{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T23:32:36Z","timestamp":1742945556222,"version":"3.40.3"},"publisher-location":"Berlin, Heidelberg","reference-count":54,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"type":"print","value":"9783662493861"},{"type":"electronic","value":"9783662493878"}],"license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"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":[[2016]]},"DOI":"10.1007\/978-3-662-49387-8_2","type":"book-chapter","created":{"date-parts":[[2016,2,17]],"date-time":"2016-02-17T14:25:41Z","timestamp":1455719141000},"page":"19-35","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Degenerate Curve Attacks"],"prefix":"10.1007","author":[{"given":"Samuel","family":"Neves","sequence":"first","affiliation":[]},{"given":"Mehdi","family":"Tibouchi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,2,18]]},"reference":[{"key":"2_CR1","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/3-540-36288-6_16","volume-title":"Public Key Cryptography - PKC 2003","author":"A Antipa","year":"2002","unstructured":"Antipa, A., Brown, D.R.L., Menezes, A., Struik, R., Vanstone, S.A.: Validation of elliptic curve public keys. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 211\u2013223. Springer, Heidelberg (2002)"},{"key":"2_CR2","unstructured":"Aranha, D.F., Barreto, P., Pereira, G., Ricardini, J.E.: A note on high-security general-purpose elliptic curves. Cryptology ePrint Archive, Report 2013\/647 (2013). \n                      http:\/\/eprint.iacr.org\/"},{"key":"2_CR3","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/978-3-540-72163-5_18","volume-title":"Information Security Practice and Experience","author":"Y-J Baek","year":"2007","unstructured":"Baek, Y.-J., Vasyltsov, I.: How to prevent DPA and fault attack in a unified way for ECC scalar multiplication \u2013 ring extension method. In: Dawson, E., Wong, D.S. (eds.) ISPEC 2007. LNCS, vol. 4464, pp. 225\u2013237. Springer, Heidelberg (2007). \n                      http:\/\/dx.doi.org\/10.1007\/978-3-540-72163-5_18"},{"key":"2_CR4","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/11745853_14","volume-title":"Public Key Cryptography - PKC 2006","author":"DJ Bernstein","year":"2006","unstructured":"Bernstein, D.J.: Curve25519: new Diffie-Hellman speed records. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 207\u2013228. Springer, Heidelberg (2006)"},{"key":"2_CR5","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/978-3-540-68164-9_26","volume-title":"Progress in Cryptology \u2013 AFRICACRYPT 2008","author":"DJ Bernstein","year":"2008","unstructured":"Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389\u2013405. Springer, Heidelberg (2008)"},{"key":"2_CR6","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/978-3-319-22174-8_15","volume-title":"Progress in Cryptology \u2013 LATINCRYPT 2015","author":"DJ Bernstein","year":"2015","unstructured":"Bernstein, D.J., Chuengsatiansup, C., Kohel, D., Lange, T.: Twisted Hessian curves. In: Lauter, K., Rodr\u00edguez-Henr\u00edquez, F. (eds.) LatinCrypt 2015. LNCS, vol. 9230, pp. 269\u2013294. Springer, Heidelberg (2015)"},{"key":"2_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/978-3-540-76900-2_3","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2007","author":"DJ Bernstein","year":"2007","unstructured":"Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29\u201350. Springer, Heidelberg (2007)"},{"key":"2_CR8","unstructured":"Bernstein, D.J., Lange, T.: Explicit-formulas database (2015). \n                      https:\/\/hyperelliptic.org\/EFD\/\n                      \n                    . Accessed 1 May 2015"},{"key":"2_CR9","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/3-540-44598-6_8","volume-title":"Advances in Cryptology - CRYPTO 2000","author":"I Biehl","year":"2000","unstructured":"Biehl, I., Meyer, B., M\u00fcller, V.: Differential fault attacks on elliptic curve cryptosystems. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 131\u2013146. Springer, Heidelberg (2000)"},{"key":"2_CR10","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/3-540-44828-4_5","volume-title":"Applied Algebra, Algebraic Algorithms and Error-Correcting Codes","author":"O Billet","year":"2003","unstructured":"Billet, O., Joye, M.: The Jacobi model of an elliptic curve and side-channel analysis. In: Fossorier, M.P.C., H\u00f8holdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 34\u201342. Springer, Heidelberg (2003). \n                      https:\/\/eprint.iacr.org\/2002\/125"},{"key":"2_CR11","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1007\/11889700_4","volume-title":"Fault Diagnosis and Tolerance in Cryptography","author":"J Bl\u00f6mer","year":"2006","unstructured":"Bl\u00f6mer, J., Otto, M., Seifert, J.-P.: Sign change fault attacks on elliptic curve cryptosystems. In: Breveglieri, L., Koren, I., Naccache, D., Seifert, J.-P. (eds.) FDTC 2006. LNCS, vol. 4236, pp. 36\u201352. Springer, Heidelberg (2006). \n                      http:\/\/dx.doi.org\/10.1007\/11889700_4"},{"key":"2_CR12","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/3-540-69053-0_4","volume-title":"Advances in Cryptology - EUROCRYPT 1997","author":"D Boneh","year":"1997","unstructured":"Boneh, D., DeMillo, R.A., Lipton, R.J.: On the importance of checking cryptographic protocols for faults. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 37\u201351. Springer, Heidelberg (1997)"},{"issue":"2","key":"2_CR13","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s001450010016","volume":"14","author":"D Boneh","year":"2001","unstructured":"Boneh, D., DeMillo, R.A., Lipton, R.J.: On the importance of eliminating errors in cryptographic computations. J. Cryptol. 14(2), 101\u2013119 (2001)","journal-title":"J. Cryptol."},{"issue":"4","key":"2_CR14","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1016\/0196-8858(86)90023-0","volume":"7","author":"DV Chudnovsky","year":"1986","unstructured":"Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Adv. Appl. Math. 7(4), 385\u2013434 (1986). \n                      http:\/\/dx.org\/10.1016\/0196-8858(86)90023-0","journal-title":"Adv. Appl. Math."},{"issue":"1","key":"2_CR15","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s10623-003-1160-8","volume":"36","author":"M Ciet","year":"2005","unstructured":"Ciet, M., Joye, M.: Elliptic curve cryptosystems in the presence of permanent and transient faults. Des. Codes Crypt. 36(1), 33\u201343 (2005). \n                      http:\/\/dx.org\/10.1007\/s10623-003-1160-8","journal-title":"Des. Codes Crypt."},{"issue":"1","key":"2_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01840433","volume":"1","author":"D Coppersmith","year":"1986","unstructured":"Coppersmith, D., Odlyzko, A.M., Schroeppel, R.: Discrete logarithms in \n                      \n                        \n                      \n                      $$GF(p)$$\n                    . Algorithmica 1(1), 1\u201315 (1986). \n                      http:\/\/dx.org\/10.1007\/BF01840433","journal-title":"Algorithmica"},{"issue":"3","key":"2_CR17","first-page":"545","volume":"5","author":"A Desboves","year":"1886","unstructured":"Desboves, A.: R\u00e9solution, en nombres entries et sous la forme la plus g\u00e9n\u00e9rale, de l\u2019\u00e9quation cubique, homog\u00e8ne, \u00e0 trois inconnues. Nouvelles annales de math\u00e9matiques, journal des candidats aux \u00e9coles polytechnique et normale 5(3), 545\u2013579 (1886). \n                      http:\/\/www.numdam.org\/item?id=NAM_1886_3_5__545_0","journal-title":"Nouvelles annales de math\u00e9matiques, journal des candidats aux \u00e9coles polytechnique et normale"},{"issue":"3","key":"2_CR18","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1090\/S0273-0979-07-01153-6","volume":"44","author":"HM Edwards","year":"2007","unstructured":"Edwards, H.M.: A normal form for elliptic curves. Bull. Am. Math. Soc. 44(3), 393\u2013422 (2007). \n                      http:\/\/dx.org\/10.1090\/S0273-0979-07-01153-6","journal-title":"Bull. Am. Math. Soc."},{"key":"2_CR19","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1109\/TIT.1985.1057074","volume":"31","author":"T ElGamal","year":"1985","unstructured":"ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inf. Theory 31, 469\u2013472 (1985)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2_CR20","unstructured":"Fouque, P., Lercier, R., R\u00e9al, D., Valette, F.: Fault attack on elliptic curve Montgomery ladder implementation. In: Breveglieri, L., Gueron, S., Koren, I., Naccache, D., Seifert, J. (eds.) 2008 Fifth International Workshop on Fault Diagnosis and Tolerance in Cryptography, FDTC 2008, Washington, DC, USA, 10 August 2008, pp. 92\u201398. IEEE Computer Society (2008). \n                      http:\/\/dx.org\/10.1109\/FDTC.2008.15"},{"issue":"1","key":"2_CR21","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 \n                      \n                        \n                      \n                      $$GF(p)$$\n                     using the number field sieve. SIAM J. Discret. Math. 6(1), 124\u2013138 (1993). \n                      http:\/\/dx.org\/10.1137\/0406010","journal-title":"SIAM J. Discret. Math."