{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T03:21:00Z","timestamp":1778296860952,"version":"3.51.4"},"publisher-location":"Berlin, Heidelberg","reference-count":27,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783540651093","type":"print"},{"value":"9783540496496","type":"electronic"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1998]]},"DOI":"10.1007\/3-540-49649-1_7","type":"book-chapter","created":{"date-parts":[[2007,11,14]],"date-time":"2007-11-14T01:21:12Z","timestamp":1195003272000},"page":"66-79","source":"Crossref","is-referenced-by-count":11,"title":["Efficient Implementation of Schoof\u2019s Algorithm"],"prefix":"10.1007","author":[{"given":"Tetsuya","family":"Izu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Kogure","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Masayuki","family":"Noro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kazuhiro","family":"Yokoyama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2002,9,24]]},"reference":[{"key":"7_CR1","unstructured":"Atkin, A.O., The number of points on an elliptic curve modulo a prime, preprint, 1988."},{"key":"7_CR2","doi-asserted-by":"publisher","first-page":"29","DOI":"10.2307\/2152935","volume":"61","author":"A.O. Atkin","year":"1993","unstructured":"Atkin, A.O., Morain, F., Elliptic curves and primality proving, Math. Comp. 61 (1993) 29\u201368.","journal-title":"Math. Comp."},{"key":"7_CR3","series-title":"Lect Notes Comput Sci","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1007\/3-540-48658-5_6","volume-title":"CRYPTO \u201894","author":"J. Chao","year":"1994","unstructured":"Chao, J., Tanada, K., Tsujii, S., Design of elliptic curves with controllable lower boundary of extension degree for reduction attacks, in CRYPTO \u201894, Y. Desmedt, Ed., Lecture Notes in Computer Science, 839, pp.50\u201355, 1994."},{"key":"7_CR4","unstructured":"Charlap, L.S., Coley, R., Robbins, D.P., Enumeration of rational points on elliptic curves over finite fields, preprint, 1991."},{"key":"7_CR5","unstructured":"Couveignes, J.-M., Dewaghe, L., Morain, F., Isogeny cycles and the Schoof-Elkies-Atkin algorithm, LIX\/RR\/96\/03, 1996."},{"key":"7_CR6","series-title":"Lect Notes Comput Sci","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/3-540-58691-1_42","volume-title":"ANT-I","author":"J.-M. Couveignes","year":"1994","unstructured":"Couveignes, J.-M., Morain, F., Schoof\u2019s algorithm and isogeny cycles, in ANT-I, L. Adleman and M.-D. Huang, Eds., Lecture Notes in Computer Science, 877, pp.43\u201358, 1994."},{"key":"7_CR7","doi-asserted-by":"crossref","unstructured":"Elkies, N.D., Explicit isogenies, preprint, 1991.","DOI":"10.1155\/S1073792891000144"},{"key":"7_CR8","unstructured":"Izu, T., Noro, M., Fast remainder calculation in polynomial multiplication, in preparation."},{"key":"7_CR9","doi-asserted-by":"publisher","first-page":"203","DOI":"10.2307\/2007884","volume":"48","author":"N. Kobilitz","year":"1987","unstructured":"Kobilitz N., Elliptic curve cryptosystems, Math, Comp. 48 (1987) 203\u2013209.","journal-title":"Math, Comp."},{"key":"7_CR10","series-title":"Lect Notes Comput Sci","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1007\/3-540-58691-1_64","volume-title":"ANT-I","author":"G.-J. Lay","year":"1994","unstructured":"Lay, G.-J., Zimmer, H.G., Constructing elliptic curves with given group order over large finite fields, in ANT-I, L. Adleman and M.-D. Huang, Eds., Lecture Notes in Computer Science, 877, pp.250\u2013263, 1994."},{"key":"7_CR11","doi-asserted-by":"publisher","first-page":"649","DOI":"10.2307\/1971363","volume":"126","author":"H.W. Lenstra Jr.","year":"1987","unstructured":"Lenstra Jr., H.W., Factoring integers with elliptic curves, Annals of Mathematics 126 (1987) 649\u2013673.","journal-title":"Annals of Mathematics"},{"key":"7_CR12","unstructured":"Lercier, R., Algorithmique des courbes elliptiques dans les corps finis, Doctoral Thesis, L\u2019\u00e9cole Polytechnique, 1997."},{"key":"7_CR13","series-title":"Lect Notes Comput Sci","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/3-540-69053-0_26","volume-title":"EURO-CRYPT \u201897","author":"R. Lercier","year":"1997","unstructured":"Lercier, R., Finding good random elliptic curves for cryptosystems defined over F2 n, in EURO-CRYPT \u201897, W. Fumy, Ed., Lecture Notes in Computer Science, 1233, pp.379\u2013392, 1997."