{"id":"https://openalex.org/W1633446687","doi":"https://doi.org/10.1090/mcom/2978","title":"The self-power map and collecting all residue classes","display_name":"The self-power map and collecting all residue classes","publication_year":2015,"publication_date":"2015-05-12","ids":{"openalex":"https://openalex.org/W1633446687","doi":"https://doi.org/10.1090/mcom/2978","mag":"1633446687"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/2978","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/2978","pdf_url":"https://www.ams.org/mcom/2016-85-297/S0025-5718-2015-02978-0/S0025-5718-2015-02978-0.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":null,"license_id":null,"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":"bronze","oa_url":"https://www.ams.org/mcom/2016-85-297/S0025-5718-2015-02978-0/S0025-5718-2015-02978-0.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020582091","display_name":"Catalina Anghel","orcid":"https://orcid.org/0000-0001-7569-5319"},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Catalina Anghel","raw_affiliation_strings":["Department of Mathematics, University of Toronto, 40 St. George St., Toronto, ON M5S 2E4, Canada","Department of Mathematics, University of Toronto, 40 St George St., Toronto, ON M5S 2E4, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Toronto, 40 St. George St., Toronto, ON M5S 2E4, Canada","institution_ids":["https://openalex.org/I185261750"]},{"raw_affiliation_string":"Department of Mathematics, University of Toronto, 40 St George St., Toronto, ON M5S 2E4, Canada","institution_ids":["https://openalex.org/I185261750"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5020582091"],"corresponding_institution_ids":["https://openalex.org/I185261750"],"apc_list":null,"apc_paid":null,"fwci":1.3096,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.85414242,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"85","issue":"297","first_page":"379","last_page":"399"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","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/T10237","display_name":"Cryptography and Data Security","score":0.9995999932289124,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9994000196456909,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.9137204885482788},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8694964051246643},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7289983630180359},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.6237853765487671},{"id":"https://openalex.org/keywords/natural-number","display_name":"Natural number","score":0.4956756830215454},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4871039390563965},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.45076924562454224},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.43622422218322754},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.4294043779373169},{"id":"https://openalex.org/keywords/prime-power","display_name":"Prime power","score":0.4268730580806732},{"id":"https://openalex.org/keywords/exponential-sum","display_name":"Exponential sum","score":0.41527581214904785},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16451489925384521}],"concepts":[{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.9137204885482788},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8694964051246643},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7289983630180359},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.6237853765487671},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.4956756830215454},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4871039390563965},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.45076924562454224},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.43622422218322754},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4294043779373169},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.4268730580806732},{"id":"https://openalex.org/C53638454","wikidata":"https://www.wikidata.org/wiki/Q4446020","display_name":"Exponential sum","level":3,"score":0.41527581214904785},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16451489925384521},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom/2978","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/2978","pdf_url":"https://www.ams.org/mcom/2016-85-297/S0025-5718-2015-02978-0/S0025-5718-2015-02978-0.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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/mcom/2978","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/2978","pdf_url":"https://www.ams.org/mcom/2016-85-297/S0025-5718-2015-02978-0/S0025-5718-2015-02978-0.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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1633446687.pdf","grobid_xml":"https://content.openalex.org/works/W1633446687.