{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T02:10:05Z","timestamp":1750299005329,"version":"3.41.0"},"reference-count":9,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2024,12,1]],"date-time":"2024-12-01T00:00:00Z","timestamp":1733011200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:p>\n            Let\n            <jats:italic>E, E<\/jats:italic>\n            \u2032 be two isogenous ordinary elliptic curves defined over a finite field. The set Hom(\n            <jats:italic>E, E<\/jats:italic>\n            \u2032) of isogenies from\n            <jats:italic>E<\/jats:italic>\n            to\n            <jats:italic>E<\/jats:italic>\n            \u2032 is a free \u2124-module of rank two. In this paper, we give an algorithm for finding an explicit \u2124-basis of Hom(\n            <jats:italic>E, E<\/jats:italic>\n            \u2032). Firstly, we find a special pair of isogenies using the structure of ordinary isogeny graphs, called isogeny volcanoes. Secondly, we construct explicit generators of Hom(\n            <jats:italic>E, E<\/jats:italic>\n            \u2032) as a \u2124-module from the pair of isogenies. Finally, we remove the \u2124-linear dependency of the generators to obtain a \u2124-basis of Hom(\n            <jats:italic>E,E<\/jats:italic>\n            \u2032).\n          <\/jats:p>","DOI":"10.1145\/3727336.3727337","type":"journal-article","created":{"date-parts":[[2025,4,18]],"date-time":"2025-04-18T10:41:56Z","timestamp":1744972916000},"page":"107-120","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Finding a Basis of the Set of Isogenies between Two Ordinary Elliptic Curves Over a Finite Field"],"prefix":"10.1145","volume":"58","author":[{"given":"Akira","family":"Katayama","sequence":"first","affiliation":[{"name":"Department of Mathematics, Rikkyo University, Tokyo, Japan"}]},{"given":"Masaya","family":"Yasuda","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rikkyo University, Tokyo, Japan"}]}],"member":"320","published-online":{"date-parts":[[2025,4,18]]},"reference":[{"issue":"5","key":"e_1_2_1_1_1","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1016\/j.jnt.2009.11.003","article-title":"Computing the endomorphism ring of an ordinary elliptic curve over a finite field","volume":"131","author":"Bisson G.","year":"2011","unstructured":"G. Bisson, A. V. Sutherland. Computing the endomorphism ring of an ordinary elliptic curve over a finite field. Journal of Number Theory, 131(5):815--831, 2011.","journal-title":"Journal of Number Theory"},{"key":"e_1_2_1_2_1","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1007\/3-540-45455-1_23","article-title":"Isogeny volcanoes and the SEA algorithm. Algorithmic Number Theory (ANTS 2002)","volume":"2369","author":"Fouquet M.","year":"2002","unstructured":"M. Fouquet, F. Morain. Isogeny volcanoes and the SEA algorithm. Algorithmic Number Theory (ANTS 2002), Springer Lecture Notes in Computer Science, 2369:276--291, 2002.","journal-title":"Springer Lecture Notes in Computer Science"},{"key":"e_1_2_1_3_1","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1007\/s13160-023-00638-y","article-title":"Computing the Brauer group of the product of two elliptic curves over a finite field","volume":"41","author":"Katayama A.","year":"2024","unstructured":"A. Katayama, M. Yasuda. Computing the Brauer group of the product of two elliptic curves over a finite field. Japan Journal of Industrial and Applied Mathematics, 41:919--943, 2024.","journal-title":"Japan Journal of Industrial and Applied Mathematics"},{"key":"e_1_2_1_4_1","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1007\/978-3-031-69070-9_11","article-title":"Computing a basis of the set of isogenies between two supersingular elliptic curves. Computer Algebra in Scientific Computing (CASC 2024)","volume":"14938","author":"Katayama A.","year":"2024","unstructured":"A. Katayama, M. Yasuda. Computing a basis of the set of isogenies between two supersingular elliptic curves. Computer Algebra in Scientific Computing (CASC 2024), Springer Lecture Notes in Computer Science, 14938:178--192, 2024.","journal-title":"Springer Lecture Notes in Computer Science"},{"issue":"170","key":"e_1_2_1_6_1","first-page":"483","article-title":"Elliptic curves over finite fields and the computation of square roots mod p","volume":"44","author":"Schoof R.","year":"1985","unstructured":"R. Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Mathematics of Computation, 44(170):483--494, 1985.","journal-title":"Mathematics of Computation"},{"key":"e_1_2_1_7_1","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-09494-6","volume-title":"The arithmetic of elliptic curves. Graduate Texts in Mathematics 106","author":"Silverman J. H.","year":"2009","unstructured":"J. H. Silverman. The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, second edition, Springer-Verlag, New York, 2009."},{"key":"e_1_2_1_8_1","volume-title":"The Open Book Series, 1: 507--530","author":"Sutherland A. V.","year":"2013","unstructured":"A. V. Sutherland. Isogeny volcanoes. Algorithmic Number Theory (ANTS X), The Open Book Series, 1:507--530, 2013."},{"key":"e_1_2_1_9_1","unstructured":"The Sage Developers. SageMath the Sage Mathematics Software System (version 10.1). available at https:\/\/www.sagemath.org\/."},{"issue":"4","key":"e_1_2_1_10_1","first-page":"521","article-title":"Abelian varieties over finite fields","volume":"2","author":"Waterhouse W. C.","year":"1969","unstructured":"W. C. Waterhouse. Abelian varieties over finite fields. Annales Scientifiques de L'\u00c9.N.S., 2(4):521--560, 1969.","journal-title":"Annales Scientifiques de L'\u00c9.N.S."}],"container-title":["ACM Communications in Computer Algebra"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3727336.3727337","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3727336.3727337","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T01:56:56Z","timestamp":1750298216000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3727336.3727337"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["10.1145\/3727336.3727337"],"URL":"https:\/\/doi.org\/10.1145\/3727336.3727337","relation":{},"ISSN":["1932-2232","1932-2240"],"issn-type":[{"type":"print","value":"1932-2232"},{"type":"electronic","value":"1932-2240"}],"subject":[],"published":{"date-parts":[[2024,12]]},"assertion":[{"value":"2025-04-18","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}