_name":"Coding theory and cryptography","score":0.006500000134110451,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10237","display_name":"Cryptography and Data Security","score":0.003700000001117587,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/elliptic-curve-discrete-logarithm-problem","display_name":"elliptic curve discrete logarithm problem","score":0.9290000200271606},{"id":"https://openalex.org/keywords/elliptic-curve-cryptography","display_name":"elliptic curve cryptography","score":0.9929999709129333},{"id":"https://openalex.org/keywords/diffie-hellman-key-exchange","display_name":"Diffie-Hellman key exchange","score":0.925000011920929},{"id":"https://openalex.org/keywords/elgamal-cryptosystem","display_name":"ElGamal encryption","score":0.9909999966621399},{"id":"https://openalex.org/keywords/functional-graphs","display_name":"functional graphs","score":0.7839999794960022},{"id":"https://openalex.org/keywords/finite-fields","display_name":"finite fields","score":0.5149999856948853},{"id":"https://openalex.org/keywords/cryptographic-hardness","display_name":"cryptographic hardness","score":0.35499998927116394}],"concepts":[{"id":"https://openalex.org/C173259116","wikidata":"https://www.wikidata.org/wiki/Q864003","display_name":"Discrete logarithm","level":4,"score":0.8934492468833923},{"id":"https://openalex.org/C11045955","wikidata":"https://www.wikidata.org/wiki/Q1162410","display_name":"ElGamal encryption","level":4,"score":0.8184909820556641},{"id":"https://openalex.org/C167615521","wikidata":"https://www.wikidata.org/wiki/Q1048911","display_name":"Elliptic curve cryptography","level":4,"score":0.7705212831497192},{"id":"https://openalex.org/C108277079","wikidata":"https://www.wikidata.org/wiki/Q7233576","display_name":"Post-quantum cryptography","level":4,"score":0.6432297229766846},{"id":"https://openalex.org/C68782407","wikidata":"https://www.wikidata.org/wiki/Q5177153","display_name":"Counting points on elliptic curves","level":5,"score":0.6398349404335022},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.6243297457695007},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5461082458496094},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.541700541973114},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.5384812951087952},{"id":"https://openalex.org/C166780011","wikidata":"https://www.wikidata.org/wiki/Q3005926","display_name":"Hyperelliptic curve cryptography","level":5,"score":0.49369415640830994},{"id":"https://openalex.org/C6295992","wikidata":"https://www.wikidata.org/wiki/Q976521","display_name":"Cryptosystem","level":3,"score":0.4872259199619293},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48510777950286865},{"id":"https://openalex.org/C99674996","wikidata":"https://www.wikidata.org/wiki/Q1414155","display_name":"Key exchange","level":4,"score":0.436750590801239},{"id":"https://openalex.org/C121444067","wikidata":"https://www.wikidata.org/wiki/Q2835817","display_name":"Schoof's algorithm","level":4,"score":0.43310368061065674},{"id":"https://openalex.org/C104993295","wikidata":"https://www.wikidata.org/wiki/Q17023345","display_name":"Hessian form of an elliptic curve","level":5,"score":0.42234593629837036},{"id":"https://openalex.org/C203062551","wikidata":"https://www.wikidata.org/wiki/Q201339","display_name":"Public-key cryptography","level":3,"score":0.36872464418411255},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26985517144203186},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2615742087364197},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.21693387627601624},{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.15428411960601807},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12317752838134766},{"id":"https://openalex.org/C197875053","wikidata":"https://www.wikidata.org/wiki/Q7269266","display_name":"Quarter period","level":3,"score":0.08521011471748352},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.06397664546966553}],"mesh":[],"locations_count":5,"locations":[{"id":"pmh:oai:scholar.rose-hulman.edu:math_mstr-1024","doi":null,"is_oa":true,"landing_page_url":"https://scholar.rose-hulman.edu/math_mstr/25","pdf_url":"https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1024&context=math_mstr","source":{"id":"https://openalex.org/S4377196619","display_name":"Rose-Hulman Scholar (Rose\u2013Hulman Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I192578771","host_organization_name":"Rose\u2013Hulman Institute of Technology","host_organization_lineage":["https://openalex.org/I192578771"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematical