{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T07:00:33Z","timestamp":1777964433209,"version":"3.51.4"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T00:00:00Z","timestamp":1609459200000},"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":[[2021,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve\n                    <jats:italic>E<\/jats:italic>\n                    defined over \ud835\udd3d\n                    <jats:sub>\n                      <jats:italic>p<\/jats:italic>\n                      <jats:sup>2<\/jats:sup>\n                    <\/jats:sub>\n                    , if an imaginary quadratic order\n                    <jats:italic>O<\/jats:italic>\n                    can be embedded in End(\n                    <jats:italic>E<\/jats:italic>\n                    ) and a prime\n                    <jats:italic>L<\/jats:italic>\n                    splits into two principal ideals in\n                    <jats:italic>O<\/jats:italic>\n                    , we construct loops or cycles in the supersingular\n                    <jats:italic>L<\/jats:italic>\n                    -isogeny graph at the vertices which are next to\n                    <jats:italic>j<\/jats:italic>\n                    (\n                    <jats:italic>E<\/jats:italic>\n                    ) in the supersingular \u2113-isogeny graph where \u2113 is a prime different from\n                    <jats:italic>L<\/jats:italic>\n                    . Next, we discuss the lengths of these cycles especially for\n                    <jats:italic>j<\/jats:italic>\n                    (\n                    <jats:italic>E<\/jats:italic>\n                    ) = 1728 and 0. Finally, we also determine an upper bound on primes\n                    <jats:italic>p<\/jats:italic>\n                    for which there are unexpected 2-cycles if \u2113 doesn\u2019t split in\n                    <jats:italic>O<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.1515\/jmc-2020-0029","type":"journal-article","created":{"date-parts":[[2021,5,15]],"date-time":"2021-05-15T17:02:02Z","timestamp":1621098122000},"page":"454-464","source":"Crossref","is-referenced-by-count":4,"title":["Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves"],"prefix":"10.1515","volume":"15","author":[{"given":"Guanju","family":"Xiao","sequence":"first","affiliation":[{"name":"Key Laboratory of Mathematics Mechanization , NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences , Beijing , China ; School of Mathematical Sciences , University of Chinese Academy of Sciences , Beijing , China"}]},{"given":"Lixia","family":"Luo","sequence":"additional","affiliation":[{"name":"Key Laboratory of Mathematics Mechanization , NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences , Beijing , China ; School of Mathematical Sciences , University of Chinese Academy of Sciences , Beijing , China"}]},{"given":"Yingpu","family":"Deng","sequence":"additional","affiliation":[{"name":"Key Laboratory of Mathematics Mechanization , NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences , Beijing , China ; School of Mathematical Sciences , University of Chinese Academy of Sciences , Beijing , China"}]}],"member":"374","published-online":{"date-parts":[[2021,5,15]]},"reference":[{"key":"2025120600293881736_j_jmc-2020-0029_ref_001","doi-asserted-by":"crossref","unstructured":"Gora Adj, Omran Ahmadi, and Alfred Menezes. 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