{"metadata":{"_oai":{"id":"oai:ipsj.ixsq.nii.ac.jp:00031859","sets":["1164:2592:2620:2625"]},"path":["2625"],"owner":"1","recid":"31859","title":["合成数のAKS witnessに関する実験的研究"],"pubdate":{"attribute_name":"公開日","attribute_value":"2004-03-19"},"_buckets":{"deposit":"ce3ac3fe-be4c-44bd-863a-3c0a305e3317"},"_deposit":{"id":"31859","pid":{"type":"depid","value":"31859","revision_id":0},"owners":[1],"status":"published","created_by":1},"item_title":"合成数のAKS witnessに関する実験的研究","author_link":["0","0"],"item_titles":{"attribute_name":"タイトル","attribute_value_mlt":[{"subitem_title":"合成数のAKS witnessに関する実験的研究"},{"subitem_title":"Experimental research on AKS witness of the number of composition","subitem_title_language":"en"}]},"item_type_id":"4","publish_date":"2004-03-19","item_4_text_3":{"attribute_name":"著者所属","attribute_value_mlt":[{"subitem_text_value":"明治大学大学院理工学研究科基礎理工学専攻"},{"subitem_text_value":"明治大学理工学部情報科学科"}]},"item_4_text_4":{"attribute_name":"著者所属(英)","attribute_value_mlt":[{"subitem_text_value":"Computer Science Course, Major in Science, Graduate School of Science and Technology, Meiji University.","subitem_text_language":"en"},{"subitem_text_value":"DepartMent of Computer Science, Meiji University","subitem_text_language":"en"}]},"item_language":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"jpn"}]},"item_publisher":{"attribute_name":"出版者","attribute_value_mlt":[{"subitem_publisher":"情報処理学会","subitem_publisher_language":"ja"}]},"publish_status":"0","weko_shared_id":-1,"item_file_price":{"attribute_name":"Billing file","attribute_type":"file","attribute_value_mlt":[{"url":{"url":"https://ipsj.ixsq.nii.ac.jp/record/31859/files/IPSJ-AL03094003.pdf"},"date":[{"dateType":"Available","dateValue":"2006-03-19"}],"format":"application/pdf","billing":["billing_file"],"filename":"IPSJ-AL03094003.pdf","filesize":[{"value":"107.5 kB"}],"mimetype":"application/pdf","priceinfo":[{"tax":["include_tax"],"price":"660","billingrole":"5"},{"tax":["include_tax"],"price":"330","billingrole":"6"},{"tax":["include_tax"],"price":"0","billingrole":"9"},{"tax":["include_tax"],"price":"0","billingrole":"44"}],"accessrole":"open_date","version_id":"2b475960-3ac7-4fea-b03d-c6effed10447","displaytype":"detail","licensetype":"license_note","license_note":"Copyright (c) 2004 by the Information Processing Society of Japan"}]},"item_4_creator_5":{"attribute_name":"著者名","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"瀬川, 紘"},{"creatorName":"玉木, 久夫"}],"nameIdentifiers":[{}]}]},"item_4_creator_6":{"attribute_name":"著者名(英)","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"Kou, Segawa","creatorNameLang":"en"},{"creatorName":"Hisao, Tamaki","creatorNameLang":"en"}],"nameIdentifiers":[{}]}]},"item_4_source_id_9":{"attribute_name":"書誌レコードID","attribute_value_mlt":[{"subitem_source_identifier":"AN1009593X","subitem_source_identifier_type":"NCID"}]},"item_4_textarea_12":{"attribute_name":"Notice","attribute_value_mlt":[{"subitem_textarea_value":"SIG Technical Reports are nonrefereed and hence may later appear in any journals, conferences, symposia, etc."}]},"item_resource_type":{"attribute_name":"資源タイプ","attribute_value_mlt":[{"resourceuri":"http://purl.org/coar/resource_type/c_18gh","resourcetype":"technical report"}]},"item_4_description_7":{"attribute_name":"論文抄録","attribute_value_mlt":[{"subitem_description":"正整数nに対して合同式(x+a)^n !≡ x^n + a  mod(n  x^r - 1)を満たしかつgcd(a  n) = 1であるような正整数rと整数aの対はnが合成数のときかつそのときに限り存在する。このような対(r a)を合成数nのAKSwitnessと呼ぶことにする。Agrawal  KayalとSaxenaの決定的多項式時間素数性判定アルゴリズムは、(1) 与えられたnに対してrがある性質を満たすとき、もしnが素数の冪乗でない合成数ならば1 ≦ a ≦ O √r log nの範囲のaで(r a)がnのAKS witnessであるものが存在する、(2)その性質を満たすr = O(log^5 n)が必ず存在する、ことに基づいている。我々は小さい値のAKS witnessの存在について実験を行った。結果の一部として、10^11以下のすべての合成数nおよび10^16以下のすべてのCarmichael数nに対して(r 1)がnのAKS witnessであるようなrの最小値は0.49log n以下、ある1 ≦ a ≦ rに対して(r  a)がnのAKS witnessであるようなrの最小値は0.37log n以下であることを見出した　","subitem_description_type":"Other"}]},"item_4_description_8":{"attribute_name":"論文抄録(英)","attribute_value_mlt":[{"subitem_description":"For any integer n ≧ 2, congruence (x+a)^n !≡ x^n +a mod(n, x^r -1) is satisfied by some positive r and a with gcd(n,a) = 1 if and only if n is a composite number. We call such a pair (r,a) an AKS witness of the composite n. The deterministic polynomial time primality algorithm of Agrawal, Kayal and Saxena is based on the facts: (1)if r satisfies some condition with respect to given n, and if n is a composite but is not a power of a prime, then (r,a) is an AKS witness of n for some 1 ≦ a ≦ O(√r log n);(2)for every n this condition is satisfied by some r = O(log^5 n). We have conducted experiments on the existences of AKS witnesses with small values. In particular, we have found that, for every composite n ≦ 10^11 and for every Carmichael number n ≦ 10^16, n has an AKS witness (r,1) with r ≦ 0.49log n and an AKS witness (r,a) for some 1 ≦ a ≦ r with r ≦ 0.37log n.","subitem_description_type":"Other"}]},"item_4_biblio_info_10":{"attribute_name":"書誌情報","attribute_value_mlt":[{"bibliographicPageEnd":"18","bibliographic_titles":[{"bibliographic_title":"情報処理学会研究報告アルゴリズム（AL）"}],"bibliographicPageStart":"13","bibliographicIssueDates":{"bibliographicIssueDate":"2004-03-19","bibliographicIssueDateType":"Issued"},"bibliographicIssueNumber":"34(2003-AL-094)","bibliographicVolumeNumber":"2004"}]},"relation_version_is_last":true,"weko_creator_id":"1"},"updated":"2025-01-22T16:26:29.834166+00:00","created":"2025-01-18T23:01:05.527518+00:00","links":{},"id":31859}