{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T03:07:59Z","timestamp":1781838479504,"version":"3.54.5"},"publisher-location":"Berlin, Heidelberg","reference-count":27,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783642041587","type":"print"},{"value":"9783642041594","type":"electronic"}],"license":[{"start":{"date-parts":[[2009,1,1]],"date-time":"2009-01-01T00:00:00Z","timestamp":1230768000000},"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":[[2009]]},"DOI":"10.1007\/978-3-642-04159-4_20","type":"book-chapter","created":{"date-parts":[[2009,8,22]],"date-time":"2009-08-22T05:19:34Z","timestamp":1250918374000},"page":"309-327","source":"Crossref","is-referenced-by-count":12,"title":["The Elliptic Curve Discrete Logarithm Problem and Equivalent Hard Problems for Elliptic Divisibility Sequences"],"prefix":"10.1007","author":[{"given":"Kristin E.","family":"Lauter","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Katherine E.","family":"Stange","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","reference":[{"key":"20_CR1","doi-asserted-by":"crossref","unstructured":"Pohlig, S.C., Hellman, M.E.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans. Information Theory IT-24, 106\u2013110 (1978)","DOI":"10.1109\/TIT.1978.1055817"},{"key":"20_CR2","unstructured":"Shipsey, R.: Elliptic Divibility Sequences. PhD thesis, Goldsmiths, University of London (2001)"},{"key":"20_CR3","unstructured":"Shipsey, R., Swart, C.: Elliptic divisibility sequences and the elliptic curve discrete logarithm problem (2008), http:\/\/eprint.iacr.org\/2008\/444"},{"key":"20_CR4","doi-asserted-by":"publisher","first-page":"31","DOI":"10.2307\/2371930","volume":"70","author":"M. Ward","year":"1948","unstructured":"Ward, M.: Memoir on elliptic divisibility sequences. Amer. J. Math.\u00a070, 31\u201374 (1948)","journal-title":"Amer. J. Math."},{"key":"20_CR5","unstructured":"Swart, C.: Elliptic curves and related sequences. PhD thesis, Royal Holloway and Bedford New College, University of London (2003)"},{"key":"20_CR6","doi-asserted-by":"publisher","first-page":"585","DOI":"10.5802\/aif.1349","volume":"43","author":"M. Ayad","year":"1993","unstructured":"Ayad, M.: P\u00e9riodicit\u00e9 (mod q) des suites elliptiques et points S-entiers sur les courbes elliptiques. Ann. Inst. Fourier (Grenoble)\u00a043, 585\u2013618 (1993)","journal-title":"Ann. Inst. Fourier (Grenoble)"},{"key":"20_CR7","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1007\/s00229-004-0468-7","volume":"114","author":"J.H. Silverman","year":"2004","unstructured":"Silverman, J.H.: Common divisors of elliptic divisibility sequences over function fields. Manuscripta Math.\u00a0114, 431\u2013446 (2004)","journal-title":"Manuscripta Math."},{"key":"20_CR8","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/s00208-004-0608-0","volume":"332","author":"J.H. Silverman","year":"2005","unstructured":"Silverman, J.H.: p-adic properties of division polynomials and elliptic divisibility sequences. Math. Ann.\u00a0332, 443\u2013471 (2005)","journal-title":"Math. Ann."},{"key":"20_CR9","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.jnt.2005.08.002","volume":"118","author":"G. Everest","year":"2006","unstructured":"Everest, G., Mclaren, G., Ward, T.: Primitive divisors of elliptic divisibility sequences. J. Number Theory\u00a0118, 71\u201389 (2006)","journal-title":"J. Number Theory"},{"key":"20_CR10","unstructured":"Stange, K.E.: Elliptic nets and elliptic curves (submitted) (2007), http:\/\/arxiv.org\/abs\/0710.1316v2"},{"key":"20_CR11","unstructured":"Stange, K.E.: Elliptic nets and elliptic curves. PhD thesis, Brown University (May 2008)"},{"key":"20_CR12","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1090\/surv\/104\/10","volume-title":"Recurrence Sequences","author":"G. Everest","year":"2003","unstructured":"Everest, G., Poorten, A.v.d., Shparlinski, I., Ward, T.: Elliptic Divisibility Sequences. In: Recurrence Sequences, pp. 163\u2013175. American Mathematical Society, Providence (2003)"},{"key":"20_CR13","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1007\/978-3-540-73489-5_19","volume-title":"Pairing-Based Cryptography \u2013 Pairing 2007","author":"K.E. Stange","year":"2007","unstructured":"Stange, K.E.: The tate pairing via elliptic nets. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol.\u00a04575, pp. 329\u2013348. Springer, Heidelberg (2007)"},{"key":"20_CR14","unstructured":"Gosper, R.W., Orman., H., Schroeppel, R.: Using somos sequences for cryptography"},{"key":"#cr-split#-20_CR15.1","unstructured":"Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol.??106. Springer, New York (1992);"},{"key":"#cr-split#-20_CR15.2","unstructured":"Corrected reprint of the, original (1986)"},{"key":"20_CR16","unstructured":"Stange, K.E.: Elliptic nets, generalised Jacobians and bi-extensions (in preparation)"},{"key":"20_CR17","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1201\/9781420034981.ch4","volume-title":"Handbook of elliptic and hyperelliptic curve cryptography. Discrete Math. Appl. (Boca Raton)","author":"G. Frey","year":"2005","unstructured":"Frey, G., Lange, T.: Background on curves and Jacobians. In: Handbook of elliptic and hyperelliptic curve cryptography. Discrete Math. Appl. (Boca Raton), pp. 45\u201385. Chapman & Hall\/CRC, Boca Raton (2005)"},{"key":"20_CR18","series-title":"Foundations of Computing Series","volume-title":"Algorithmic number theory, Efficient algorithms","author":"E. Bach","year":"1996","unstructured":"Bach, E., Shallit, J.: Algorithmic number theory, Efficient algorithms. Foundations of Computing Series, vol.\u00a01. MIT Press, Cambridge (1996)"},{"key":"20_CR19","first-page":"115","volume-title":"Handbook of elliptic and hyperelliptic curve cryptography. Discrete Math. Appl. (Boca Raton)","author":"S. Duquesne","year":"2006","unstructured":"Duquesne, S., Frey, G.: Background on pairings. In: Handbook of elliptic and hyperelliptic curve cryptography. Discrete Math. Appl. (Boca Raton), pp. 115\u2013124. Chapman & Hall\/CRC, Boca Raton (2006)"},{"key":"20_CR20","series-title":"London Math. Soc. Lecture Note Ser.","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1017\/CBO9780511546570.011","volume-title":"Advances in elliptic curve cryptography","author":"S.D. Galbraith","year":"2005","unstructured":"Galbraith, S.D.: Pairings. In: Advances in elliptic curve cryptography. London Math. Soc. Lecture Note Ser., vol.\u00a0317, pp. 183\u2013213. Cambridge Univ. Press, Cambridge (2005)"},{"key":"20_CR21","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1109\/18.259647","volume":"39","author":"A.J. Menezes","year":"1993","unstructured":"Menezes, A.J., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inform. Theory\u00a039, 1639\u20131646 (1993)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"20_CR22","first-page":"865","volume":"62","author":"G. Frey","year":"1994","unstructured":"Frey, G., R\u00fcck, H.G.: A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. Comp.\u00a062, 865\u2013874 (1994)","journal-title":"Math. Comp."},{"key":"20_CR23","series-title":"CRC Press Series on Discrete Mathematics and its Applications","volume-title":"Handbook of applied cryptography","author":"A.J. Menezes","year":"1997","unstructured":"Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of applied cryptography. CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton (1997) (With a foreword by Ronald L. Rivest)"},{"key":"20_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9316-0","volume-title":"Prime numbers.","author":"R. Crandall","year":"2001","unstructured":"Crandall, R., Pomerance, C.: Prime numbers. Springer, New York (2001) (A computational perspective)"},{"key":"20_CR25","unstructured":"K. Fong, D. Hankerson, J.L\u00f3pez., Menezes, A.: Field inversion and point halving revisited. Technical Report, CORR 2003-18, Department of Combinatorics and Optimization, University of Waterloo, Canada (2003)"},{"key":"20_CR26","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/0020-0190(89)90127-0","volume":"30","author":"T. Itoh","year":"1989","unstructured":"Itoh, T., Tsujii, S.: An efficient algorithm for deciding quadratic residuosity in finite fields GF(p m ). Inform. Process. Lett.\u00a030, 111\u2013114 (1989)","journal-title":"Inform. Process. Lett."}],"container-title":["Lecture Notes in Computer Science","Selected Areas in Cryptography"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-642-04159-4_20","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T03:54:39Z","timestamp":1558497279000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-642-04159-4_20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009]]},"ISBN":["9783642041587","9783642041594"],"references-count":27,"URL":"https:\/\/doi.org\/10.1007\/978-3-642-04159-4_20","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009]]}}}