},{"key":"7_CR14","series-title":"Lect Notes Comput Sci","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/3-540-49264-X_7","volume-title":"EURO-CRYPT \u201895","author":"R. Lercier","year":"1995","unstructured":"Lercier, R., Morain, F., Counting the number of points on elliptic curves over finite fields: strategy and performances, in EURO-CRYPT \u201895, L.C. Guillou and J.-J. Quisquater, Eds., Lecture Notes in Computer Science, 921, pp.79\u201394, 1995."},{"key":"7_CR15","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4615-3198-2","volume-title":"Elliptic curve public key cryptosystems","author":"A. Menezes","year":"1993","unstructured":"Menezes, A., Elliptic curve public key cryptosystems, Kluwer Academic Publishers, Boston, 1993."},{"key":"7_CR16","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1145\/103418.103434","volume-title":"STOC \u201891","author":"A. Menezes","year":"1991","unstructured":"Menezes, A., Okamoto, T., Vanstone, S.E., Reducing elliptic curves logarithms to logarithms in a finite field, in STOC \u201891, ACM Press, New York, pp.80\u201389, 1991."},{"key":"7_CR17","series-title":"Lect Notes Comput Sci","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/3-540-39799-X_31","volume-title":"CRYPTO \u201885","author":"V.S. Miller","year":"1986","unstructured":"Miller, V.S., Uses of elliptic curves in cryptography, in CRYPTO \u201885, Lecture Notes in Computer Science, 218, pp.417\u2013426, 1986."},{"key":"7_CR18","series-title":"Lect Notes Comput Sci","first-page":"479","volume-title":"AUSCRYPT \u201892","author":"A. Miyaji","year":"1992","unstructured":"Miyaji, A., Elliptic curves over F p suitable for cryptosystems, in AUSCRYPT \u201892, J. Seberry and Y. Zhengs, Eds., Lecture Notes in Computer Science, 718, pp.479\u2013491, 1992."},{"key":"7_CR19","doi-asserted-by":"crossref","first-page":"255","DOI":"10.5802\/jtnb.143","volume":"7","author":"F. Morain","year":"1995","unstructured":"Morain, F., Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques, J. Th\u00e9or. Nombres Bordeaux 7 (1995) 255\u2013282.","journal-title":"J. Th\u00e9or. Nombres Bordeaux"},{"key":"7_CR20","unstructured":"Risa\/ Asir, ( ftp:\/\/endeavor.fujitsu.co.jp\/pub\/isis\/asir )."},{"key":"7_CR21","first-page":"81","volume":"47","author":"T. Satoh","year":"1998","unstructured":"Satoh, T., Araki, K., Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Commentarii Mathematici Universitatis Sancti Pauli 47 (1998) 81\u201392.","journal-title":"Commentarii Mathematici Universitatis Sancti Pauli"},{"key":"7_CR22","doi-asserted-by":"publisher","first-page":"483","DOI":"10.2307\/2007968","volume":"44","author":"R. Schoof","year":"1985","unstructured":"Schoof, R., Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985) 483\u2013494.","journal-title":"Math. Comp."},{"key":"7_CR23","doi-asserted-by":"crossref","first-page":"219","DOI":"10.5802\/jtnb.142","volume":"7","author":"R. Schoof","year":"1995","unstructured":"Schoof, R., Counting points on elliptic curves over finite fields, J. Th\u00e9or. Nombres Bordeaux 7 (1995) 219\u2013254.","journal-title":"J. Th\u00e9or. Nombres Bordeaux"},{"key":"7_CR24","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1006\/jsco.1995.1055","volume":"20","author":"V. Shoup","year":"1995","unstructured":"Shoup, V., A new polynomial factorization algorithm and its implementation, J. Symbolic Computation. 20 (1995) 364\u2013397.","journal-title":"J. Symbolic Computation"},{"key":"7_CR25","doi-asserted-by":"crossref","unstructured":"Silverman, J.H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, 1986.","DOI":"10.1007\/978-1-4757-1920-8"},{"key":"7_CR26","doi-asserted-by":"crossref","unstructured":"Silverman, J.H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151, Springer-Verlag, 1994.","DOI":"10.1007\/978-1-4612-0851-8"},{"key":"7_CR27","unstructured":"Smart, N.P., The discrete logarithm problem on elliptic curves of trace one, preprint, 1997."}],"container-title":["Lecture Notes in Computer Science","Advances in Cryptology \u2014 ASIACRYPT\u201998"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/3-540-49649-1_7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,4]],"date-time":"2019-05-04T11:29:33Z","timestamp":1556969373000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/3-540-49649-1_7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"ISBN":["9783540651093","9783540496496"],"references-count":27,"URL":"https:\/\/doi.org\/10.1007\/3-540-49649-1_7","relation":{},"ISSN":["0302-9743"],"issn-type":[{"value":"0302-9743","type":"print"}],"subject":[],"published":{"date-parts":[[1998]]}}}