{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:37:36Z","timestamp":1776724656976,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We show that to solve the discrete log problem in a subgroup of order\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of an elliptic curve over the finite field of characteristic\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n one needs\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\ln p)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n operations in this field.\n <\/p>","DOI":"10.1090\/s0025-5718-98-00887-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"353-356","source":"Crossref","is-referenced-by-count":136,"title":["Evaluation of discrete logarithms in a group of \ud835\udc5d-torsion points of an elliptic curve in characteristic \ud835\udc5d"],"prefix":"10.1090","volume":"67","author":[{"given":"I.","family":"Semaev","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1007\/3-540-39799-X_31","article-title":"Use of elliptic curves in cryptography","author":"Miller, Victor S.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540164634"},{"issue":"177","key":"2","doi-asserted-by":"publisher","first-page":"203","DOI":"10.2307\/2007884","article-title":"Elliptic curve cryptosystems","volume":"48","author":"Koblitz, Neal","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"A. Menezes, S. Vanstone, and O. Tatsuaki, Reducing elliptic curve logarithms to logarithms in a finite field, Proc. 23rd ACM Sympos. Theory of Computing, 1991, pp. 80\u201389.","DOI":"10.1145\/103418.103434"},{"key":"4","unstructured":"I. A. Semaev, Bystry\u012d algoritm vyqisleniya sparivaniya A. Ve\u012dlya na \u00e8lliptiqesko\u012d krivo\u012d, International Conference \u201cModern Problems in Number Theory\u201d, Russia, Tula, Sept. 20\u201325, 1993, Abstracts of papers."},{"issue":"206","key":"5","doi-asserted-by":"publisher","first-page":"865","DOI":"10.2307\/2153546","article-title":"A remark concerning \ud835\udc5a-divisibility and the discrete logarithm in the divisor class group of curves","volume":"62","author":"Frey, Gerhard","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1109\/tit.1978.1055817","article-title":"An improved algorithm for computing logarithms over \ud835\udc3a\ud835\udc39(\ud835\udc5d) and its cryptographic significance","volume":"IT-24","author":"Pohlig, Stephen C.","year":"1978","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"issue":"143","key":"7","doi-asserted-by":"publisher","first-page":"918","DOI":"10.2307\/2006496","article-title":"Monte Carlo methods for index computation (\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc5d)","volume":"32","author":"Pollard, J. M.","year":"1978","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1920-8","volume-title":"The arithmetic of elliptic curves","volume":"106","author":"Silverman, Joseph H.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0387962034"},{"key":"9","first-page":"24","article-title":"Sur la topologie des vari\u00e9t\u00e9s alg\u00e9briques en caract\u00e9ristique \ud835\udc5d","author":"Serre, Jean-Pierre","year":"1958"},{"key":"10","unstructured":"H.-G. Ruck, A remark on the paper \u201cEvaluation of discrete logarithms on some elliptic curves, by I. A. Semaev\u201d, communication to \u201cMathematics of Computation\u201d."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00887-4\/S0025-5718-98-00887-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00887-4\/S0025-5718-98-00887-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:41:22Z","timestamp":1776721282000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00887-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":10,"journal-issue":{"issue":"221","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["S0025-5718-98-00887-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00887-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1998]]}}}