{"id":"https://openalex.org/W2028320175","doi":"https://doi.org/10.1090/s0025-5718-05-01768-0","title":"Some heuristics and results for small cycles of the discrete logarithm","display_name":"Some heuristics and results for small cycles of the discrete logarithm","publication_year":2005,"publication_date":"2005-06-28","ids":{"openalex":"https://openalex.org/W2028320175","doi":"https://doi.org/10.1090/s0025-5718-05-01768-0","mag":"2028320175"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-05-01768-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01768-0","pdf_url":"https://www.ams.org/mcom/2006-75-253/S0025-5718-05-01768-0/S0025-5718-05-01768-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.ams.org/mcom/2006-75-253/S0025-5718-05-01768-0/S0025-5718-05-01768-0.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043053001","display_name":"Joshua Holden","orcid":"https://orcid.org/0000-0003-2847-3777"},"institutions":[{"id":"https://openalex.org/I192578771","display_name":"Rose\u2013Hulman Institute of Technology","ror":"https://ror.org/00mp6e841","country_code":"US","type":"education","lineage":["https://openalex.org/I192578771"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Joshua Holden","raw_affiliation_strings":["Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana, 47803-3999"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana, 47803-3999","institution_ids":["https://openalex.org/I192578771"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009645792","display_name":"Pieter Moree","orcid":"https://orcid.org/0000-0002-5318-2587"},"institutions":[{"id":"https://openalex.org/I149899117","display_name":"Max Planck Society","ror":"https://ror.org/01hhn8329","country_code":"DE","type":"nonprofit","lineage":["https://openalex.org/I149899117"]},{"id":"https://openalex.org/I4210119712","display_name":"Max Planck Institute for Mathematics","ror":"https://ror.org/02dh8ja68","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210119712"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Pieter Moree","raw_affiliation_strings":["Max-Planck-Institut f\u00c3\u00bcr Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany","Max\u2010Planck\u2010Institut f\u00fcr Mathematik Vivatsgasse 7 D-53111 Bonn Germany"],"affiliations":[{"raw_affiliation_string":"Max-Planck-Institut f\u00c3\u00bcr Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany","institution_ids":["https://openalex.org/I4210119712"]},{"raw_affiliation_string":"Max\u2010Planck\u2010Institut f\u00fcr Mathematik Vivatsgasse 7 D-53111 Bonn Germany","institution_ids":["https://openalex.org/I149899117"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5043053001"],"corresponding_institution_ids":["https://openalex.org/I192578771"],"apc_list":null,"apc_paid":null,"fwci":0.5629,"has_fulltext":true,"cited_by_count":25,"citation_normalized_percentile":{"value":0.62278481,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"75","issue":"253","first_page":"419","last_page":"449"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.996999979019165,"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":0.996999979019165,"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/T11130","display_name":"Coding theory and cryptography","score":0.9922999739646912,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9861000180244446,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.6460993885993958},{"id":"https://openalex.org/keywords/annotation","display_name":"Annotation","score":0.5922892093658447},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.48063769936561584},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.44291484355926514},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4194786548614502},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.34948503971099854},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.0658479630947113}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6460993885993958},{"id":"https://openalex.org/C2776321320","wikidata":"https://www.wikidata.org/wiki/Q857525","display_name":"Annotation","level":2,"score":0.5922892093658447},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.48063769936561584},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.44291484355926514},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4194786548614502},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.34948503971099854},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0658479630947113},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1090/s0025-5718-05-01768-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01768-0","pdf_url":"https://www.ams.org/mcom/2006-75-253/S0025-5718-05-01768-0/S0025-5718-05-01768-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:scholar.rose-hulman.edu:math_fac-1104","is_oa":false,"landing_page_url":"https://scholar.rose-hulman.edu/math_fac/194","pdf_url":null,"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,"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":"Faculty Publications - Mathematics","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.240.4510","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.240.4510","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0401013v2.pdf","raw_type":"text"},{"id":"pmh:oai:dare.uva.nl:openaire_cris_publications/1449421d-761f-4fe9-87b3-46999d2fce2b","is_oa":false,"landing_page_url":"https://handle.uba.uva.nl/personal/pure/en/publications/some-heuristics-and-results-for-small-cycles-of-the-discrete-logarithm(1449421d-761f-4fe9-87b3-46999d2fce2b).