{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,15]],"date-time":"2024-09-15T14:02:28Z","timestamp":1726408948897},"publisher-location":"Cham","reference-count":23,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783319321608"},{"type":"electronic","value":"9783319321622"}],"license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016]]},"DOI":"10.1007\/978-3-319-32162-2_12","type":"book-chapter","created":{"date-parts":[[2016,7,5]],"date-time":"2016-07-05T10:20:27Z","timestamp":1467714027000},"page":"403-416","source":"Crossref","is-referenced-by-count":0,"title":["The Discrete Logarithm Problem"],"prefix":"10.1007","author":[{"given":"Ren\u00e9","family":"Schoof","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,7,6]]},"reference":[{"key":"12_CR1","doi-asserted-by":"crossref","unstructured":"Barbulescu, R., Gaudry, P., Joux, A. and Thom\u00e9, E.: A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic, In Nguyen, P., Oswald, E. (Eds) Eurocrypto 2014, LNCS 8441, 1\u201316, Springer 2014.","DOI":"10.1007\/978-3-642-55220-5_1"},{"key":"12_CR2","doi-asserted-by":"crossref","first-page":"1874","DOI":"10.1016\/j.jnt.2008.02.002","volume":"128","author":"J.V. Belding","year":"2008","unstructured":"Belding, J.V.: A Weil pairing on the p-torsion of ordinary elliptic curves over K [ \ud835\udf00 ] $$K[\\varepsilon ]$$ , J. of Number Theory, 128 (2008), 1874\u20131888.","journal-title":"J. of Number Theory"},{"key":"12_CR3","doi-asserted-by":"crossref","unstructured":"Buchmann, J., Jacobson, M. and Teske, E.: On some computational problems in finite abelian groups, Math. Comp. 66 (1997), 1663\u20131687.","DOI":"10.1090\/S0025-5718-97-00880-6"},{"key":"12_CR4","doi-asserted-by":"crossref","unstructured":"Canfield, E.R., Pomerance C. and Erd\u0151s, P.: On a problem of Oppenheim concerning \u2018Factorisation Numerorum\u2019, J. Number Theory 17 (1981), 1\u201328.","DOI":"10.1016\/0022-314X(83)90002-1"},{"key":"12_CR5","doi-asserted-by":"crossref","unstructured":"Cheng, Q., Wan, D. and Zhuang, J: Traps to the BGJT-algorithm for discrete logarithms, LMS Journal of Computation and Mathematics 17 (2014), 218\u2013229.","DOI":"10.1112\/S1461157014000242"},{"key":"12_CR6","doi-asserted-by":"crossref","unstructured":"Diffie, W. and Hellman, M.: New directions in cryptography. IEEE Transactions on Information Theory 22 (1976), 587\u2013594.","DOI":"10.1109\/TIT.1976.1055638"},{"key":"12_CR7","doi-asserted-by":"crossref","unstructured":"G\u00f6lo\u011flu, F., Granger, R., McGuire, G. and Zumbr\u00e4gel, J.: On the function field sieve and the impact of higher splitting probabilities. In Canetti, R. and Garay, J. editors, Advances in Cryptology\u2014CRYPTO 2013, LNCS 8043, 109\u2013128. Springer 2013.","DOI":"10.1007\/978-3-642-40084-1_7"},{"key":"12_CR8","unstructured":"Granger, R., Kleinjung, T. and Zumbr\u00e4gel, J.: On the discrete logarithm problem in finite fields of fixed characteristic, Cryptology ePrint Archive: Report 2015\/685."},{"key":"12_CR9","volume-title":"The Art of Computer Programming, vol","author":"Donald E. Knuth","year":"1969","unstructured":"Knuth, Donald E.: The Art of Computer Programming, vol. II: Seminumerical Algorithms, Addison-Wesley 1969."},{"key":"12_CR10","doi-asserted-by":"crossref","unstructured":"Lenstra, H.W.: Finding isomorphisms between finite fields, Math. Comp. 56 (1991), 329\u2013347.","DOI":"10.1090\/S0025-5718-1991-1052099-2"},{"key":"12_CR11","volume-title":"Roots of unity in orders, Foundations of Computational Mathematics, to appear","author":"H.W. Lenstra","year":"2016","unstructured":"Lenstra, H.W. and Silverberg, A.: Roots of unity in orders, Foundations of Computational Mathematics, to appear (2016)."