{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T17:11:21Z","timestamp":1694625081563},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2005,11,7]],"date-time":"2005-11-07T00:00:00Z","timestamp":1131321600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2005,11,7]],"date-time":"2005-11-07T00:00:00Z","timestamp":1131321600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2006,1]]},"DOI":"10.1007\/s00200-005-0184-x","type":"journal-article","created":{"date-parts":[[2005,11,7]],"date-time":"2005-11-07T12:38:13Z","timestamp":1131367093000},"page":"397-404","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On the bit security of the Diffie-Hellman key"],"prefix":"10.1007","volume":"16","author":[{"given":"Ian F.","family":"Blake","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Theo","family":"Garefalakis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Igor E.","family":"Shparlinski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2005,11,7]]},"reference":[{"key":"184_CR1","doi-asserted-by":"publisher","first-page":"148","DOI":"10.1016\/j.jco.2004.01.002","volume":"20","author":"Blake","year":"2004","unstructured":"Blake, I.F., Garefalakis, T.: On the complexity of the discrete logarithm and Diffie-Hellman problems. J. Compl. 20, 148\u2013170 (2004)","journal-title":"J. Compl."},{"key":"184_CR2","doi-asserted-by":"crossref","unstructured":"Blake, I.F., Seroussi, G., Smart, N.: Elliptic curves in cryptography. London Math. Soc., Lecture Note Series, Vol.265, Cambridge University Press 1999","DOI":"10.1017\/CBO9781107360211"},{"key":"184_CR3","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/3-540-44647-8_12","volume":"2139","author":"Boneh","year":"2001","unstructured":"Boneh, D., Shparlinski, I.E.: On the unpredictability of bits of the elliptic curve Diffie\u2013Hellman scheme. Lect. Notes in Comp. Sci. Springer-Verlag, Berlin, 2139, 201\u2013212 (2001)","journal-title":"Lect. Notes in Comp. Sci. Springer-Verlag, Berlin,"},{"key":"184_CR4","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/3-540-68697-5_11","volume":"1109","author":"Boneh","year":"1996","unstructured":"Boneh, D., Venkatesan, R.: Hardness of computing the most significant bits of secret keys in Diffie\u2013Hellman and related schemes. Lect. Notes in Comp. Sci. Springer-Verlag, Berlin, 1109, 129\u2013142 (1996)","journal-title":"Lect. Notes in Comp. Sci. Springer-Verlag, Berlin,"},{"key":"184_CR5","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/S1631-073X(03)00281-4","volume":"337","author":"Bourgain","year":"2003","unstructured":"Bourgain, J., Konyagin, S.V.: Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order. Comptes Rendus Mathematique 337, 75\u201380 (2003)","journal-title":"Comptes Rendus Mathematique"},{"key":"184_CR6","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/978-3-540-24676-3_22","volume":"3027","author":"Gennaro","year":"2004","unstructured":"Gennaro, R., Krawczyk, H., Rabin, T.: Hashed Diffie-Hellman over non-DDH groups. Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 3027, 361\u2013381 (2004)","journal-title":"Lect. Notes in Comp. Sci., Springer-Verlag, Berlin,"},{"key":"184_CR7","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/3-540-45995-2_14","volume":"2286","author":"Vasco","year":"2002","unstructured":"Gonz\u00e1lez Vasco, M.I., N\u00e4slund, M., Shparlinski, I.E.: The hidden number problem in extension fields and its applications. Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 2286, 105\u2013117 (2002)","journal-title":"Lect. Notes in Comp. Sci., Springer-Verlag, Berlin,"},{"key":"184_CR8","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez Vasco M.I., N\u00e4slund, M., Shparlinski I.E.: New results on the hardness of Diffie-Hellman bits. Proc. Intern. Workshop on Public Key Cryptography, Singapore, 2004. Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 2947, 159\u2013172 (2004)","DOI":"10.1007\/978-3-540-24632-9_12"},{"key":"184_CR9","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez Vasco, M.I., Shparlinski, I.E.: On the security of Diffie-Hellman bits. Proc. Workshop on Cryptography and Computational Number Theory, Singapore 1999, Birkh\u00e4user, 2001, 257\u2013268","DOI":"10.1007\/978-3-0348-8295-8_19"},{"key":"184_CR10","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1093\/qjmath\/51.2.221","volume":"51","author":"Heath-Brown","year":"2000","unstructured":"Heath-Brown, D.R., Konyagin, S.V.: New bounds for Gauss sums derived from kth powers, and for Heilbronn's exponential sum. Ouart. J. Math. 51, 221\u2013235 (2000)","journal-title":"Ouart. J. Math."},{"key":"184_CR11","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s00145-003-0052-4","volume":"16","author":"Joux","year":"2003","unstructured":"Joux, A., Nguyen, K.: Separating decision Diffie-Hellman from Diffie-Hellman in cryptographic groups. J. Cryptology 16, 239\u2013247 (2003)","journal-title":"J. Cryptology"},{"key":"184_CR12","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/10722028_24","volume":"1838","author":"Kohel","year":"2000","unstructured":"Kohel, D.R., Shparlinski, I.E.: Exponential sums and group generators for elliptic curves over finite fields. Lect. Notes in Comp. Sci. Springer-Verlag, Berlin, 1838, 395\u2013404 (2000)","journal-title":"Lect. Notes in Comp. Sci. Springer-Verlag, Berlin,"},{"key":"184_CR13","doi-asserted-by":"crossref","unstructured":"Konyagin, S.V., Shparlinski, I.: Character sums with exponential functions and their applications. Cambridge Univ. Press, Cambridge, 1999","DOI":"10.1017\/CBO9780511542930"},{"key":"184_CR14","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/3-540-45708-9_28","volume":"2442","author":"Li","year":"2002","unstructured":"Li, W.-C.W., N\u00e4slund, M., Shparlinski, I.E.: The hidden number problem with the trace and bit security of XTR and LUC. Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 2442, 433\u2013448 (2002)","journal-title":"Lect. Notes in Comp. Sci., Springer-Verlag, Berlin,"},{"key":"184_CR15","doi-asserted-by":"crossref","unstructured":"Shparlinski, I.E.: Cryptographic applications of analytic number theory. Birkh\u00e4user, 2003","DOI":"10.1007\/978-3-0348-8037-4"},{"key":"184_CR16","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/S1071-5797(03)00048-0","volume":"10","author":"Shparlinski","year":"2004","unstructured":"Shparlinski, I.E.: Security of polynomial transformations of the Diffie-Hellman key. Finite Fields and Their Appl. 10, 123\u2013131 (2004)","journal-title":"Finite Fields and Their Appl."},{"key":"184_CR17","first-page":"2073","volume":"74","author":"Shparlinski","year":"2005","unstructured":"Shparlinski, I.E., Winterhof, A.: A hidden number problem in smal subgroups, Math. Comp., 74, 2073\u20132080 (2005), 1\u201312","journal-title":"Comp.,"},{"key":"184_CR18","unstructured":"Silverman, J.H.: The arithmetic of elliptic curves. Springer-Verlag, Berlin, 1995"}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-005-0184-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-005-0184-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-005-0184-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-005-0184-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,14]],"date-time":"2022-05-14T00:17:24Z","timestamp":1652487444000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-005-0184-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,11,7]]},"references-count":18,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["184"],"URL":"https:\/\/doi.org\/10.1007\/s00200-005-0184-x","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,11,7]]},"assertion":[{"value":"2 October 2004","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 November 2005","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}