{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T22:41:35Z","timestamp":1777675295398,"version":"3.51.4"},"reference-count":39,"publisher":"American Mathematical Society (AMS)","issue":"249","license":[{"start":{"date-parts":[[2005,5,21]],"date-time":"2005-05-21T00:00:00Z","timestamp":1116633600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation.<\/p>","DOI":"10.1090\/s0025-5718-04-01656-4","type":"journal-article","created":{"date-parts":[[2004,9,20]],"date-time":"2004-09-20T09:57:53Z","timestamp":1095674273000},"page":"499-518","source":"Crossref","is-referenced-by-count":19,"title":["Construction of CM Picard curves"],"prefix":"10.1090","volume":"74","author":[{"given":"Kenji","family":"Koike","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Annegret","family":"Weng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2004,5,21]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/10722028_5","article-title":"Construction of secure \ud835\udc36_{\ud835\udc4e\ud835\udc4f} curves using modular curves","author":"Arita, Seigo","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3540676953"},{"key":"2","unstructured":"A.O.L. Atkin. The number of points on an elliptic curve modulo a prime. \\textsl{Unpublished manuscript}, 1991."},{"issue":"203","key":"3","doi-asserted-by":"publisher","first-page":"29","DOI":"10.2307\/2152935","article-title":"Elliptic curves and primality proving","volume":"61","author":"Atkin, A. O. L.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"M. Bauer. The arithmetic of certain cubic function fields. Math. Comp., 73:387\u2013413, 2003.","DOI":"10.1090\/S0025-5718-03-01559-X"},{"key":"5","unstructured":"C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier. User\u2019s guide to pari-gp. 2000."},{"issue":"165","key":"6","first-page":"210","article-title":"Point sets in projective spaces and theta functions","author":"Dolgachev, Igor","year":"1988","journal-title":"Ast\\'{e}risque","ISSN":"https:\/\/id.crossref.org\/issn\/0303-1179","issn-type":"print"},{"issue":"237","key":"7","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1090\/S0025-5718-00-01297-7","article-title":"Arithmetic on superelliptic curves","volume":"71","author":"Galbraith, S. D.","year":"2002","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"P. Gaudry. An algorithm for solving the discrete logarithm problem on hyperelliptic curves. Eurocrypt 2000, LNCS 1807, Springer, pages 19\u201334, 2000.","DOI":"10.1007\/3-540-45539-6_2"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1007\/3-540-45682-1_28","article-title":"An extension of Kedlaya\u2019s point-counting algorithm to superelliptic curves","author":"Gaudry, Pierrick","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/3540429875"},{"key":"10","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/BF01342938","article-title":"Explizite Bestimmung der Randfl\u00e4chen des Fundamentalbereiches der Modulgruppe zweiten Grades","volume":"138","author":"Gottschling, Erhard","year":"1959","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"11","series-title":"Graduate Texts in Mathematics, No. 52","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3849-0","volume-title":"Algebraic geometry","author":"Hartshorne, Robin","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902449"},{"key":"12","series-title":"Lectures in Mathematics ETH Z\\\"{u}rich","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-9051-9","volume-title":"The ball and some Hilbert problems","author":"Holzapfel, Rolf-Peter","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/3764328355"},{"key":"13","doi-asserted-by":"publisher","first-page":"83","DOI":"10.2969\/jmsj\/02010083","article-title":"Isogeny classes of abelian varieties over finite fields","volume":"20","author":"Honda, Taira","year":"1968","journal-title":"J. Math. Soc. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5645","issn-type":"print"},{"issue":"177","key":"14","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"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/BF02252872","article-title":"Hyperelliptic cryptosystems","volume":"1","author":"Koblitz, Neal","year":"1989","journal-title":"J. Cryptology","ISSN":"https:\/\/id.crossref.org\/issn\/0933-2790","issn-type":"print"},{"key":"16","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-5740-0","volume-title":"Introduction to algebraic and abelian functions","volume":"89","author":"Lang, Serge","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387907106","edition":"2"},{"key":"17","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5485-0","volume-title":"Complex multiplication","volume":"255","author":"Lang, Serge","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387907866"},{"key":"18","unstructured":"H. Lange and Ch. Birkenhake. Complex Abelian varieties. Springer, 1982."},{"key":"19","unstructured":"MAGMA. http:\/\/magma.maths.usyd.edu.au\/magma\/. University of Sydney, 2002."},{"key":"20","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"},{"key":"21","doi-asserted-by":"crossref","unstructured":"J.