{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:30:55Z","timestamp":1764981055625,"version":"3.46.0"},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,2,14]]},"abstract":"Abstract<\/jats:title>\n \n We characterize the possible groups\n \n \n \n \n E<\/m:mi>\n \n (<\/m:mo>\n \n Z<\/m:mi>\n \u2215<\/m:mo>\n N<\/m:mi>\n Z<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n E\\left({\\mathbb{Z}}\/N{\\mathbb{Z}})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n arising from elliptic curves over\n \n \n \n \n Z<\/m:mi>\n \u2215<\/m:mo>\n N<\/m:mi>\n Z<\/m:mi>\n <\/m:math>\n {\\mathbb{Z}}\/N{\\mathbb{Z}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in terms of the groups\n \n \n \n \n E<\/m:mi>\n \n (<\/m:mo>\n \n \n \n F<\/m:mi>\n <\/m:mrow>\n \n p<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n E\\left({{\\mathbb{F}}}_{p})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , with\n \n \n \n \n p<\/m:mi>\n <\/m:math>\n p<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n varying among the prime divisors of\n \n \n \n \n N<\/m:mi>\n <\/m:math>\n N<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . This classification is achieved by showing that the infinity part of any elliptic curves over\n \n \n \n \n Z<\/m:mi>\n \u2215<\/m:mo>\n \n \n p<\/m:mi>\n <\/m:mrow>\n \n e<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n Z<\/m:mi>\n <\/m:math>\n {\\mathbb{Z}}\/{p}^{e}{\\mathbb{Z}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a\n \n \n \n \n Z<\/m:mi>\n \u2215<\/m:mo>\n \n \n p<\/m:mi>\n <\/m:mrow>\n \n e<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n Z<\/m:mi>\n <\/m:math>\n {\\mathbb{Z}}\/{p}^{e}{\\mathbb{Z}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -torsor, of which a generator is exhibited. As a first consequence, when\n \n \n \n \n E<\/m:mi>\n \n (<\/m:mo>\n \n Z<\/m:mi>\n \u2215<\/m:mo>\n N<\/m:mi>\n Z<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n E\\left({\\mathbb{Z}}\/N{\\mathbb{Z}})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a\n \n \n \n \n p<\/m:mi>\n <\/m:math>\n p<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -group, we provide an explicit and sharp bound on its rank. As a second consequence, when\n \n \n \n \n N<\/m:mi>\n =<\/m:mo>\n \n \n p<\/m:mi>\n <\/m:mrow>\n \n e<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n <\/m:math>\n N={p}^{e}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a prime power and the projected curve\n \n \n \n \n E<\/m:mi>\n \n (<\/m:mo>\n \n \n \n F<\/m:mi>\n <\/m:mrow>\n \n p<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n E\\left({{\\mathbb{F}}}_{p})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n has trace one, we provide an isomorphism attack to the elliptic curve discrete logarithm problem, which works only by means of finite ring arithmetic.\n <\/jats:p>","DOI":"10.1515\/jmc-2023-0025","type":"journal-article","created":{"date-parts":[[2024,2,14]],"date-time":"2024-02-14T08:12:26Z","timestamp":1707898346000},"source":"Crossref","is-referenced-by-count":0,"title":["Group structure of elliptic curves over\n \u2124<\/i>\n \/\n N\u2124<\/i>"],"prefix":"10.1515","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7266-5146","authenticated-orcid":false,"given":"Massimiliano","family":"Sala","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Trento , Via Sommarive 14 , 38123 Povo , Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3402-4863","authenticated-orcid":false,"given":"Daniele","family":"Taufer","sequence":"additional","affiliation":[{"name":"KU Leuven, Numerical Analysis and Applied Mathematics (NUMA), Celestijnenlaan 200a , 3001 Leuven , Belgium"}]}],"member":"374","published-online":{"date-parts":[[2024,2,14]]},"reference":[{"key":"2025120600265547518_j_jmc-2023-0025_ref_001","doi-asserted-by":"crossref","unstructured":"Breuil C, Conrad B, Diamond F, Taylor R. 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