{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T08:01:44Z","timestamp":1790496104441,"version":"4.1.0"},"reference-count":11,"publisher":"Wiley","issue":"A","license":[{"start":{"date-parts":[[2014,8,1]],"date-time":"2014-08-01T00:00:00Z","timestamp":1406851200000},"content-version":"unspecified","delay-in-days":212,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000151_inline3\"\/><jats:tex-math>$\\mathcal{O}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be a maximal order in a definite quaternion algebra over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000151_inline4\"\/><jats:tex-math>$\\mathbb{Q}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of prime discriminant <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000151_inline5\"\/><jats:tex-math>$p$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000151_inline6\"\/><jats:tex-math>$\\ell $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> a small prime. We describe a probabilistic algorithm which, for a given left <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000151_inline7\"\/><jats:tex-math>$\\mathcal{O}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-ideal, computes a representative in its left ideal class of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000151_inline8\"\/><jats:tex-math>$\\ell $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-power norm. In practice the algorithm is efficient and, subject to heuristics on expected distributions of primes, runs in expected polynomial time. This solves the underlying problem for a quaternion analog of the Charles\u2013Goren\u2013Lauter hash function, and has security implications for the original CGL construction in terms of supersingular elliptic curves.<\/jats:p>","DOI":"10.1112\/s1461157014000151","type":"journal-article","created":{"date-parts":[[2014,8,5]],"date-time":"2014-08-05T06:34:15Z","timestamp":1407220455000},"page":"418-432","source":"Crossref","is-referenced-by-count":112,"title":["On the quaternion -isogeny path problem"],"prefix":"10.1112","volume":"17","author":[{"given":"David","family":"Kohel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kristin","family":"Lauter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christophe","family":"Petit","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Pierre","family":"Tignol","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2014,8,1]]},"reference":[{"key":"S1461157014000151_r1","doi-asserted-by":"publisher","DOI":"10.2307\/1969420"},{"key":"S1461157014000151_r11","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0091027"},{"key":"S1461157014000151_r7","volume-title":"Handbook of Magma functions","author":"Bosma","year":"2013"},{"key":"S1461157014000151_r4","doi-asserted-by":"publisher","DOI":"10.1007\/s00145-007-9002-x"},{"key":"S1461157014000151_r6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02940746"},{"key":"S1461157014000151_r3","first-page":"42","volume-title":"Algebraic number theory","author":"Cassels","year":"1967"},{"key":"S1461157014000151_r8","unstructured":"8. D. Kohel , (1996) \u2018Endomorphism rings of elliptic curves over finite fields\u2019, PhD Thesis, University of California, Berkeley."},{"key":"S1461157014000151_r5","first-page":"33","article-title":"Su di un metodo per la risoluzione in numeri interi dell\u2019equazione \u2211h=0nchxn\u2212hyh = p","volume":"46","author":"Cornacchia","year":"1903","journal-title":"Giornale di Matematiche di Battaglini"},{"key":"S1461157014000151_r10","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(80)90151-9"},{"key":"S1461157014000151_r9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-85855-3_18"},{"key":"S1461157014000151_r2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1990-1023756-8"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157014000151","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T21:48:11Z","timestamp":1555969691000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157014000151\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":11,"journal-issue":{"issue":"A","published-print":{"date-parts":[[2014]]}},"alternative-id":["S1461157014000151"],"URL":"https:\/\/doi.org\/10.1112\/s1461157014000151","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}