Abstract
In this paper, we present a new model of elliptic curves over finite fields of characteristic 2. We first describe the group law on this new binary curve. Furthermore, this paper presents the unified addition formulas for new binary elliptic curves, that is the point addition formulas which can be used for almost all doubling and addition. Finally, this paper presents explicit addition formulas for differential addition.
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References
Bernstein, D.J., Lange, T.: Explicit-formulas database, http://www.hyperelliptic.org/EFD
Bernstein, D.J., Lange, T., Farashahi, R.R.: Binary Edwards Curves. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 244–265. Springer, Heidelberg (2008)
Brier, E., Joye, M.: Weierstraß Elliptic Curves and Side-Channel Attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 335–345. Springer, Heidelberg (2002)
Farashahi, R.R., Joye, M.: Efficient Arithmetic on Hessian Curves. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 243–260. Springer, Heidelberg (2010)
Gaudry, P., Lubicz, D.: The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines. Finite Fields and Their Applications 15(2), 246–260 (2009)
Järvinen, K.U., Skytta, J.: Fast Point Multiplication on Koblitz Curves: Parallelization Method and Implementations. Microproc. Microsyst. 33(2), 106–116 (2009)
Kim, K.H., Kim, S.I.: A new method for speeding up arithmetic on elliptic curves over binary fields (2007), http://eprint.iacr.org/2007/181
Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48(177), 203–209 (1987)
Lange, T.: A note on Lápez-Dahab coordinates. Tatra Mountains Mathematical Publications 33, 75-81 (2006), http://eprint.iacr.org/2004/323
Devigne, J., Joye, M.: Binary Huff Curves. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 340–355. Springer, Heidelberg (2011)
López, J., Dahab, R.: Fast Multiplication on Elliptic Curves over GF(2m) without Precomputation. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 316–327. Springer, Heidelberg (1999)
Menezes, A.J.: Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers (1993)
Miller, V.S.: Use of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)
Roy, S.S., Rebeiro, C., Mukhopadhyay, D., Takahashi, J., Fukunaga, T.: Scalar Multiplication on Koblitz Curves using Ï„ 2-NAF. Cryptology ePrint Archive, Report 2011/318, http://eprint.iacr.org/2011/318
Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987)
Stam, M.: On Montgomery-Like Representationsfor Elliptic Curves over GF(2k). In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 240–253. Springer, Heidelberg (2002)
Stein, W.A. (ed.): Sage Mathematics Software (Version 4.6), The Sage Group (2010), http://www.sagemath.org
Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer (1986)
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Wu, H., Tang, C., Feng, R. (2012). A New Model of Binary Elliptic Curves. In: Galbraith, S., Nandi, M. (eds) Progress in Cryptology - INDOCRYPT 2012. INDOCRYPT 2012. Lecture Notes in Computer Science, vol 7668. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34931-7_23
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DOI: https://doi.org/10.1007/978-3-642-34931-7_23
Publisher Name: Springer, Berlin, Heidelberg
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