{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:53:27Z","timestamp":1776848007405,"version":"3.51.2"},"publisher-location":"Berlin, Heidelberg","reference-count":36,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783662487990","type":"print"},{"value":"9783662488003","type":"electronic"}],"license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015]]},"DOI":"10.1007\/978-3-662-48800-3_2","type":"book-chapter","created":{"date-parts":[[2015,11,26]],"date-time":"2015-11-26T07:12:21Z","timestamp":1448521941000},"page":"31-55","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["The Tower Number Field Sieve"],"prefix":"10.1007","author":[{"given":"Razvan","family":"Barbulescu","sequence":"first","affiliation":[]},{"given":"Pierrick","family":"Gaudry","sequence":"additional","affiliation":[]},{"given":"Thorsten","family":"Kleinjung","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,12,30]]},"reference":[{"key":"2_CR1","doi-asserted-by":"crossref","unstructured":"Adleman, L.M., Lenstra, H.W.: Finding irreducible polynomials over finite fields. In: Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, pp. 350\u2013355. ACM (1986)","DOI":"10.1145\/12130.12166"},{"key":"2_CR2","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-540-76900-2_1","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2007","author":"K Aoki","year":"2007","unstructured":"Aoki, K., Franke, J., Kleinjung, T., Lenstra, A.K., Osvik, D.A.: A kilobit special number field sieve factorization. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 1\u201312. Springer, Heidelberg (2007)"},{"key":"2_CR3","unstructured":"Bai, S.: Polynomial selection for the number field sieve. Ph.D. thesis, Australian National University (2011)"},{"key":"2_CR4","doi-asserted-by":"crossref","unstructured":"Bai, S., Bouvier, C., Kruppa, A., Zimmermann, P.: Better polynomials for GNFS. Preprint (2014)","DOI":"10.1090\/mcom3048"},{"key":"2_CR5","unstructured":"Barbulescu, R.: Algorithmes de logarithmes discrets dans les corps finis. Ph.D. thesis, Universit\u00e9 de Lorraine (2013)"},{"key":"2_CR6","unstructured":"Barbulescu, R., Gaudry, P., Guillevic, A., Morain, F.: (Algebraic) improvements to the number field sieve for non-prime finite fields. Preprint \n                      http:\/\/hal.inria.fr\/hal-01052449"},{"key":"2_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/978-3-662-46800-5_6","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2015","author":"R Barbulescu","year":"2015","unstructured":"Barbulescu, R., Gaudry, P., Guillevic, A., Morain, F.: Improving NFS for the discrete logarithm problem in non-prime finite fields. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 129\u2013155. Springer, Heidelberg (2015)"},{"key":"2_CR8","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1112\/S1461157014000369","volume":"17","author":"R Barbulescu","year":"2014","unstructured":"Barbulescu, R., Pierrot, C.: The multiple number field sieve for medium- and high-characteristic finite fields. LMS J. Comput. Math. 17, 230\u2013246 (2014)","journal-title":"LMS J. Comput. Math."},{"key":"2_CR9","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)"},{"issue":"8","key":"2_CR10","doi-asserted-by":"publisher","first-page":"1995","DOI":"10.1016\/j.laa.2009.08.012","volume":"432","author":"Y Bistritz","year":"2010","unstructured":"Bistritz, Y., Lifshitz, A.: Bounds for resultants of univariate and bivariate polynomials. Linear Algebra Appl. 432(8), 1995\u20132005 (2010)","journal-title":"Linear Algebra Appl."},{"issue":"2","key":"2_CR11","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1137\/0605029","volume":"5","author":"IF Blake","year":"1984","unstructured":"Blake, I.F., Fuji-Hara, R., Mullin, R.C., Vanstone, S.A.: Computing logarithms in finite fields of characteristic two. SIAM J. Algebraic Discrete Methods 5(2), 276\u2013285 (1984)","journal-title":"SIAM J. Algebraic Discrete Methods"},{"issue":"1","key":"2_CR12","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/s10623-004-3808-4","volume":"37","author":"F Brezing","year":"2005","unstructured":"Brezing, F., Weng, A.: Elliptic curves suitable for pairing based cryptography. Des. Codes Crypt. 37(1), 133\u2013141 (2005)","journal-title":"Des. Codes Crypt."