Sciences Technical Reports (MSTR)","raw_type":"text"},{"id":"pmh:oai:scholar.rose-hulman.edu:rhumj-1101","doi":null,"is_oa":true,"landing_page_url":"https://scholar.rose-hulman.edu/rhumj/vol12/iss1/3","pdf_url":"https://scholar.rose-hulman.edu/rhumj/vol12/iss1/3","source":{"id":"https://openalex.org/S4377196619","display_name":"Rose-Hulman Scholar (Rose\u2013Hulman Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I192578771","host_organization_name":"Rose\u2013Hulman Institute of Technology","host_organization_lineage":["https://openalex.org/I192578771"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Rose-Hulman Undergraduate Mathematics Journal","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.188.1798","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.188.1798","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.rose-hulman.edu/math/MSTR/MSTRpubs/2010/RHIT-MSTR-2010-04.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.641.3522","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.641.3522","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.rose-hulman.edu/mathjournal/archives/2011/vol12-n1/paper3/v12n1-3pd.pdf","raw_type":"text"},{"id":"mag:2198520256","doi":null,"is_oa":false,"landing_page_url":"https://www.rose-hulman.edu/mathjournal/archives/2011/vol12-n1/paper3/v12n1-3pd.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S2764715623","display_name":"Rose\u2013Hulman Undergraduate Mathematics Journal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":"Rose\u2013Hulman Undergraduate Mathematics Journal","raw_type":null}],"best_oa_location":{"id":"pmh:oai:scholar.rose-hulman.edu:math_mstr-1024","doi":null,"is_oa":true,"landing_page_url":"https://scholar.rose-hulman.edu/math_mstr/25","pdf_url":"https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1024&context=math_mstr","source":{"id":"https://openalex.org/S4377196619","display_name":"Rose-Hulman Scholar (Rose\u2013Hulman Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I192578771","host_organization_name":"Rose\u2013Hulman Institute of Technology","host_organization_lineage":["https://openalex.org/I192578771"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematical Sciences Technical Reports (MSTR)","raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2198520256.pdf","grobid_xml":"https://content.openalex.org/works/W2198520256.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W1498993177","https://openalex.org/W1547407391","https://openalex.org/W1857130924","https://openalex.org/W1957940770","https://openalex.org/W2127703673","https://openalex.org/W2226303357","https://openalex.org/W2400908863"],"related_works":["https://openalex.org/W7097336165","https://openalex.org/W2735610865","https://openalex.org/W2378553768","https://openalex.org/W2046569422","https://openalex.org/W7046525709","https://openalex.org/W4298312512","https://openalex.org/W4236294898","https://openalex.org/W2116283421","https://openalex.org/W2349430786","https://openalex.org/W109935170"],"abstract_inverted_index":{"Abstract.":[0],"Cryptographic":[1],"protocols":[2],"often":[3],"make":[4],"use":[5],"of":[6,10,29,78],"the":[7,11,36,95,106,124,129],"inherent":[8],"hardness":[9,28],"classical":[12],"discrete":[13,52,107],"logarithm":[14,53,108],"problem,":[15],"which":[16,89],"is":[17,49,75,85],"to":[18],"solve":[19,58],"gx":[20],"\u2261":[21],"y":[22],"(mod":[23],"p)":[24],"for":[25,62,71,109,114,131],"x.":[26],"The":[27],"this":[30,102],"problem":[31,54],"has":[32],"been":[33,69],"exploited":[34],"in":[35,43],"Diffie-Hellman":[37,65],"key":[38],"exchange,":[39],"as":[40,42,46],"well":[41],"cryptosystems":[44],"such":[45],"ElGamal.":[47],"There":[48,74],"a":[50,120],"similar":[51],"on":[55],"elliptic":[56,72,82,110],"curves:":[57],"kB":[59],"=":[60],"P":[61],"k.":[63],"Therefore,":[64],"and":[66,122,128],"ElGamal":[67],"have":[68],"adapted":[70],"curves.":[73],"an":[76],"abundance":[77],"evidence":[79],"suggesting":[80],"that":[81,91],"curve":[83],"cryptography":[84],"even":[86],"more":[87],"secure,":[88],"means":[90],"we":[92,104],"can":[93],"obtain":[94],"same":[96],"security":[97],"with":[98],"fewer":[99],"bits.":[100],"In":[101],"paper,":[103],"investigate":[105],"curves":[111],"over":[112],"Fp":[113],"p":[115],"\u2265":[116],"5":[117],"by":[118],"constructing":[119],"function":[121],"considering":[123],"induced":[125],"functional":[126],"graph":[127],"implications":[130],"cryptography.":[132]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2016-06-24T00:00:00"}