{"id":"https://openalex.org/W2167334956","doi":"https://doi.org/10.1007/978-3-642-04159-4_20","title":"The Elliptic Curve Discrete Logarithm Problem and Equivalent Hard Problems for Elliptic Divisibility Sequences","display_name":"The Elliptic Curve Discrete Logarithm Problem and Equivalent Hard Problems for Elliptic Divisibility Sequences","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2167334956","doi":"https://doi.org/10.1007/978-3-642-04159-4_20","mag":"2167334956"},"language":"en","primary_location":{"id":"doi:10.1007/978-3-642-04159-4_20","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-642-04159-4_20","pdf_url":"https://link.springer.com/content/pdf/10.1007/978-3-642-04159-4_20.pdf","source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"type":"book-chapter","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://link.springer.com/content/pdf/10.1007/978-3-642-04159-4_20.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002850656","display_name":"Kristin Lauter","orcid":"https://orcid.org/0000-0002-1320-696X"},"institutions":[{"id":"https://openalex.org/I1290206253","display_name":"Microsoft (United States)","ror":"https://ror.org/00d0nc645","country_code":"US","type":"company","lineage":["https://openalex.org/I1290206253"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kristin E. Lauter","raw_affiliation_strings":["Microsoft Research, One Microsoft Way, Redmond, WA, 98052"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Microsoft Research, One Microsoft Way, Redmond, WA, 98052","institution_ids":["https://openalex.org/I1290206253"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062267978","display_name":"Katherine E. Stange","orcid":"https://orcid.org/0000-0003-2294-0397"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Katherine E. Stange","raw_affiliation_strings":["Department of Mathematics, Harvard University, Cambridge, MA, 02138"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harvard University, Cambridge, MA, 02138","institution_ids":["https://openalex.org/I136199984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":3.2004,"has_fulltext":true,"cited_by_count":28,"citation_normalized_percentile":{"value":0.92119779,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":null,"issue":null,"first_page":"309","last_page":"327"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9976999759674072,"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":0.9976999759674072,"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":0.9919000267982483,"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":0.988099992275238,"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/divisibility-rule","display_name":"Divisibility rule","score":0.8549596071243286},{"id":"https://openalex.org/keywords/discrete-logarithm","display_name":"Discrete logarithm","score":0.7757298946380615},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.7597448825836182},{"id":"https://openalex.org/keywords/elliptic-curve","display_name":"Elliptic curve","score":0.6566978096961975},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5940302014350891},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.591419517993927},{"id":"https://openalex.org/keywords/elliptic-curve-point-multiplication","display_name":"Elliptic curve point multiplication","score":0.5777996778488159},{"id":"https://openalex.org/keywords/supersingular-elliptic-curve","display_name":"Supersingular elliptic curve","score":0.5587627291679382},{"id":"https://openalex.org/keywords/tripling-oriented-doche\u2013icart\u2013kohel-curve","display_name":"Tripling-oriented Doche\u2013Icart\u2013Kohel curve","score":0.5023484230041504},{"id":"https://openalex.org/keywords/schoofs-algorithm","display_name":"Schoof's algorithm","score":0.4826103448867798},{"id":"https://openalex.org/keywords/hessian-form-of-an-elliptic-curve","display_name":"Hessian form of an elliptic curve","score":0.4403315484523773},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4021151065826416},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3509747385978699},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3285256624221802},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23579248785972595},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19356757402420044},{"id":"https://openalex.org/keywords/quarter-period","display_name":"Quarter period","score":0.15069013833999634},{"id":"https://openalex.org/keywords/public-key-cryptography","display_name":"Public-key cryptography","score":0.054045170545578}],"concepts":[{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.8549596071243286},{"id":"https://openalex.org/C173259116","wikidata":"https://www.wikidata.org/wiki/Q864003","display_name":"Discrete logarithm","level":4,"score":0.7757298946380615},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.7597448825836182},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.6566978096961975},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5940302014350891},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.591419517993927},{"id":"https://openalex.org/C117121985","wikidata":"https://www.wikidata.org/wiki/Q78380999","display_name":"Elliptic curve point multiplication","level":3,"score":0.5777996778488159},{"id":"https://openalex.org/C157567686","wikidata":"https://www.wikidata.org/wiki/Q7644139","display_name":"Supersingular elliptic curve","level":3,"score":0.5587627291679382},{"id":"https://openalex.org/C37055786","wikidata":"https://www.wikidata.org/wiki/Q7843607","display_name":"Tripling-oriented Doche\u2013Icart\u2013Kohel curve","level":5,"score":0.5023484230041504},{"id":"https://openalex.org/C121444067","wikidata":"https://www.wikidata.org/wiki/Q2835817","display_name":"Schoof's algorithm","level":4,"score":0.4826103448867798},{"id":"https://openalex.org/C104993295","wikidata":"https://www.wikidata.org/wiki/Q17023345","display_name":"Hessian form of an elliptic curve","level":5,"score":0.4403315484523773},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4021151065826416},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3509747385978699},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3285256624221802},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23579248785972595},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19356757402420044},{"id":"https://openalex.org/C197875053","wikidata":"https://www.wikidata.org/wiki/Q7269266","display_name":"Quarter period","level":3,"score":0.15069013833999634},{"id":"https://openalex.org/C203062551","wikidata":"https://www.wikidata.org/wiki/Q201339","display_name":"Public-key cryptography","level":3,"score":0.054045170545578},{"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/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/978-3-642-04159-4_20","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-642-04159-4_20","pdf_url":"https://link.springer.com/content/pdf/10.1007/978-3-642-04159-4_20.pdf","source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},{"id":"pmh:oai:arXiv.org:0803.0728","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0803.0728","pdf_url":"https://arxiv.org/pdf/0803.0728","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"doi:10.1007/978-3-642-04159-4_20","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-642-04159-4_20","pdf_url":"https://link.springer.com/content/pdf/10.1007/978-3-642-04159-4_20.pdf","source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"sustainable_development_goals":[{"score":0.4300000071525574,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2167334956.pdf","grobid_xml":"https://content.openalex.org/works/W2167334956.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W239595814","https://openalex.org/W1482385240","https://openalex.org/W1486898453","https://openalex.org/W1562630396","https://openalex.org/W1592482282","https://openalex.org/W1660562555","https://openalex.org/W1981455414","https://openalex.org/W1983731997","https://openalex.org/W2007439687","https://openalex.org/W2035285509","https://openalex.org/W2065774251","https://openalex.org/W2073145552","https://openalex.org/W2091908303","https://openalex.org/W2094556471","https://openalex.org/W2111310412","https://openalex.org/W2121371977","https://openalex.org/W2330044538","https://openalex.org/W2336582372","https://openalex.org/W2408156040","https://openalex.org/W2490684999","https://openalex.org/W2501812802","https://openalex.org/W2950961150","https://openalex.org/W3105909371","https://openalex.org/W4252543992","https://openalex.org/W4253109663"],"related_works":["https://openalex.org/W1921672120","https://openalex.org/W2121110380","https://openalex.org/W1851253658","https://openalex.org/W2548082687","https://openalex.org/W2355104288","https://openalex.org/W3138493780","https://openalex.org/W2071707911","https://openalex.org/W3211097549","https://openalex.org/W1976871117","https://openalex.org/W3003881355"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-19T17:40:00.097472","created_date":"2025-10-10T00:00:00"}