{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,7]],"date-time":"2025-12-07T21:15:29Z","timestamp":1765142129263,"version":"3.40.3"},"publisher-location":"Cham","reference-count":35,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783031584107"},{"type":"electronic","value":"9783031584114"}],"license":[{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024]]},"DOI":"10.1007\/978-3-031-58411-4_12","type":"book-chapter","created":{"date-parts":[[2024,5,11]],"date-time":"2024-05-11T12:01:44Z","timestamp":1715428904000},"page":"263-281","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Effective Lower Bound on\u00a0the\u00a0Number of\u00a0Orientable Supersingular Elliptic Curves"],"prefix":"10.1007","author":[{"given":"Antonin","family":"Leroux","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,5,12]]},"reference":[{"key":"12_CR1","doi-asserted-by":"crossref","unstructured":"Arpin, S., Chen, M., Lauter, K.E., Scheidler, R., Stange, K.E., Tran, H.T.N.: Orienteering with one endomorphism. arXiv preprint arXiv:2201.11079 (2022)","DOI":"10.1007\/s44007-023-00053-2"},{"key":"12_CR2","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/978-3-030-64834-3_14","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2020","author":"N Alamati","year":"2020","unstructured":"Alamati, N., De Feo, L., Montgomery, H., Patranabis, S.: Cryptographic group actions and applications. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12492, pp. 411\u2013439. Springer, Cham (2020). https:\/\/doi.org\/10.1007\/978-3-030-64834-3_14"},{"key":"12_CR3","doi-asserted-by":"crossref","unstructured":"Brzezinski, J., Eichler, M.: On the imbeddings of imaginary quadratic orders in definite quaternion orders (1992)","DOI":"10.1515\/crll.1992.426.91"},{"key":"12_CR4","unstructured":"Belding, J.V.: Number theoretic algorithms for elliptic curves. University of Maryland, College Park (2008)"},{"key":"12_CR5","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/978-3-030-34578-5_9","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2019","author":"W Beullens","year":"2019","unstructured":"Beullens, W., Kleinjung, T., Vercauteren, F.: CSI-FiSh: efficient isogeny based signatures through class group computations. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11921, pp. 227\u2013247. Springer, Cham (2019). https:\/\/doi.org\/10.1007\/978-3-030-34578-5_9"},{"issue":"5","key":"12_CR6","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1080\/00927878308822861","volume":"11","author":"J Brzezinski","year":"1983","unstructured":"Brzezinski, J.: On orders in quaternion algebras. Commun. Algebra 11(5), 501\u2013522 (1983)","journal-title":"Commun. Algebra"},{"key":"12_CR7","doi-asserted-by":"crossref","unstructured":"Castryck, W., Decru, T.: An efficient key recovery attack on SIDH (preliminary version). Cryptology ePrint Archive (2022)","DOI":"10.1007\/978-3-031-30589-4_15"},{"key":"12_CR8","doi-asserted-by":"crossref","unstructured":"Col\u00f2, L., Kohel, D.: Orienting supersingular isogeny graphs. Number-Theoretic Methods in Cryptology 2019 (2019)","DOI":"10.1515\/jmc-2019-0034"},{"issue":"1","key":"12_CR9","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s00145-007-9002-x","volume":"22","author":"DX Charles","year":"2009","unstructured":"Charles, D.X., Lauter, K.E., Goren, E.Z.: Cryptographic hash functions from expander graphs. J. Cryptol. 22(1), 93\u2013113 (2009)","journal-title":"J. Cryptol."},{"key":"12_CR10","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/978-3-030-03332-3_15","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2018","author":"W Castryck","year":"2018","unstructured":"Castryck, W., Lange, T., Martindale, C., Panny, L., Renes, J.: CSIDH: an efficient post-quantum commutative group action. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11274, pp. 395\u2013427. Springer, Cham (2018). https:\/\/doi.org\/10.1007\/978-3-030-03332-3_15"},{"key":"12_CR11","unstructured":"Couveignes, J.-M.: Hard homogeneous spaces. Cryptology ePrint Archive, Report 2006\/291 (2006)"},{"key":"12_CR12","unstructured":"Chenu, M., Smith, B.: Higher-degree supersingular group actions. Math. Cryptol. (2021)"},{"issue":"2","key":"12_CR13","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1090\/bproc\/68","volume":"8","author":"S Chari","year":"2021","unstructured":"Chari, S., Smertnig, D., Voight, J.: On basic and bass quaternion orders. Proc. Am. Math. Soc. Ser. B 8(2), 11\u201326 (2021)","journal-title":"Proc. Am. Math. Soc. Ser. B"},{"issue":"1","key":"12_CR14","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/BF02940746","volume":"14","author":"M Deuring","year":"1941","unstructured":"Deuring, M.: Die typen der multiplikatorenringe elliptischer funktionenk\u00f6rper. Abh. Math. Semin. Univ. Hambg. 14(1), 197\u2013272 (1941)","journal-title":"Abh. Math. Semin. Univ. Hambg."},{"key":"12_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/978-3-030-92068-5_9","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2021","author":"L De Feo","year":"2021","unstructured":"De Feo, L., et al.: S\u00e9ta: supersingular encryption from\u00a0torsion attacks. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13093, pp. 249\u2013278. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-92068-5_9"},{"key":"12_CR16","unstructured":"De\u00a0Koninck, J.-M., Letendre, P.: New upper bounds for the number of divisors function. arXiv preprint arXiv:1812.09950 (2018)"},{"key":"12_CR17","unstructured":"Dorman, D.R.: Global orders in definite quaternion algebras as endomorphism rings for reduced cm elliptic curves. Th\u00e9orie des nombres (Quebec, PQ, 1987), pp. 108\u2013116 (1987)"},{"issue":"1","key":"12_CR18","doi-asserted-by":"publisher","first-page":"215","DOI":"10.2140\/obs.2020.4.215","volume":"4","author":"K Eisentr\u00e4ger","year":"2020","unstructured":"Eisentr\u00e4ger, K., Hallgren, S., Leonardi, C., Morrison, T., Park, J.: Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs. Open Book Ser. 4(1), 215\u2013232 (2020)","journal-title":"Open Book Ser."