},{"key":"2_CR22","unstructured":"Hamburg, M.: Ed448-Goldilocks. In: Workshop on Elliptic Curve Cryptography Standards (2015)"},{"key":"2_CR23","unstructured":"Harris, B., et al.: The Pareto frontiers of sleeveless primes. The Curves mailing list, October 2014. \n                      https:\/\/moderncrypto.org\/mail-archive\/curves\/2014\/000324.html"},{"key":"2_CR24","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1515\/crll.1844.28.68","volume":"28","author":"O Hesse","year":"1844","unstructured":"Hesse, O.: \u00dcber die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln. Journal f\u00fcr die reine und angewandte Mathematik 28, 68\u201396 (1844). \n                      http:\/\/resolver.sub.uni-goettingen.de\/purl?GDZPPN002144069","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"key":"2_CR25","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"326","DOI":"10.1007\/978-3-540-89255-7_20","volume-title":"Advances in Cryptology - ASIACRYPT 2008","author":"H Hisil","year":"2008","unstructured":"Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Twisted Edwards curves revisited. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 326\u2013343. Springer, Heidelberg (2008)"},{"issue":"1","key":"2_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/jmc.2011.005","volume":"5","author":"H Hisil","year":"2011","unstructured":"Hisil, H., Wong, K.K., Carter, G., Dawson, E.: An exploration of affine group laws for elliptic curves. J. Math. Cryptol. 5(1), 1\u201350 (2011). \n                      http:\/\/dx.org\/10.1515\/jmc.2011.005","journal-title":"J. Math. Cryptol."},{"issue":"2","key":"2_CR27","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1215\/S0012-7094-48-01543-9","volume":"15","author":"GB Huff","year":"1948","unstructured":"Huff, G.B.: Diophantine problems in geometry and elliptic ternary forms. Duke Math. J. 15(2), 443\u2013453 (1948)","journal-title":"Duke Math. J."},{"key":"2_CR28","unstructured":"Husem\u00f6ller, D.: Elliptic Curves, Graduate Texts in Mathematics, vol. 111, 2nd edn. Springer, New York (2004)"},{"key":"2_CR29","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/978-3-319-24174-6_21","volume-title":"Computer Security \u2013 ESORICS 2015","author":"T Jager","year":"2015","unstructured":"Jager, T., Schwenk, J., Somorovsky, J.: Practical invalid curve attacks on TLS-ECDH. In: Pernul, G., Y A Ryan, P., Weippl, E. (eds.) ESORICS. LNCS, vol. 9326, pp. 407\u2013425. Springer, Heidelberg (2015). doi:\n                      10.1007\/978-3-319-24174-6_21"},{"key":"2_CR30","unstructured":"Joye, M.: Fault-resistant calculcations on elliptic curves, June 2013. \n                      http:\/\/www.google.com\/patents\/US8457303\n                      \n                    , US Patent 8,457,303"},{"key":"2_CR31","unstructured":"Joye, M.: On the security of a unified countermeasure. In: Breveglieri, L., Gueron, S., Koren, I., Naccache, D., Seifert, J. (eds.) 2008 Fifth International Workshop on Fault Diagnosis and Tolerance in Cryptography, FDTC 2008, Washington, DC, USA, 10 August 2008, pp. 87\u201391. IEEE Computer Society (2008). \n                      http:\/\/dx.org\/10.1109\/FDTC.2008.8"},{"key":"2_CR32","unstructured":"Joye, M.: Elliptic curve cryptosystems in the presence of faults. In: Fischer, W., Schmidt, J. (eds.) 2013 Workshop on Fault Diagnosis and Tolerance in Cryptography, Los Alamitos, CA, USA, 20 August 2013, p. 73. IEEE Computer Society (2013). \n                      http:\/\/conferenze.dei.polimi.it\/FDTC13\/shared\/FDTC-2013-keynote-2.pdf"},{"key":"2_CR33","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1007\/3-540-44709-1_33","volume-title":"Cryptographic Hardware and Embedded Systems - CHES 2001","author":"M Joye","year":"2001","unstructured":"Joye, M., Quisquater, J.