},{"key":"12_CR12","doi-asserted-by":"crossref","unstructured":"Menezes, A., Okamoto, T., Vanstone, S. A.: Reducing elliptic curve logarithms to logarithms in a finite field, IEEE Transactions on Information Theory 39 (1993), 1639\u20131646.","DOI":"10.1109\/18.259647"},{"key":"12_CR13","doi-asserted-by":"crossref","unstructured":"Miller, G.: Riemann\u2019s Hypothesis and tests for primality J. of Computer and System science 13 (1976), 300\u2013317.","DOI":"10.1016\/S0022-0000(76)80043-8"},{"key":"12_CR14","doi-asserted-by":"crossref","unstructured":"Miller, V.: The Weil pairing, and its efficient calculation, J. Cryptology 17 (2004), 235\u2013261.","DOI":"10.1007\/s00145-004-0315-8"},{"key":"12_CR15","doi-asserted-by":"crossref","unstructured":"Panario, D., Gourdon, X. and Flajolet, P.: An analytic approach to smooth polynomials over finite fields. In J. Buhler, editor, Algorithmic Number Theory, Proceedings of the ANTS-III conference, 1423, 226\u2013236. Springer 1998.","DOI":"10.1007\/BFb0054865"},{"key":"12_CR16","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1109\/TIT.1978.1055817","volume":"IT-24","author":"S. Pohlig","year":"1978","unstructured":"Pohlig, S. and Hellman, M.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans. Inform. Theory IT-24 (1978), 106\u2013110.","journal-title":"IEEE Trans. Inform. Theory"},{"key":"12_CR17","first-page":"918","volume":"32","author":"J. Pollard","year":"1978","unstructured":"Pollard, J.: Monte Carlo methods for index computation mod p, Mathematics of Computation, 32 (1978), 918\u2013924.","journal-title":"Mathematics of Computation"},{"key":"12_CR18","unstructured":"Satoh T. and Araki, K.: Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Comment. Math. Univ. St. Paul. (1998), 81\u201392."},{"key":"12_CR19","doi-asserted-by":"crossref","unstructured":"Semaev, I.A.: Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p, Math. Comp. 67 (1998), 353\u2013356.","DOI":"10.1090\/S0025-5718-98-00887-4"},{"key":"12_CR20","doi-asserted-by":"crossref","unstructured":"Shanks, D.: Class number, a theory of factorization and genera. In Proc. Symp. Pure Math. 20 (1971), 415\u2013440. AMS, Providence, R.I.","DOI":"10.1090\/pspum\/020\/0316385"},{"key":"12_CR21","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-09494-6","volume-title":"The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, 2 nd Ed","author":"J.H. Silverman","year":"2009","unstructured":"Silverman, J.H.: The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, 2 nd Ed. Springer\u2013Verlag, 2009."},{"key":"12_CR22","doi-asserted-by":"crossref","unstructured":"Smart, N.: The discrete logarithm problem on elliptic curves of trace one, J. Cryptology 12 (1999), 193\u2013196.","DOI":"10.1007\/s001459900052"},{"key":"12_CR23","unstructured":"Xiao, D., Zhuang, J. and Cheng, Q.: Factor base discrete logarithms in Kummer Extensions, Cryptology ePrint Archive: Report 2015\/859."}],"container-title":["Open Problems in Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-32162-2_12","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,10]],"date-time":"2019-09-10T18:24:14Z","timestamp":1568139854000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-319-32162-2_12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"ISBN":["9783319321608","9783319321622"],"references-count":23,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-32162-2_12","relation":{},"subject":[],"published":{"date-parts":[[2016]]}}}