{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T07:30:09Z","timestamp":1742974209207,"version":"3.40.3"},"publisher-location":"Cham","reference-count":24,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783319159423"},{"type":"electronic","value":"9783319159430"}],"license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"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":[[2015]]},"DOI":"10.1007\/978-3-319-15943-0_5","type":"book-chapter","created":{"date-parts":[[2015,3,16]],"date-time":"2015-03-16T10:27:17Z","timestamp":1426501637000},"page":"69-83","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Discrete Logarithms for Torsion Points on Elliptic Curve of Embedding Degree $$1$$"],"prefix":"10.1007","author":[{"given":"Yasuyuki","family":"Nogami","sequence":"first","affiliation":[]},{"given":"Hwajeong","family":"Seo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,3,17]]},"reference":[{"key":"5_CR1","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/11693383_22","volume-title":"Selected Areas in Cryptography","author":"PSLM Barreto","year":"2006","unstructured":"Barreto, P.S.L.M., Naehrig, M.: Pairing-friendly elliptic curves of prime order. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 319\u2013331. Springer, Heidelberg (2006)"},{"key":"5_CR2","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1007\/978-3-540-30576-7_18","volume-title":"Theory of Cryptography","author":"D Boneh","year":"2005","unstructured":"Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325\u2013341. Springer, Heidelberg (2005)"},{"key":"5_CR3","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/11761679_34","volume-title":"Advances in Cryptology - EUROCRYPT 2006","author":"D Boneh","year":"2006","unstructured":"Boneh, D., Sahai, A., Waters, B.: Fully collusion resistant traitor tracing with short ciphertexts and private keys. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 573\u2013592. Springer, Heidelberg (2006)"},{"key":"5_CR4","unstructured":"Boneh, D., Rabin, K., Silverberg, A.: Finding composite order ordinary elliptic curves using the cocks-pinch method. In: Cryptology ePrint Archive, Report 2009\/533 (2009)"},{"key":"5_CR5","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"374","DOI":"10.1007\/978-3-642-32928-9_21","volume-title":"Security and Cryptography for Networks","author":"G Castagnos","year":"2012","unstructured":"Castagnos, G., Laguillaumie, F.: Homomorphic encryption for multiplications and pairing evaluation. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 374\u2013392. Springer, Heidelberg (2012)"},{"key":"5_CR6","unstructured":"Charles, D.: On the existence of distortion maps on ordinary elliptic curves. In: Cryptology ePrint Archive, Report 2006\/128 (2006)"},{"key":"5_CR7","series-title":"Discrete Mathematics and Its Applications","doi-asserted-by":"publisher","DOI":"10.1201\/9781420034981","volume-title":"Handbook of Elliptic and Hyperelliptic Curve Cryptography","author":"H Cohen","year":"2005","unstructured":"Cohen, H., Frey, G.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and Its Applications. Chapman & Hall CRC, Boca Raton (2005)"},{"key":"5_CR8","doi-asserted-by":"publisher","first-page":"161","DOI":"10.2307\/1967224","volume":"11","author":"LE Dickson","year":"1897","unstructured":"Dickson, L.E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group. Ann. Math. 11, 161\u2013183 (1897)","journal-title":"Ann. Math."},{"key":"5_CR9","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/978-3-540-85538-5_15","volume-title":"Pairing-Based Cryptography \u2013 Pairing 2008","author":"SD Galbraith","year":"2008","unstructured":"Galbraith, S.D., Scott, M.: Exponentiation in pairing-friendly groups using homomorphisms. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 211\u2013224. Springer, Heidelberg (2008)"},{"key":"5_CR10","unstructured":"GNU MP. http:\/\/gmplib.org\/"},{"key":"5_CR11","volume-title":"Guide to Elliptic Curves Cryptography","author":"D Hankerson","year":"2004","unstructured":"Hankerson, D., Vanstone, S., Menezes, A.: Guide to Elliptic Curves Cryptography. Springer, New York (2004)"},{"key":"5_CR12","unstructured":"Hitt, L.: On the minimal embedding field. In: Cryptology ePrint Archive, Report 2006\/415 (2006)"},{"key":"5_CR13","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/0890-5401(88)90024-7","volume":"78","author":"T Itoh","year":"1988","unstructured":"Itoh, T., Tsujii, S.: A fast algorithm for computing multiplicative inverses in GF$$(2^m)$$ using normal bases. Inf. Comp. 78, 171\u2013177 (1988)","journal-title":"Inf. Comp."