{"id":"https://openalex.org/W128615905","doi":"https://doi.org/10.1007/bfb0054116","title":"Equivalence of counting the number of points on elliptic curve over the ring Zn and factoring n","display_name":"Equivalence of counting the number of points on elliptic curve over the ring Zn and factoring n","publication_year":1998,"publication_date":"1998-01-01","ids":{"openalex":"https://openalex.org/W128615905","doi":"https://doi.org/10.1007/bfb0054116","mag":"128615905"},"language":"en","primary_location":{"id":"doi:10.1007/bfb0054116","is_oa":true,"landing_page_url":"https://doi.org/10.1007/bfb0054116","pdf_url":"https://link.springer.com/content/pdf/10.1007/BFb0054116.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":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://link.springer.com/content/pdf/10.1007/BFb0054116.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000629127","display_name":"Noboru Kunihiro","orcid":"https://orcid.org/0000-0003-1822-7476"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Noboru Kunihiro","raw_affiliation_strings":["NTT Communication Science Laboratories, 2-4, Hikaridai, Seika-cho, Soraku-gun, 619-0237, Kyoto, Japan","NTT Communication Science Laboratories"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"NTT Communication Science Laboratories, 2-4, Hikaridai, Seika-cho, Soraku-gun, 619-0237, Kyoto, Japan","institution_ids":[]},{"raw_affiliation_string":"NTT Communication Science Laboratories","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010911834","display_name":"Kenji Koyama","orcid":"https://orcid.org/0000-0002-1467-7050"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kenji Koyama","raw_affiliation_strings":["NTT Communication Science Laboratories, 2-4, Hikaridai, Seika-cho, Soraku-gun, 619-0237, Kyoto, Japan","NTT Communication Science Laboratories"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"NTT Communication Science Laboratories, 2-4, Hikaridai, Seika-cho, Soraku-gun, 619-0237, Kyoto, Japan","institution_ids":[]},{"raw_affiliation_string":"NTT Communication Science Laboratories","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":1.9793,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.83963134,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":null,"issue":null,"first_page":"47","last_page":"58"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9993000030517578,"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":0.9993000030517578,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11166","display_name":"Analytic Number Theory Research","score":0.9914000034332275,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/elliptic-curve","display_name":"Elliptic curve","score":0.7405111193656921},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.6308454871177673},{"id":"https://openalex.org/keywords/discrete-logarithm","display_name":"Discrete logarithm","score":0.6175858378410339},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5876367092132568},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5499170422554016},{"id":"https://openalex.org/keywords/schoofs-algorithm","display_name":"Schoof's algorithm","score":0.5464698672294617},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5460238456726074},{"id":"https://openalex.org/keywords/factoring","display_name":"Factoring","score":0.5381056070327759},{"id":"https://openalex.org/keywords/supersingular-elliptic-curve","display_name":"Supersingular elliptic curve","score":0.5229723453521729},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.5196189284324646},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4678179919719696},{"id":"https://openalex.org/keywords/counting-points-on-elliptic-curves","display_name":"Counting points on elliptic curves","score":0.4669448137283325},{"id":"https://openalex.org/keywords/elliptic-curve-point-multiplication","display_name":"Elliptic curve point multiplication","score":0.414201021194458},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3773215115070343},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.28736358880996704},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.27020013332366943},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1658496856689453},{"id":"https://openalex.org/keywords/quarter-period","display_name":"Quarter period","score":0.12864023447036743},{"id":"https://openalex.org/keywords/public-key-cryptography","display_name":"Public-key cryptography","score":0.09047380089759827}],"concepts":[{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.7405111193656921},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6308454871177673},{"id":"https://openalex.org/C173259116","wikidata":"https://www.wikidata.org/wiki/Q864003","display_name":"Discrete logarithm","level":4,"score":0.6175858378410339},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5876367092132568},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5499170422554016},{"id":"https://openalex.org/C121444067","wikidata":"https://www.wikidata.org/wiki/Q2835817","display_name":"Schoof's algorithm","level":4,"score":0.5464698672294617},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5460238456726074},{"id":"https://openalex.org/C177225278","wikidata":"https://www.wikidata.org/wiki/Q192674","display_name":"Factoring","level":2,"score":0.5381056070327759},{"id":"https://openalex.org/C157567686","wikidata":"https://www.wikidata.org/wiki/Q7644139","display_name":"Supersingular elliptic curve","level":3,"score":0.5229723453521729},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.5196189284324646},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4678179919719696},{"id":"https://openalex.org/C68782407","wikidata":"https://www.wikidata.org/wiki/Q5177153","display_name":"Counting points on elliptic curves","level":5,"score":0.4669448137283325},{"id":"https://openalex.org/C117121985","wikidata":"https://www.wikidata.org/wiki/Q78380999","display_name":"Elliptic curve point multiplication","level":3,"score":0.414201021194458},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3773215115070343},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28736358880996704},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27020013332366943},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1658496856689453},{"id":"https://openalex.org/C197875053","wikidata":"https://www.wikidata.org/wiki/Q7269266","display_name":"Quarter period","level":3,"score":0.12864023447036743},{"id":"https://openalex.org/C203062551","wikidata":"https://www.wikidata.org/wiki/Q201339","display_name":"Public-key cryptography","level":3,"score":0.09047380089759827},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C148730421","wikidata":"https://www.wikidata.org/wiki/Q141090","display_name":"Encryption","level":2,"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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/bfb0054116","is_oa":true,"landing_page_url":"https://doi.org/10.1007/bfb0054116","pdf_url":"https://link.springer.com/content/pdf/10.1007/BFb0054116.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"}],"best_oa_location":{"id":"doi:10.1007/bfb0054116","is_oa":true,"landing_page_url":"https://doi.org/10.1007/bfb0054116","pdf_url":"https://link.springer.com/content/pdf/10.1007/BFb0054116.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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W128615905.pdf","grobid_xml":"https://content.openalex.org/works/W128615905.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1486898453","https://openalex.org/W1515644111","https://openalex.org/W1557014463","https://openalex.org/W1567892694","https://openalex.org/W1982647225","https://openalex.org/W2003736153","https://openalex.org/W2005317356","https://openalex.org/W2012857639","https://openalex.org/W2016718197","https://openalex.org/W2036378739","https://openalex.org/W2075252632","https://openalex.org/W2128718974","https://openalex.org/W4242424230","https://openalex.org/W4245047333","https://openalex.org/W6630669390","https://openalex.org/W6651121328"],"related_works":["https://openalex.org/W3138493780","https://openalex.org/W3043110944","https://openalex.org/W203192885","https://openalex.org/W2548082687","https://openalex.org/W3172172454","https://openalex.org/W2559355736","https://openalex.org/W1921672120","https://openalex.org/W1546865581","https://openalex.org/W1851253658","https://openalex.org/W2613818326"],"abstract_inverted_index":null,"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}