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W186403072","https://openalex.org/W196531003","https://openalex.org/W1534159642","https://openalex.org/W1660562555","https://openalex.org/W1788867460","https://openalex.org/W1990997832","https://openalex.org/W1998375233","https://openalex.org/W2005830269","https://openalex.org/W2057182494","https://openalex.org/W2080017601","https://openalex.org/W2135070632","https://openalex.org/W2164284862","https://openalex.org/W2185169205","https://openalex.org/W2324953118","https://openalex.org/W2340541605","https://openalex.org/W2590819028","https://openalex.org/W2997707806","https://openalex.org/W4289126832","https://openalex.org/W6822497661"],"related_works":["https://openalex.org/W3087194925","https://openalex.org/W2467357782","https://openalex.org/W3152691454","https://openalex.org/W4288118777","https://openalex.org/W4324098889","https://openalex.org/W4323049945","https://openalex.org/W1634118047","https://openalex.org/W1986279035","https://openalex.org/W2177966018","https://openalex.org/W4302423208"],"abstract_inverted_index":{"The":[0],"self-power":[1],"map":[2,60],"is":[3,229,267],"the":[4,7,16,56,79,253,338,347],"function":[5],"from":[6],"set":[8],"of":[9,58,81],"natural":[10],"numbers":[11],"to":[12,30,52,236,250],"itself":[13],"which":[14],"sends":[15],"number":[17],"<inline-formula":[18,31,64,84,100,139,174,187,201,255,269,349],"content-type=\"math/mathml\">":[19,32,65,85,101,140,175,188,202,256,270,350],"<mml:math":[20,33,66,86,102,141,176,189,203,257,271,351],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[21,34,67,87,103,142,177,190,204,258,272,352],"alttext=\"n\">":[22],"<mml:semantics>":[23,38,69,89,112,148,179,192,211,260,289,360],"<mml:mi>n</mml:mi>":[24,40,41,115,116,154,214,215],"<mml:annotation":[25,43,71,91,127,160,181,194,222,262,328],"encoding=\"application/x-tex\">n</mml:annotation>":[26],"</mml:semantics>":[27,45,73,93,133,166,183,196,226,264,333],"</mml:math>":[28,46,74,94,134,167,184,197,227,265,334],"</inline-formula>":[29,47,75,95,135,168,185,198,228,266,335],"alttext=\"n":[35,104,205],"Superscript":[36,105,206,356],"n\">":[37],"<mml:msup>":[39,114,213,291,364],"</mml:msup>":[42,117,216,294,376],"encoding=\"application/x-tex\">n^n</mml:annotation>":[44],".":[48,76,199],"Motivated":[49],"by":[50],"applications":[51],"cryptography,":[53],"we":[54],"consider":[55],"image":[57],"this":[59],"modulo":[61,186],"a":[62,109,130,137,242,247,343],"prime":[63],"alttext=\"p\">":[68,191],"<mml:mi>p</mml:mi>":[70,125,193,220,292,304,320,365],"encoding=\"application/x-tex\">p</mml:annotation>":[72,195],"We":[77,245,345],"study":[78],"question":[80],"how":[82],"large":[83],"alttext=\"x\">":[88,259],"<mml:mi>x</mml:mi>":[90,158,261,362],"encoding=\"application/x-tex\">x</mml:annotation>":[92,263],"must":[96],"be":[97],"so":[98],"that":[99,252],"n":[106,145,163,207],"Baseline":[107,208],"identical-to":[108],"mod":[110,209],"p\">":[111,210],"<mml:mrow>":[113,149,212,290,361],"<mml:mo>":[118,151,155,296,305,321,369],"\u2261":[119],"</mml:mo>":[120,153,157,298,307,323,371],"<mml:mi>a</mml:mi>":[121,180],"<mml:mo":[122,217,302,309,318,325],"lspace=\"thickmathspace\"":[123,218],"rspace=\"thickmathspace\">mod</mml:mo>":[124,219],"</mml:mrow>":[126,159,221,314,327,375,377],"encoding=\"application/x-tex\">n^n":[128,223],"\\equiv":[129],"\\bmod":[131,224],"p</mml:annotation>":[132,225],"has":[136],"solution":[138],"alttext=\"1":[143],"less-than-or-equal-to":[144,146],"x\">":[147],"<mml:mn>1</mml:mn>":[150,308,324],"\u2264":[152,156],"encoding=\"application/x-tex\">1":[161],"\\le":[162,164],"x</mml:annotation>":[165],",":[169,336],"for":[170],"every":[171],"residue":[172],"class":[173],"alttext=\"a\">":[178],"encoding=\"application/x-tex\">a</mml:annotation>":[182],"While":[200],"not":[230],"uniformly":[231],"distributed,":[232],"it":[233],"does":[234],"appear":[235],"behave":[237],"in":[238],"certain":[239],"ways":[240],"as":[241,342],"random":[243],"function.":[244],"give":[246],"heuristic":[248],"argument":[249],"show":[251],"expected":[254],"approximately":[268],"alttext=\"p":[273],"squared":[274],"log":[275],"phi":[276,283],"left-parenthesis":[277,284],"p":[278,285,355],"minus":[279,286,358],"1":[280,287],"right-parenthesis":[281],"slash":[282],"right-parenthesis\">":[288],"<mml:mn>2</mml:mn>":[293,368],"<mml:mi>log</mml:mi>":[295],"\u2061":[297],"<mml:mi>":[299,315,372],"\u03d5":[300,316],"</mml:mi>":[301,317,374],"stretchy=\"false\">(</mml:mo>":[303,319],"\u2212":[306,322,370],"stretchy=\"false\">)</mml:mo>":[310,326],"<mml:mrow":[311,366],"class=\"MJX-TeXAtom-ORD\">":[312,367],"<mml:mo>/</mml:mo>":[313],"encoding=\"application/x-tex\">p^2\\log":[329],"\\phi":[330],"(p-1)/\\phi":[331],"(p-1)</mml:annotation>":[332],"using":[337],"coupon":[339],"collector":[340],"problem":[341],"model.":[344],"prove":[346],"bound":[348],"alttext=\"x":[353],"greater-than":[354],"2":[357],"alpha\">":[359],"<mml:mo>&gt;</mml:mo>":[363],"\u03b1":[373]},"counts_by_year":[{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2016,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}