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400088","display_name":"UvA-DARE (University of Amsterdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887064364","host_organization_name":"University of Amsterdam","host_organization_lineage":["https://openalex.org/I887064364"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Holden, J & Moree, P 2006, 'Some heuristics and results for small cycles of the discrete logarithm.', Mathematics of Computation, vol. 75, no. 253, pp. 419-449. < http://www.ams.org/mcom >","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:dare.uva.nl:publications/1449421d-761f-4fe9-87b3-46999d2fce2b","is_oa":false,"landing_page_url":"http://dare.uva.nl/personal/pure/en/publications/some-heuristics-and-results-for-small-cycles-of-the-discrete-logarithm(1449421d-761f-4fe9-87b3-46999d2fce2b).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400088","display_name":"UvA-DARE (University of Amsterdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887064364","host_organization_name":"University of Amsterdam","host_organization_lineage":["https://openalex.org/I887064364"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Holden, J & Moree, P 2006, 'Some heuristics and results for small cycles of the discrete logarithm.', Mathematics of Computation, vol. 75, no. 253, pp. 419-449. < http://www.ams.org/mcom >","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:uvapub:oai:dare.uva.nl:publications/1449421d-761f-4fe9-87b3-46999d2fce2b","is_oa":false,"landing_page_url":"https://dare.uva.nl/personal/pure/en/publications/some-heuristics-and-results-for-small-cycles-of-the-discrete-logarithm(1449421d-761f-4fe9-87b3-46999d2fce2b).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics of Computation, 75(253), 419 - 449. American Mathematical Society","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-05-01768-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01768-0","pdf_url":"https://www.ams.org/mcom/2006-75-253/S0025-5718-05-01768-0/S0025-5718-05-01768-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321008","display_name":"Universiteit van Amsterdam","ror":"https://ror.org/04dkp9463"},{"id":"https://openalex.org/F4320321800","display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","ror":"https://ror.org/04jsz6e67"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2028320175.pdf","grobid_xml":"https://content.openalex.org/works/W2028320175.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W1555475515","https://openalex.org/W1664064177","https://openalex.org/W1788867460","https://openalex.org/W1965046890","https://openalex.org/W2002645954","https://openalex.org/W2028962322","https://openalex.org/W2046465034","https://openalex.org/W2059115081","https://openalex.org/W2101230898","https://openalex.org/W2104209721","https://openalex.org/W2556959701","https://openalex.org/W2569778053","https://openalex.org/W2774692261"],"related_works":["https://openalex.org/W151193258","https://openalex.org/W1892467659","https://openalex.org/W1529400504","https://openalex.org/W2348710178","https://openalex.org/W3201926073","https://openalex.org/W1871911958","https://openalex.org/W73525116","https://openalex.org/W2349865494","https://openalex.org/W2998403542","https://openalex.org/W2808586768"],"abstract_inverted_index":{"Brizolis":[0],"asked":[1],"the":[2,65,115,167,182],"question:":[3],"does":[4],"every":[5],"prime":[6],"<inline-formula":[7,22,48,70,124,137],"content-type=\"math/mathml\">":[8,23,49,71,125,138],"<mml:math":[9,24,50,72,126,139],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[10,25,51,73,127,140],"alttext=\"p\">":[11],"<mml:semantics>":[12,31,53,75,129,142],"<mml:mi>p</mml:mi>":[13],"<mml:annotation":[14,41,55,77,131,144],"encoding=\"application/x-tex\">p</mml:annotation>":[15],"</mml:semantics>":[16,43,57,79,133,146],"</mml:math>":[17,44,58,80,134,147],"</inline-formula>":[18,45,59,81,135,148],"have":[19],"a":[20,61],"pair":[21],"alttext=\"left-parenthesis":[26],"g":[27],"comma":[28],"h":[29],"right-parenthesis\">":[30],"<mml:mrow>":[32],"<mml:mo":[33,38],"stretchy=\"false\">(</mml:mo>":[34],"<mml:mi>g</mml:mi>":[35,76,130],"<mml:mo>,</mml:mo>":[36],"<mml:mi>h</mml:mi>":[37,54,143],"stretchy=\"false\">)</mml:mo>":[39],"</mml:mrow>":[40],"encoding=\"application/x-tex\">(g,h)</mml:annotation>":[42],"such":[46,118],"that":[47],"alttext=\"h\">":[52,141],"encoding=\"application/x-tex\">h</mml:annotation>":[56,145],"is":[60],"fixed":[62,95],"point":[63],"for":[64,113,160],"discrete":[66],"logarithm":[67],"with":[68],"base":[69],"alttext=\"g\">":[74,128],"encoding=\"application/x-tex\">g</mml:annotation>":[78,132],"?":[82],"The":[83],"first":[84],"author":[85],"previously":[86],"extended":[87],"this":[88,151],"question":[89],"to":[90],"ask":[91],"about":[92],"not":[93],"only":[94],"points":[96],"but":[97],"also":[98,171],"two-cycles,":[99],"and":[100,111,136,157,176],"gave":[101],"heuristics":[102,156],"(building":[103],"on":[104,123,166],"work":[105],"of":[106,117,162,181],"Zhang,":[107],"Cobeli,":[108],"Zaharescu,":[109],"Campbell,":[110],"Pomerance)":[112],"estimating":[114],"number":[116],"pairs":[119],"given":[120],"certain":[121],"conditions":[122],".":[149],"In":[150],"paper":[152],"we":[153],"extend":[154],"these":[155],"prove":[158,177],"results":[159],"some":[161,173,178],"them,":[163],"building":[164],"again":[165],"aforementioned":[168],"work.":[169],"We":[170],"make":[172],"new":[174],"conjectures":[175],"average":[179],"versions":[180],"results.":[183]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2025-10-10T00:00:00"}