-S. Milne. Jacobian varieties. In Cornell G. and J.H. Silverman, editors, Arithmetic Geometry, pages 167\u2013212. Springer, 1986.","DOI":"10.1007\/978-1-4613-8655-1_7"},{"key":"22","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-2843-6","volume-title":"Tata lectures on theta. I","volume":"28","author":"Mumford, David","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3764331097"},{"key":"23","series-title":"Pure and Applied Mathematics, Vol. 45","volume-title":"Integral matrices","author":"Newman, Morris","year":"1972"},{"key":"24","first-page":"377","article-title":"Principally polarized abelian varieties of dimension two or three are Jacobian varieties","volume":"20","author":"Oort, Frans","year":"1973","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-8980","issn-type":"print"},{"key":"25","doi-asserted-by":"crossref","unstructured":"E. Picard. Sur les fonctions de deux variables ind\u00e9pendantes analogues aux fonctions modulaires. Acta math., 2:114-135, 1883.","DOI":"10.1007\/BF02612158"},{"key":"26","doi-asserted-by":"crossref","unstructured":"E. Picard. Sur les formes quadratiques ternaire ind\u00e9finies et sur les fonctions hyperfuchsiennes, Acta math., 5:121-182, 1884.","DOI":"10.1007\/BF02421555"},{"issue":"1","key":"27","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"},{"key":"28","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511661952","volume-title":"Algorithmic algebraic number theory","volume":"30","author":"Pohst, M.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0521330602"},{"key":"29","isbn-type":"print","first-page":"13","article-title":"Efficient reduction on the Jacobian variety of Picard curves","author":"Reinaldo Barreiro, Ernesto","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3540662480"},{"issue":"3","key":"30","doi-asserted-by":"publisher","first-page":"311","DOI":"10.2977\/prims\/1195175031","article-title":"On the representation of the Picard modular function by \ud835\udf03 constants. I, II","volume":"24","author":"Shiga, Hironori","year":"1988","journal-title":"Publ. Res. Inst. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0034-5318","issn-type":"print"},{"key":"31","series-title":"Princeton Mathematical Series","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1515\/9781400883943","volume-title":"Abelian varieties with complex multiplication and modular functions","volume":"46","author":"Shimura, Goro","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0691016569"},{"key":"32","series-title":"Kan\\^{o} Memorial Lectures, No. 1","volume-title":"Introduction to the arithmetic theory of automorphic functions","author":"Shimura, Goro","year":"1971"},{"key":"33","unstructured":"C.L.  Siegel. Topics in Complex Function Theory. Vol. II. John Wiley and Sons, 1972"},{"key":"34","unstructured":"A.-M. Spallek. Kurven vom Geschlecht 2 und ihre Anwendung in Public-Key-Kryptosystemen. PhD thesis, Institut f\u00fcr Experimentelle Mathematik, Universit\u00e4t GH Essen, 1994."},{"key":"35","doi-asserted-by":"crossref","unstructured":"J. Tate. Classes d\u2019isog\u00e9nie des vari\u00e9t\u00e8s ab\u00e9liennes sur un corps fini (d\u2019apr\u00e8s T. Honda). Seminaire Bourbaki, Soc. Math. France., 352, 95-110. 1968","DOI":"10.1007\/BFb0058807"},{"issue":"225","key":"36","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1090\/S0025-5718-99-01020-0","article-title":"Examples of genus two CM curves defined over the rationals","volume":"68","author":"van Wamelen, Paul","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"241","key":"37","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1090\/S0025-5718-02-01422-9","article-title":"Constructing hyperelliptic curves of genus 2 suitable for cryptography","volume":"72","author":"Weng, Annegret","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"38","first-page":"339","article-title":"A class of hyperelliptic CM-curves of genus three","volume":"16","author":"Weng, Annegret","year":"2001","journal-title":"J. Ramanujan Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0970-1249","issn-type":"print"},{"issue":"2","key":"39","first-page":"471","article-title":"On unramified Galois extensions of real quadratic number fields","volume":"23","author":"Yamamura, Ken","year":"1986","journal-title":"Osaka J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-6126","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2005-74-249\/S0025-5718-04-01656-4\/S0025-5718-04-01656-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-249\/S0025-5718-04-01656-4\/S0025-5718-04-01656-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T13:58:51Z","timestamp":1776779931000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-249\/S0025-5718-04-01656-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,5,21]]},"references-count":39,"journal-issue":{"issue":"249","published-print":{"date-parts":[[2005,1]]}},"alternative-id":["S0025-5718-04-01656-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-04-01656-4","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":[[2004,5,21]]}}}