},{"key":"12_CR19","doi-asserted-by":"crossref","unstructured":"Eichler, M.: Untersuchungen in der zahlentheorie der rationalen quaternionenalgebren (1936)","DOI":"10.1515\/crll.1936.174.129"},{"key":"12_CR20","unstructured":"Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. American Mathematical Society (2021)"},{"key":"12_CR21","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/978-3-642-25405-5_2","volume-title":"Post-Quantum Cryptography","author":"D Jao","year":"2011","unstructured":"Jao, D., De Feo, L.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 19\u201334. Springer, Heidelberg (2011). https:\/\/doi.org\/10.1007\/978-3-642-25405-5_2"},{"key":"12_CR22","unstructured":"Kaneko, M.: Supersingular $$j$$-invariants as singular moduli$$\\text{mod} \\, p$$ (1989)"},{"key":"12_CR23","unstructured":"Kohel, D.: Endomorphism rings of elliptic curves over finite fields. Ph.D. thesis, University of California at Berkeley (1996)"},{"issue":"1","key":"12_CR24","doi-asserted-by":"publisher","first-page":"7","DOI":"10.2140\/obs.2020.4.7","volume":"4","author":"J Love","year":"2020","unstructured":"Love, J., Boneh, D.: Supersingular curves with small noninteger endomorphisms. Open Book Ser. 4(1), 7\u201322 (2020)","journal-title":"Open Book Ser."},{"issue":"1","key":"12_CR25","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1112\/plms\/s2-27.1.358","volume":"2","author":"JE Littlewood","year":"1928","unstructured":"Littlewood, J.E.: On the class-number of the corpus p ( $$\\sqrt{- k}$$). Proc. London Math. Soc. 2(1), 358\u2013372 (1928)","journal-title":"Proc. London Math. Soc."},{"issue":"19","key":"12_CR26","doi-asserted-by":"publisher","first-page":"9206","DOI":"10.1093\/imrn\/rnu223","volume":"2015","author":"K Lauter","year":"2015","unstructured":"Lauter, K., Viray, B.: On singular moduli for arbitrary discriminants. Int. Math. Res. Not. 2015(19), 9206\u20139250 (2015)","journal-title":"Int. Math. Res. Not."},{"key":"12_CR27","unstructured":"Maino, L., Martindale, C.: An attack on SIDH with arbitrary starting curve. Cryptology ePrint Archive (2022)"},{"key":"12_CR28","doi-asserted-by":"publisher","first-page":"101777","DOI":"10.1016\/j.ffa.2020.101777","volume":"69","author":"H Onuki","year":"2021","unstructured":"Onuki, H.: On oriented supersingular elliptic curves. Finite Fields Appl. 69, 101777 (2021)","journal-title":"Finite Fields Appl."},{"key":"12_CR29","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1007\/978-3-030-84252-9_15","volume-title":"Advances in Cryptology \u2013 CRYPTO 2021","author":"V de Quehen","year":"2021","unstructured":"de Quehen, V., et al.: Improved torsion-point attacks on SIDH variants. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12827, pp. 432\u2013470. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-84252-9_15"},{"key":"12_CR30","doi-asserted-by":"crossref","unstructured":"Robert, D.: Breaking SIDH in polynomial time. Cryptology ePrint Archive (2022)","DOI":"10.1007\/978-3-031-30589-4_17"},{"key":"12_CR31","unstructured":"Rostovtsev, A., Stolbunov, A.: Public-key cryptosystem based on isogenies. Cryptology ePrint Archive, Report 2006\/145 (2006)"},{"key":"12_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-56694-4","volume-title":"Quaternion Algebras","author":"J Voight","year":"2021","unstructured":"Voight, J.: Quaternion Algebras. Springer, Heidelberg (2021). https:\/\/doi.org\/10.1007\/978-3-030-56694-4"},{"key":"12_CR33","unstructured":"Wesolowski, B.: Orientations and the supersingular endomorphism ring problem. Cryptology ePrint Archive, Report 2021\/1583 (2021). https:\/\/ia.cr\/2021\/1583"},{"key":"12_CR34","unstructured":"Wigert, C.S.: Sur l\u2019ordre de grandeur du nombre des diviseurs d\u2019un entier. Almqvist & Wiksell (1907)"},{"key":"12_CR35","first-page":"1985","volume":"191\u2013220","author":"D Zagier","year":"1985","unstructured":"Zagier, D., Gross, B.: On singular moduli. J. Fur Die Reine Angewandte Math. 191\u2013220, 1985 (1985)","journal-title":"J. Fur Die Reine Angewandte Math."}],"container-title":["Lecture Notes in Computer Science","Selected Areas in Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-58411-4_12","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,11]],"date-time":"2024-05-11T12:03:14Z","timestamp":1715428994000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-58411-4_12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024]]},"ISBN":["9783031584107","9783031584114"],"references-count":35,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-58411-4_12","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2024]]},"assertion":[{"value":"12 May 2024","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"SAC","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Selected Areas in Cryptography","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Windsor, ON","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Canada","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2022","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"24 August 2022","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"26 August 2022","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"29","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"sacrypt2022","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/www.sac2022.ca\/home","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}