-J.: Hessian elliptic curves and side-channel attacks. In: Ko\u00e7, \u00c7.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 402\u2013410. Springer, Heidelberg (2001)"},{"key":"2_CR34","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1007\/978-3-642-14518-6_20","volume-title":"Algorithmic Number Theory","author":"M Joye","year":"2010","unstructured":"Joye, M., Tibouchi, M., Vergnaud, D.: Huff\u2019s model for elliptic curves. In: Hanrot, G., Morain, F., Thom\u00e9, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 234\u2013250. Springer, Heidelberg (2010). \n                      http:\/\/dx.doi.org\/10.1007\/978-3-642-14518-6_20"},{"issue":"3","key":"2_CR35","doi-asserted-by":"publisher","first-page":"307","DOI":"10.3934\/amc.2010.4.307","volume":"4","author":"K Karabina","year":"2010","unstructured":"Karabina, K., Ustao\u011flu, B.: Invalid-curve attacks on (hyper)elliptic curve cryptosystems. Adv. Math. Commun. 4(3), 307\u2013321 (2010). \n                      http:\/\/cryptolounge.net\/pdf\/KarUst10.pdf","journal-title":"Adv. Math. Commun."},{"key":"2_CR36","series-title":"Lecture Notes in Computer Science","first-page":"163","volume-title":"Applied Cryptography and Network Security","author":"T Kim","year":"2014","unstructured":"Kim, T., Tibouchi, M.: Bit-flip faults on elliptic curve base fields, revisited. In: Boureanu, I., Owesarski, P., Vaudenay, S. (eds.) ACNS 2014. LNCS, vol. 8479, pp. 163\u2013180. Springer, Heidelberg (2014)"},{"key":"2_CR37","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/978-3-319-22425-1_3","volume-title":"Advances in Information and Computer Security","author":"T Kim","year":"2015","unstructured":"Kim, T., Tibouchi, M.: Invalid curve attacks in a GLS setting. In: Tanaka, K., Suga, Y. (eds.) IWSEC 2015. LNCS, vol. 9241, pp. 41\u201355. Springer, Heidelberg (2015)"},{"key":"2_CR38","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1090\/S0025-5718-1987-0866109-5","volume":"48","author":"N Koblitz","year":"1987","unstructured":"Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48, 203\u2013209 (1987). \n                      http:\/\/dx.org\/10.1090\/S0025-5718-1987-0866109-5","journal-title":"Math. Comput."},{"issue":"3","key":"2_CR39","doi-asserted-by":"crossref","first-page":"649","DOI":"10.2307\/1971363","volume":"126","author":"H. W. Lenstra","year":"1987","unstructured":"Lenstra Jr., H.W.: Factoring integers with elliptic curves. Ann. Math. 126(3), 649\u2013673 (1987). \n                      http:\/\/www.jstor.org\/stable\/1971363","journal-title":"The Annals of Mathematics"},{"key":"2_CR40","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/3-540-44709-1_32","volume-title":"Cryptographic Hardware and Embedded Systems - CHES 2001","author":"P Liardet","year":"2001","unstructured":"Liardet, P., Smart, N.P.: Preventing SPA\/DPA in ECC systems using the Jacobi form. In: Ko\u00e7, \u00c7.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 391\u2013401. Springer, Heidelberg (2001)"},{"key":"2_CR41","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/BFb0052240","volume-title":"Advances in Cryptology - CRYPTO 1997","author":"CH Lim","year":"1997","unstructured":"Lim, C.H., Lee, P.J.: A key recovery attack on discrete log-based schemes using a prime order subgroup. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 249\u2013263. Springer, Heidelberg (1997)"},{"key":"2_CR42","unstructured":"Lochter, M., Merkle, J., Schmidt, J.M., Sch\u00fctze, T.: Requirements for standard elliptic curves. Cryptology ePrint Archive, Report 2014\/832 (2014). \n                      http:\/\/eprint.iacr.org\/2014\/832"},{"key":"2_CR43","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1515\/JMC.2007.004","volume":"1","author":"A Menezes","year":"2007","unstructured":"Menezes, A.: Another look at HMQV. J. Math. Cryptol. 1, 47\u201364 (2007). \n                      http:\/\/dx.org\/10.1515\/JMC.2007.004","journal-title":"J. Math. Cryptol."},{"key":"2_CR44","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/3-540-39799-X_31","volume-title":"Advances in Cryptology of CRYPTO 1985","author":"VS Miller","year":"1986","unstructured":"Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417\u2013426. Springer, Heidelberg (1986)"},{"issue":"177","key":"2_CR45","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1090\/S0025-5718-1987-0866113-7","volume":"48","author":"PL Montgomery","year":"1987","unstructured":"Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comput. 