},{"key":"2_CR13","doi-asserted-by":"crossref","unstructured":"Buhler, J.P., Lenstra Jr., H.W., Pomerance, C.: Factoring integers with the number field sieve. In: Lenstra, A.K., Lenstra Jr., H.W. (eds.) The Development of the Number Field Sieve. Lecture Notes in Mathematics, vol. 1554, pp. 50\u201394. Springer, Heidelberg (1993)","DOI":"10.1007\/BFb0091539"},{"key":"2_CR14","series-title":"Graduate Texts in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8489-0","volume-title":"Advanced Topics in Computational Number Theory","author":"H Cohen","year":"2000","unstructured":"Cohen, H.: Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics, vol. 193. Springer, New York (2000)"},{"key":"2_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1007\/11745853_12","volume-title":"Public Key Cryptography - PKC 2006","author":"A Commeine","year":"2006","unstructured":"Commeine, A., Semaev, I.A.: An algorithm to solve the discrete logarithm problem with the number field sieve. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 174\u2013190. Springer, Heidelberg (2006)"},{"issue":"3","key":"2_CR16","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/BF00198464","volume":"6","author":"D Coppersmith","year":"1993","unstructured":"Coppersmith, D.: Modifications to the number field sieve. J. Cryptol. 6(3), 169\u2013180 (1993)","journal-title":"J. Cryptol."},{"issue":"4","key":"2_CR17","doi-asserted-by":"crossref","first-page":"1621","DOI":"10.1215\/ijm\/1373636699","volume":"55","author":"K Foster","year":"2011","unstructured":"Foster, K.: HT90 and \u201csimplest\u201d number fields. Illinois J. Math. 55(4), 1621\u20131655 (2011)","journal-title":"Illinois J. Math."},{"issue":"2","key":"2_CR18","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1007\/s00145-009-9048-z","volume":"23","author":"D Freeman","year":"2010","unstructured":"Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptol. 23(2), 224\u2013280 (2010)","journal-title":"J. Cryptol."},{"issue":"1","key":"2_CR19","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1137\/0406010","volume":"6","author":"DM Gordon","year":"1993","unstructured":"Gordon, D.M.: Discrete logarithms in GF(p) using the number field sieve. SIAM J. Discrete Math. 6(1), 124\u2013138 (1993)","journal-title":"SIAM J. Discrete Math."},{"key":"2_CR20","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1007\/3-540-45455-1_34","volume-title":"Algorithmic Number Theory","author":"A Joux","year":"2002","unstructured":"Joux, A., Lercier, R.: The function field sieve is quite special. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 431\u2013445. Springer, Heidelberg (2002)"},{"key":"2_CR21","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"326","DOI":"10.1007\/11818175_19","volume-title":"Advances in Cryptology - CRYPTO 2006","author":"A Joux","year":"2006","unstructured":"Joux, A., Lercier, R., Smart, N.P., Vercauteren, F.: The number field sieve in the medium prime case. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 326\u2013344. Springer, Heidelberg (2006)"},{"key":"2_CR22","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/978-3-319-04873-4_3","volume-title":"Pairing-Based Cryptography \u2013 Pairing 2013","author":"A Joux","year":"2014","unstructured":"Joux, A., Pierrot, C.: The special number field sieve in \n                      \n                        \n                      \n                      $$\\mathbb{F}_{p^{n}}$$\n                    . In: Cao, Z., Zhang, F. (eds.) Pairing 2013. LNCS, vol. 8365, pp. 45\u201361. Springer, Heidelberg (2014)"},{"issue":"256","key":"2_CR23","doi-asserted-by":"publisher","first-page":"2037","DOI":"10.1090\/S0025-5718-06-01870-9","volume":"75","author":"T Kleinjung","year":"2006","unstructured":"Kleinjung, T.: On polynomial selection for the general number field sieve. Math. Comput. 75(256), 2037\u20132047 (2006)","journal-title":"Math. Comput."},{"key":"2_CR24","unstructured":"Kleinjung, T.: Polynomial selection. Slides at CADO workshop (2008). \n                      http:\/\/cado.gforge.inria.fr\/workshop\/slides\/kleinjung.pdf"},{"key":"2_CR25","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/978-3-642-14623-7_18","volume-title":"Advances in Cryptology \u2013 CRYPTO 2010","author":"T Kleinjung","year":"2010","unstructured":"Kleinjung, T., Aoki, K., Franke, J., Lenstra, A.K., Thom\u00e9, E., Bos, J.W., Gaudry, P., Kruppa, A., Montgomery, P.L., Osvik, D.A., te Riele, H., Timofeev, A., Zimmermann, P.: Factorization of a 768-Bit RSA modulus. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 333\u2013350. Springer, Heidelberg (2010)"},{"key":"2_CR26","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1007\/978-3-662-45611-8_19","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2014","author":"T Kleinjung","year":"2014","unstructured":"Kleinjung, T., Bos, J.W., Lenstra, A.K.: Mersenne factorization factory. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8873, pp. 358\u2013377. Springer, Heidelberg (2014)"},{"key":"2_CR27","doi-asserted-by":"crossref","unstructured":"Lenstra, A.K., Lenstra Jr., H.W., Manasse, M., Pollard, J.: The number field sieve. The Development of the Number Field Sieve. Lecture Notes in Mathematics, vol. 1554, pp. 11\u201342. Springer, Heidelberg (1993)","DOI":"10.1007\/BFb0091537"},{"issue":"1","key":"2_CR28","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1515\/156939203321669546","volume":"13","author":"DV Matyukhin","year":"2003","unstructured":"Matyukhin, D.V.: On asymptotic complexity of computing discrete logarithms over GF(p). Discrete Math. Appl. 13(1), 27\u201350 (2003)","journal-title":"Discrete Math. Appl."},{"issue":"5","key":"2_CR29","first-page":"1234","volume":"84","author":"A Miyaji","year":"2001","unstructured":"Miyaji, A., Nakabayashi, M., Takano, S.: New explicit conditions of elliptic curve traces for FR-reduction. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 84(5), 1234\u20131243 (2001)","journal-title":"IEICE Trans. Fundam. Electron. Commun. Comput. Sci."},{"key":"2_CR30","unstructured":"Murphy, A., Fitzpatrick, N.: Elliptic curves for pairing applications. Cryptology ePrint Archive, Report 2005\/302 (2005). \n                      http:\/\/eprint.iacr.org\/"},{"key":"2_CR31","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1007\/978-3-662-46800-5_7","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2015","author":"C Pierrot","year":"2015","unstructured":"Pierrot, C.: The multiple number field sieve with conjugation and generalized joux-lercier methods. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 156\u2013170. Springer, Heidelberg (2015)"},{"key":"2_CR32","doi-asserted-by":"crossref","unstructured":"Pollard, J.M.: The lattice sieve. In: Lenstra, A.K., Lenstra, Jr., H.W.: The development of the number field sieve, vol. 1554 of Lecture Notes in Mathematics, pp. 43\u201349. Springer (1993)","DOI":"10.1007\/BFb0091538"},{"issue":"1676","key":"2_CR33","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1098\/rsta.1993.0139","volume":"345","author":"O Schirokauer","year":"1993","unstructured":"Schirokauer, O.: Discrete logarithms and local units. Philos. Trans. Roy. Soc. London Ser. A 345(1676), 409\u2013423 (1993)","journal-title":"Philos. Trans. Roy. Soc. London Ser. A"},{"issue":"231","key":"2_CR34","doi-asserted-by":"publisher","first-page":"1267","DOI":"10.1090\/S0025-5718-99-01137-0","volume":"69","author":"O Schirokauer","year":"2000","unstructured":"Schirokauer, O.: Using number fields to compute logarithms in finite fields. Math. Comp. 69(231), 1267\u20131283 (2000)","journal-title":"Math. Comp."},{"issue":"237","key":"2_CR35","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1090\/S0025-5718-00-01308-9","volume":"71","author":"I Semaev","year":"2002","unstructured":"Semaev, I.: Special prime numbers and discrete logs in finite prime fields. Math. Comp. 71(237), 363\u2013377 (2002)","journal-title":"Math. Comp."},{"issue":"1","key":"2_CR36","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1109\/TIT.1986.1057137","volume":"32","author":"DH Wiedemann","year":"1986","unstructured":"Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory 32(1), 54\u201362 (1986)","journal-title":"IEEE Trans. Inform. Theory"}],"container-title":["Lecture Notes in Computer Science","Advances in Cryptology \u2013 ASIACRYPT 2015"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-662-48800-3_2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,11,27]],"date-time":"2019-11-27T20:16:10Z","timestamp":1574885770000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-662-48800-3_2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"ISBN":["9783662487990","9783662488003"],"references-count":36,"URL":"https:\/\/doi.org\/10.1007\/978-3-662-48800-3_2","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015]]},"assertion":[{"value":"30 December 2015","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}