{"id":"https://openalex.org/W2000952383","doi":"https://doi.org/10.1090/s0025-5718-05-01799-0","title":"Two efficient algorithms for the computation of ideal sums in quadratic orders","display_name":"Two efficient algorithms for the computation of ideal sums in quadratic orders","publication_year":2005,"publication_date":"2005-12-08","ids":{"openalex":"https://openalex.org/W2000952383","doi":"https://doi.org/10.1090/s0025-5718-05-01799-0","mag":"2000952383"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-05-01799-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01799-0","pdf_url":"https://www.ams.org/mcom/2006-75-254/S0025-5718-05-01799-0/S0025-5718-05-01799-0.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/2006-75-254/S0025-5718-05-01799-0/S0025-5718-05-01799-0.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012425156","display_name":"Andr\u00e9 Weilert","orcid":null},"institutions":[{"id":"https://openalex.org/I135140700","display_name":"University of Bonn","ror":"https://ror.org/041nas322","country_code":"DE","type":"education","lineage":["https://openalex.org/I135140700"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Andr\u00e9 Weilert","raw_affiliation_strings":["Department of Computer Science II, University of Bonn, R\u00c3\u00b6merstra\u00c3\u009fe 164, 53117 Bonn, Germany","Liliencronstr. 8, 12167 Berlin, Germany"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science II, University of Bonn, R\u00c3\u00b6merstra\u00c3\u009fe 164, 53117 Bonn, Germany","institution_ids":["https://openalex.org/I135140700"]},{"raw_affiliation_string":"Liliencronstr. 8, 12167 Berlin, Germany","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5012425156"],"corresponding_institution_ids":["https://openalex.org/I135140700"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.10422289,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"75","issue":"254","first_page":"941","last_page":"981"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9997000098228455,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9991999864578247,"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/algorithm","display_name":"Algorithm","score":0.7474175691604614},{"id":"https://openalex.org/keywords/ideal","display_name":"Ideal (ethics)","score":0.4760432243347168},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3882104754447937},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.37004485726356506},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.35200339555740356}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.7474175691604614},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.4760432243347168},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3882104754447937},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.37004485726356506},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.35200339555740356},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-05-01799-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01799-0","pdf_url":"https://www.ams.org/mcom/2006-75-254/S0025-5718-05-01799-0/S0025-5718-05-01799-0.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"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-05-01799-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-05-01799-0","pdf_url":"https://www.ams.org/mcom/2006-75-254/S0025-5718-05-01799-0/S0025-5718-05-01799-0.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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2000952383.pdf","grobid_xml":"https://content.openalex.org/works/W2000952383.grobid-xml"},"referenced_works_count":50,"referenced_works":["https://openalex.org/W134647183","https://openalex.org/W1486771829","https://openalex.org/W1494010285","https://openalex.org/W1496801598","https://openalex.org/W1501162952","https://openalex.org/W1509364989","https://openalex.org/W1550016006","https://openalex.org/W1562115776","https://openalex.org/W1562183207","https://openalex.org/W1571785349","https://openalex.org/W1575834508","https://openalex.org/W1581260235","https://openalex.org/W1583935850","https://openalex.org/W1584026429","https://openalex.org/W1588160094","https://openalex.org/W1592261758","https://openalex.org/W1596400416","https://openalex.org/W1606822455","https://openalex.org/W1965555407","https://openalex.org/W1985138225","https://openalex.org/W1986121109","https://openalex.org/W2008130084","https://openalex.org/W2014355030","https://openalex.org/W2024918898","https://openalex.org/W2029377897","https://openalex.org/W2044036580","https://openalex.org/W2057357264","https://openalex.org/W2075488055","https://openalex.org/W2083368929","https://openalex.org/W2092004383","https://openalex.org/W2094096244","https://openalex.org/W2120223570","https://openalex.org/W2123923527","https://openalex.org/W2148676594","https://openalex.org/W2313012327","https://openalex.org/W2333102661","https://openalex.org/W2487302930","https://openalex.org/W2489377303","https://openalex.org/W2752853835","https://openalex.org/W2888871578","https://openalex.org/W3007836994","https://openalex.org/W3039948875","https://openalex.org/W3043060850","https://openalex.org/W3143801195","https://openalex.org/W4206919233","https://openalex.org/W4233185505","https://openalex.org/W4239676575","https://openalex.org/W4252614677","https://openalex.org/W4293193836","https://openalex.org/W6629248823"],"related_works":["https://openalex.org/W2748952813","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644"],"abstract_inverted_index":{"This":[0],"paper":[1],"deals":[2],"with":[3],"two":[4],"different":[5],"asymptotically":[6,206],"fast":[7,62,207],"algorithms":[8,32],"for":[9,185,209],"the":[10,19,23,38,41,47,150,153,186],"computation":[11],"of":[12,22,49,152,189],"ideal":[13,51],"sums":[14],"in":[15,40,53,64,90],"quadratic":[16,24,42,56],"orders.":[17],"If":[18],"class":[20],"number":[21,25],"field":[26],"is":[27],"equal":[28],"to":[29,36,85],"1,":[30],"these":[31],"can":[33,58],"be":[34,59],"used":[35],"calculate":[37],"GCD":[39],"order.":[43],"We":[44],"show":[45],"that":[46],"calculation":[48],"an":[50,182],"sum":[52],"a":[54,86],"fixed":[55],"order":[57],"done":[60],"as":[61,63],"<inline-formula":[65,91,137,156,190,210],"content-type=\"math/mathml\">":[66,92,138,157,191,211],"<mml:math":[67,93,139,158,192,212],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[68,94,140,159,193,213],"alttext=\"bold":[69],"upper":[70,183],"Z\">":[71],"<mml:semantics>":[72,106,142,164,195,215],"<mml:mrow":[73],"class=\"MJX-TeXAtom-ORD\">":[74],"<mml:mi":[75],"mathvariant=\"bold\">Z</mml:mi>":[76],"</mml:mrow>":[77,127,174],"<mml:annotation":[78,128,144,175,197,217],"encoding=\"application/x-tex\">\\mathbf":[79],"{Z}</mml:annotation>":[80],"</mml:semantics>":[81,133,146,178,199,219],"</mml:math>":[82,134,147,179,200,220],"</inline-formula>":[83,135,148,180,201,221],"up":[84],"constant":[87],"factor,":[88],"i.e.,":[89],"alttext=\"upper":[95],"O":[96],"left-parenthesis":[97,99,161],"mu":[98],"n":[100,103,162],"right-parenthesis":[101,104],"log":[102],"comma\">":[105],"<mml:mrow>":[107,165],"<mml:mi>O</mml:mi>":[108],"<mml:mo":[109,114,117,124,169,172],"stretchy=\"false\">(</mml:mo>":[110,115,170],"<mml:mi>":[111,166],"\u03bc":[112,167],"</mml:mi>":[113,168],"<mml:mi>n</mml:mi>":[116,123,143,171,196,216],"stretchy=\"false\">)</mml:mo>":[118,125,173],"<mml:mi>log</mml:mi>":[119],"<mml:mo>":[120],"\u2061":[121],"</mml:mo>":[122],"<mml:mo>,</mml:mo>":[126],"encoding=\"application/x-tex\">O(\\mu":[129],"(n)":[130],"\\log":[131],"n),</mml:annotation>":[132],"where":[136],"alttext=\"n\">":[141,194,214],"encoding=\"application/x-tex\">n</mml:annotation>":[145,198,218],"bounds":[149],"size":[151],"operands":[154],"and":[155],"alttext=\"mu":[160],"right-parenthesis\">":[163],"encoding=\"application/x-tex\">\\mu":[176],"(n)</mml:annotation>":[177],"denotes":[181],"bound":[184],"multiplication":[187,208],"time":[188],"-bit":[202,222],"integers.":[203],"Using":[204],"Sch\u00f6nhage\u2013Strassen\u2019s":[205],"integers,":[223],"we":[224],"achieve":[225],"\ud835\udf07(\ud835\udc5b)=\ud835\udc42(\ud835\udc5blog\ud835\udc5bloglog\ud835\udc5b).":[226]},"counts_by_year":[],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2025-10-10T00:00:00"}