48(177), 243\u2013264 (1987). \n                      http:\/\/www.ams.org\/journals\/mcom\/1987-48-177\/S0025-5718-1987-0866113-7\/","journal-title":"Math. Comput."},{"issue":"4","key":"2_CR46","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/BF01389737","volume":"1","author":"D Mumford","year":"1966","unstructured":"Mumford, D.: On the equations defining Abelian varieties. I. Inventiones Math. 1(4), 287\u2013354 (1966). \n                      http:\/\/dash.harvard.edu\/handle\/1\/3597241","journal-title":"I. Inventiones Math."},{"key":"2_CR47","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1109\/TIT.1978.1055817","volume":"24","author":"SC Pohlig","year":"1978","unstructured":"Pohlig, S.C., Hellman, M.E.: An improved algorithm for computing logarithms over GF\n                      \n                        \n                      \n                      $$(p)$$\n                     and its crytographic significance. IEEE Trans. Inf. Theory 24, 106\u2013110 (1978)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2_CR48","unstructured":"Rauzy, P., Moreau, M., Guilley, S., Najm, Z.: Using modular extension to provably protect ECC against fault attacks. Cryptology ePrint Archive, Report 2015\/882 (2015). \n                      http:\/\/eprint.iacr.org\/2015\/882"},{"issue":"5","key":"2_CR49","doi-asserted-by":"publisher","first-page":"1401","DOI":"10.1137\/060676155","volume":"37","author":"K Rubin","year":"2008","unstructured":"Rubin, K., Silverberg, A.: Compression in finite fields and torus-based cryptography. SIAM J. Comput. 37(5), 1401\u20131428 (2008)","journal-title":"SIAM J. Comput."},{"key":"2_CR50","unstructured":"Shamir, A.: How to check modular exponentiation, May 1997. (presented at the rump session of EUROCRYPT 1997)"},{"key":"2_CR51","series-title":"Proceedings of Symposia in Pure Mathematics","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1090\/pspum\/020\/0316385","volume-title":"1969 Number Theory Institute","author":"D Shanks","year":"1971","unstructured":"Shanks, D.: Class number, a theory of factorization, and genera. In: Lewis, D.J. (ed.) 1969 Number Theory Institute. Proceedings of Symposia in Pure Mathematics, vol. 20, pp. 415\u2013440. American Mathematical Society, Providence, Rhode Island (1971)"},{"key":"2_CR52","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-09494-6","volume-title":"The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics","author":"JH Silverman","year":"2009","unstructured":"Silverman, J.H.: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106, 2nd edn. Springer, New York (2009). \n                      http:\/\/www.math.brown.edu\/jhs\/AECHome.html","edition":"2"},{"key":"2_CR53","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1007\/3-540-44709-1_11","volume-title":"Cryptographic Hardware and Embedded Systems - CHES 2001","author":"NP Smart","year":"2001","unstructured":"Smart, N.P.: The Hessian form of an elliptic curve. In: Ko\u00e7, \u00c7.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 118\u2013125. Springer, Heidelberg (2001)"},{"issue":"7","key":"2_CR54","first-page":"806","volume":"71","author":"EG Straus","year":"1964","unstructured":"Straus, E.G.: Addition chains of vectors (problem 5125). Am. Math. Monthly 71(7), 806\u2013808 (1964). \n                      http:\/\/www.jstor.org\/stable\/2310929","journal-title":"Am. Math. Monthly"}],"container-title":["Lecture Notes in Computer Science","Public-Key Cryptography \u2013 PKC 2016"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-662-49387-8_2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,2,18]],"date-time":"2020-02-18T11:20:08Z","timestamp":1582024808000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-662-49387-8_2"}},"subtitle":["Extending Invalid Curve Attacks to Edwards Curves and Other Models"],"short-title":[],"issued":{"date-parts":[[2016]]},"ISBN":["9783662493861","9783662493878"],"references-count":54,"URL":"https:\/\/doi.org\/10.1007\/978-3-662-49387-8_2","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2016]]},"assertion":[{"value":"18 February 2016","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}