},{"key":"5_CR14","unstructured":"Izuta, T., Takeuchi, S., Nishii, K., Nogami, Y., Morikawa, Y.: GLV subgroups on non-supersingular pairing-friendly curves of embedding degree 1. In: Computer Security Symposium 2010, pp. 249\u2013254 (2010)"},{"issue":"4","key":"5_CR15","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/s00145-004-0312-y","volume":"17","author":"A Joux","year":"2004","unstructured":"Joux, A.: A one round protocol for tripartite diffie-hellman. J. Cryptol. 17(4), 263\u2013276 (2004)","journal-title":"J. Cryptol."},{"key":"5_CR16","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1007\/11586821_2","volume-title":"Cryptography and Coding","author":"N Koblitz","year":"2005","unstructured":"Koblitz, N., Menezes, A.: Pairing-based cryptography at high security levels. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13\u201336. Springer, Heidelberg (2005)"},{"key":"5_CR17","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1007\/BFb0054855","volume-title":"Algorithmic Number Theory","author":"F Morain","year":"1998","unstructured":"Morain, F.: Primality proving using elliptic curves: an update. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 111\u2013127. Springer, Heidelberg (1998)"},{"key":"5_CR18","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/11593447_29","volume-title":"Advances in Cryptology - ASIACRYPT 2005","author":"T Nakanishi","year":"2005","unstructured":"Nakanishi, T., Funabiki, N.: Verifier-local revocation group signature schemes with backward unlinkability from bilinear maps. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 533\u2013548. Springer, Heidelberg (2005)"},{"key":"5_CR19","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1007\/978-3-540-85538-5_13","volume-title":"Pairing-Based Cryptography \u2013 Pairing 2008","author":"Y Nogami","year":"2008","unstructured":"Nogami, Y., Akane, M., Sakemi, Y., Kato, H., Morikawa, Y.: Integer variable $$\\chi $$\u2013based ate pairing. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 178\u2013191. Springer, Heidelberg (2008)"},{"key":"5_CR20","unstructured":"Ohta, K., Shiota, K.: Construction of CM Curves Suitable for Cryptosystem from the Weil Pairing. Memoirs of the Faculty of Science, Kochi Univ., Vol. 27, No. 1 (2007)"},{"key":"5_CR21","unstructured":"Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystems based on pairing. In: SCIS 2000 (2000)"},{"key":"5_CR22","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"226","DOI":"10.1007\/978-3-540-89641-8_16","volume-title":"Cryptology and Network Security","author":"Y Sakemi","year":"2008","unstructured":"Sakemi, Y., Nogami, Y., Okeya, K., Kato, H., Morikawa, Y.: Skew frobenius map and efficient scalar multiplication for pairing\u2013based cryptography. In: Franklin, M.K., Hui, L.C.K., Wong, D.S. (eds.) CANS 2008. LNCS, vol. 5339, pp. 226\u2013239. Springer, Heidelberg (2008)"},{"key":"5_CR23","series-title":"LMS Lecture Note Series","volume-title":"Elliptic Curves in Cryptography","author":"N Smart","year":"1999","unstructured":"Smart, N., Blake, I.F., Seroussi, G.: Elliptic Curves in Cryptography. LMS Lecture Note Series. Cambridge University Press, New York (1999)"},{"issue":"1","key":"5_CR24","doi-asserted-by":"publisher","first-page":"188","DOI":"10.1587\/transfun.E93.A.188","volume":"E93\u2013A","author":"M Yoshida","year":"2010","unstructured":"Yoshida, M., Mitsunari, S., Fujiwara, T.: The vector decomposition problem. IEICE Trans. Fundamentals E93\u2013A(1), 188\u2013193 (2010)","journal-title":"IEICE Trans. Fundamentals"}],"container-title":["Lecture Notes in Computer Science","Information Security and Cryptology - ICISC 2014"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-15943-0_5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T22:46:50Z","timestamp":1676933210000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-319-15943-0_5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"ISBN":["9783319159423","9783319159430"],"references-count":24,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-15943-0_5","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2015]]},"assertion":[{"value":"17 March 2015","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}