{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T14:06:25Z","timestamp":1768917985583,"version":"3.49.0"},"publisher-location":"Cham","reference-count":40,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783319783710","type":"print"},{"value":"9783319783727","type":"electronic"}],"license":[{"start":{"date-parts":[[2018,1,1]],"date-time":"2018-01-01T00:00:00Z","timestamp":1514764800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2018,1,1]],"date-time":"2018-01-01T00:00:00Z","timestamp":1514764800000},"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":[[2018]]},"DOI":"10.1007\/978-3-319-78372-7_11","type":"book-chapter","created":{"date-parts":[[2018,3,30]],"date-time":"2018-03-30T06:13:04Z","timestamp":1522390384000},"page":"329-368","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":83,"title":["Supersingular Isogeny Graphs and Endomorphism Rings: Reductions and Solutions"],"prefix":"10.1007","author":[{"given":"Kirsten","family":"Eisentr\u00e4ger","sequence":"first","affiliation":[]},{"given":"Sean","family":"Hallgren","sequence":"additional","affiliation":[]},{"given":"Kristin","family":"Lauter","sequence":"additional","affiliation":[]},{"given":"Travis","family":"Morrison","sequence":"additional","affiliation":[]},{"given":"Christophe","family":"Petit","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,3,31]]},"reference":[{"key":"11_CR1","unstructured":"Azarderakhsh, R., Campagna, M., Costello, C., De Feo, L., Hess, B., Jalali, A., Jao, D., Koziel, B., LaMacchia, B., Longa, P., Naehrig, M., Renes, J., Soukharev, V., Urbanik, D.: Supersingular isogeny key encapsulation. Submission to the NIST Post-Quantum Standardization Project (2017). https:\/\/csrc.nist.gov\/Projects\/Post-Quantum-Cryptography\/Round-1-Submissions"},{"key":"11_CR2","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1007\/978-3-319-13039-2_25","volume-title":"Progress in Cryptology \u2013 INDOCRYPT 2014","author":"J-F Biasse","year":"2014","unstructured":"Biasse, J.-F., Jao, D., Sankar, A.: A quantum algorithm for computing isogenies between supersingular elliptic curves. In: Meier, W., Mukhopadhyay, D. (eds.) INDOCRYPT 2014. LNCS, vol. 8885, pp. 428\u2013442. Springer, Cham (2014). https:\/\/doi.org\/10.1007\/978-3-319-13039-2_25"},{"issue":"3","key":"11_CR3","first-page":"269","volume":"1","author":"R Br\u00f6ker","year":"2009","unstructured":"Br\u00f6ker, R.: Constructing supersingular elliptic curves. J. Comb. Number Theory 1(3), 269\u2013273 (2009)","journal-title":"J. Comb. Number Theory"},{"key":"11_CR4","unstructured":"Cervi\u00f1o, J.M.: Supersingular elliptic curves and maximal quaternionic orders. Mathematisches Institut. Georg-August-Universit\u00e4t G\u00f6ttingen: Seminars Summer Term 2004, pp. 53\u201360. Universit\u00e4tsdrucke G\u00f6ttingen, G\u00f6ttingen (2004)"},{"issue":"suppl. A","key":"11_CR5","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1112\/S1461157014000254","volume":"1","author":"I Chevyrev","year":"2014","unstructured":"Chevyrev, I., Galbraith, S.D.: Constructing supersingular elliptic curves with a given endomorphism ring. LMS J. Comput. Math. 1(suppl. A), 71\u201391 (2014)","journal-title":"LMS J. Comput. Math."},{"key":"11_CR6","unstructured":"Charles, D., Goren, E., Lauter, K.: Cryptographic hash functions from expander graphs. Cryptology ePrint Archive, Report 2006\/021 (2006). https:\/\/eprint.iacr.org\/2006\/021"},{"issue":"1","key":"11_CR7","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., Goren, E.Z., Lauter, K.: Cryptographic hash functions from expander graphs. J. Cryptol. 22(1), 93\u2013113 (2009)","journal-title":"J. Cryptol."},{"key":"11_CR8","first-page":"33","volume":"46","author":"G Cornacchia","year":"1908","unstructured":"Cornacchia, G.: Su di un metodo per la risoluzione in numeri interi dell\u2019 equazione \n$$\\sum _{h=0}^nc_hx^{n-h}y^h=p$$\n. Giornale di Matematiche di Battaglini 46, 33\u201390 (1908)","journal-title":"Giornale di Matematiche di Battaglini"},{"issue":"1","key":"11_CR9","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. Sem. Univ. Hambg. 14(1), 197\u2013272 (1941)","journal-title":"Abh. Math. Sem. Univ. Hambg."},{"issue":"3","key":"11_CR10","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1515\/jmc-2012-0015","volume":"3","author":"L De Feo","year":"2014","unstructured":"De Feo, L., Jao, D., Pl\u00fbt, J.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. J. Math. Cryptol. 3(3), 209\u2013247 (2014)","journal-title":"J. Math. Cryptol."},{"issue":"2","key":"11_CR11","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1007\/s10623-014-0010-1","volume":"78","author":"C Delfs","year":"2016","unstructured":"Delfs, C., Galbraith, S.D.: Computing isogenies between supersingular elliptic curves over \n$${\\mathbb{F}}_p$$\n. Des. Codes Cryptogr. 78(2), 425\u2013440 (2016)","journal-title":"Des. Codes Cryptogr."},{"key":"11_CR12","unstructured":"Eisentr\u00e4ger, K., Hallgren, S., Morrison, T.: On the hardness of computing endomorphism rings of supersingular elliptic curves. Cryptology ePrint Archive, Report 2017\/986 (2017). https:\/\/eprint.iacr.org\/2017\/986"},{"key":"11_CR13","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1112\/S1461157000000097","volume":"2","author":"SD Galbraith","year":"1999","unstructured":"Galbraith, S.D.: Constructing isogenies between elliptic curves over finite fields. LMS J. Comput. Math. 2, 118\u2013138 (1999)","journal-title":"LMS J. Comput. Math."},{"key":"11_CR14","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/978-3-319-70694-8_1","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2017","author":"SD Galbraith","year":"2017","unstructured":"Galbraith, S.D., Petit, C., Silva, J.: Identification protocols and signature schemes based on supersingular isogeny problems. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 3\u201333. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-70694-8_1"},{"key":"11_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/978-3-662-53887-6_3","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2016","author":"SD Galbraith","year":"2016","unstructured":"Galbraith, S.D., Petit, C., Shani, B., Ti, Y.B.: On the security of supersingular isogeny cryptosystems. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016, Part I. LNCS, vol. 10031, pp. 63\u201391. Springer, Heidelberg (2016). https:\/\/doi.org\/10.1007\/978-3-662-53887-6_3"},{"key":"11_CR16","unstructured":"Gross, B.H.: Heights and the special values of \n$$L$$\n-series. In: Number Theory, Montreal, QC, 1985. CMS Conference Proceedings, vol. 7, pp. 115\u2013187. American Mathematical Society, Providence (1987)"},{"key":"11_CR17","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/978-3-319-59879-6_6","volume-title":"Post-Quantum Cryptography","author":"A G\u00e9lin","year":"2017","unstructured":"G\u00e9lin, A., Wesolowski, B.: Loop-abort faults on supersingular isogeny cryptosystems. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 93\u2013106. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-59879-6_6"},{"issue":"4","key":"11_CR18","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1090\/S0273-0979-06-01126-8","volume":"43","author":"S Hoory","year":"2006","unstructured":"Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43(4), 439\u2013561 (2006)","journal-title":"Bull. Amer. Math. Soc. (N.S.)"},{"key":"11_CR19","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":"11_CR20","doi-asserted-by":"publisher","first-page":"418","DOI":"10.1112\/S1461157014000151","volume":"17","author":"D Kohel","year":"2014","unstructured":"Kohel, D., Lauter, K., Petit, C., Tignol, J.-P.: On the quaternion \n$$\\ell $$\n-isogeny path problem. LMS J. Comput. Math. 17, 418\u2013432 (2014)","journal-title":"LMS J. Comput. Math."},{"key":"11_CR21","unstructured":"Kohel, D.: Endomorphism rings of elliptic curves over finite fields. Ph.D. thesis, University of California, Berkeley (1996)"},{"issue":"5","key":"11_CR22","doi-asserted-by":"publisher","first-page":"1714","DOI":"10.1137\/080734467","volume":"39","author":"M Kirschmer","year":"2010","unstructured":"Kirschmer, M., Voight, J.: Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. 39(5), 1714\u20131747 (2010)","journal-title":"SIAM J. Comput."},{"key":"11_CR23","series-title":"Graduate Texts in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4752-4","volume-title":"Elliptic Functions","author":"S Lang","year":"1987","unstructured":"Lang, S.: Elliptic Functions. Graduate Texts in Mathematics, vol. 112, 2nd edn. Springer, New York (1987). https:\/\/doi.org\/10.1007\/978-1-4612-4752-4. With an appendix by J. Tate","edition":"2"},{"key":"11_CR24","unstructured":"Lauter, K., McMurdy, K.: Explicit generators of endomorphism rings of supersingular elliptic curves. Preprint (2004)"},{"key":"11_CR25","unstructured":"Lagarias, J.C., Odlyzko, A.M.: Effective versions of the Chebotarev density theorem. In: Algebraic Number Fields: \n$$L$$\n-functions and Galois Properties: Proceedings of Symposium, Durham University, Durham, 1975, pp. 409\u2013464. Academic Press, London (1977)"},{"key":"11_CR26","unstructured":"Mestre, J.-F.: La m\u00e9thode des graphes. Exemples et applications. In: Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields, Katata, 1986, pp. 217\u2013242. Nagoya University, Nagoya (1986)"},{"key":"11_CR27","unstructured":"NIST: Post-quantum cryptography (2016). http:\/\/csrc.nist.gov\/Projects\/Post-Quantum-Cryptography. Accessed 30 Sept 2017"},{"issue":"4","key":"11_CR28","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1145\/1597036.1597050","volume":"5","author":"PQ Nguyen","year":"2009","unstructured":"Nguyen, P.Q., Stehl\u00e9, D.: Low-dimensional lattice basis reduction revisited. ACM Trans. Algorithms 5(4), 48 (2009). Article No. 46","journal-title":"ACM Trans. Algorithms"},{"key":"11_CR29","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"330","DOI":"10.1007\/978-3-319-70697-9_12","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2017","author":"C Petit","year":"2017","unstructured":"Petit, C.: Faster algorithms for isogeny problems using torsion point images. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10625, pp. 330\u2013353. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-70697-9_12"},{"issue":"2","key":"11_CR30","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1016\/0021-8693(80)90151-9","volume":"64","author":"A Pizer","year":"1980","unstructured":"Pizer, A.: An algorithm for computing modular forms on \n$$\\varGamma _{0}(N)$$\n. J. Algebra 64(2), 340\u2013390 (1980)","journal-title":"J. Algebra"},{"key":"11_CR31","unstructured":"Petit, C., Lauter, K.: Hard and easy problems for supersingular isogeny graphs. Cryptology ePrint Archive, Report 2017\/962 (2017). https:\/\/eprint.iacr.org\/2017\/962"},{"issue":"3","key":"11_CR32","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/BF01272075","volume":"2","author":"L R\u00f3nyai","year":"1992","unstructured":"R\u00f3nyai, L.: Algorithmic properties of maximal orders in simple algebras over \n$${ Q}$$\n. Comput. Complex. 2(3), 225\u2013243 (1992)","journal-title":"Comput. Complex."},{"key":"11_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-09494-6","volume-title":"The Arithmetic of Elliptic Curves","author":"JH Silverman","year":"2009","unstructured":"Silverman, J.H.: The Arithmetic of Elliptic Curves. Springer, New York (2009). https:\/\/doi.org\/10.1007\/978-0-387-09494-6"},{"key":"11_CR34","unstructured":"Simon, D.: Quadratic equations in dimensions 4, 5 and more. Preprint (2005)"},{"key":"11_CR35","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/978-3-319-59879-6_7","volume-title":"Post-Quantum Cryptography","author":"YB Ti","year":"2017","unstructured":"Ti, Y.B.: Fault attack on supersingular isogeny cryptosystems. In: Lange, T., Takagi, T. (eds.) PQCrypto 2017. LNCS, vol. 10346, pp. 107\u2013122. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-59879-6_7"},{"key":"11_CR36","first-page":"A238","volume":"273","author":"J V\u00e9lu","year":"1971","unstructured":"V\u00e9lu, J.: Isog\u00e9nies entre courbes elliptiques. C. R. Acad. Sci. Paris S\u00e9r. A-B 273, A238\u2013A241 (1971)","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A-B"},{"key":"11_CR37","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0091027","volume-title":"Arithm\u00e9tique des Alg\u00e8bres de Quaternions","author":"M-F Vign\u00e9ras","year":"1980","unstructured":"Vign\u00e9ras, M.-F.: Arithm\u00e9tique des Alg\u00e8bres de Quaternions. LNM, vol. 800. Springer, Heidelberg (1980). https:\/\/doi.org\/10.1007\/BFb0091027"},{"key":"11_CR38","unstructured":"Voight, J.: Quaternion Algebras. Version v0.9.7, 3 September 2017"},{"issue":"2","key":"11_CR39","doi-asserted-by":"publisher","first-page":"521","DOI":"10.24033\/asens.1183","volume":"4","author":"WC Waterhouse","year":"1969","unstructured":"Waterhouse, W.C.: Abelian varieties over finite fields. Ann. Sci. \u00c9cole Norm. Sup. 4(2), 521\u2013560 (1969)","journal-title":"Ann. Sci. \u00c9cole Norm. Sup."},{"key":"11_CR40","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/978-3-319-70972-7_9","volume-title":"Financial Cryptography and Data Security","author":"Y Yoo","year":"2017","unstructured":"Yoo, Y., Azarderakhsh, R., Jalali, A., Jao, D., Soukharev, V.: A post-quantum digital signature scheme based on supersingular isogenies. In: Kiayias, A. (ed.) FC 2017. LNCS, vol. 10322, pp. 163\u2013181. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-70972-7_9"}],"container-title":["Lecture Notes in Computer Science","Advances in Cryptology \u2013 EUROCRYPT 2018"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-78372-7_11","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T15:12:57Z","timestamp":1709824377000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-319-78372-7_11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018]]},"ISBN":["9783319783710","9783319783727"],"references-count":40,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-78372-7_11","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018]]},"assertion":[{"value":"31 March 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"EUROCRYPT","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Annual International Conference on the Theory and Applications of Cryptographic Techniques","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Tel Aviv","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Israel","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2018","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"29 April 2018","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"3 May 2018","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"37","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"eurocrypt2018","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/eurocrypt.iacr.org\/2018\/index.html","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}