Abstract
Koblitz curves are a special family of binary elliptic curves satisfying equation \(y^2+xy=x^3+ax^2+1\), \(a\in \{0,1\}\). Scalar multiplication on Koblitz curves can be achieved with point addition and fast Frobenius endomorphism. We show a new point representation system \(\mu _4\) coordinates for Koblitz curves. When \(a=0\), \(\mu _4\) coordinates derive basic group operations—point addition and mixed-addition with complexities \(7\mathbf{M}+2\mathbf{S}\) and \(6\mathbf{M}+2\mathbf{S}\), respectively. Moreover, Frobenius endomorphism on \(\mu _4\) coordinates requires \(4\mathbf{S}\). Compared with the state-of-the-art \(\lambda \) representation system, the timings obtained using \(\mu _4\) coordinates show speed-ups of \(28.6\%\) to \(32.2\%\) for NAF algorithms, of \(13.7\%\) to \(20.1\%\) for \(\tau \)NAF and of \(18.4\%\) to \(23.1\%\) for regular \(\tau \)NAF on four NIST-recommended Koblitz curves K-233, K-283, K-409 and K-571.
This work is supported by the National Natural Science Foundation of China (No. 61872442, No. 61802401, No. 61502487) and the National Cryptography Development Fund (No. MMJJ20180216).
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References
Bernstein, D.J.: Explicit-formulas database (2007)
Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Twisted Edwards curves revisited. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 326–343. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89255-7_20
Joye, M., Tunstall, M.: Exponent recoding and regular exponentiation algorithms. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 334–349. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02384-2_21
Kerry, C.F., Director, C.R.: FIPS PUB 186–4 federal information processing standards publication digital signature standard (DSS) (2013)
Koblitz, N.: CM-curves with good cryptographic properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-46766-1_22
Kohel, D.: Twisted \({\mu }_4\)-normal form for elliptic curves. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 659–678. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_23
López, J., Dahab, R.: Improved algorithms for elliptic curve arithmetic in GF(2n). In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 201–212. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48892-8_16
Oliveira, T., Aranha, D.F., LĂłpez, J., RodrĂguez-HenrĂquez, F.: Fast point multiplication algorithms for binary elliptic curves with and without precomputation. In: Joux, A., Youssef, A. (eds.) SAC 2014. LNCS, vol. 8781, pp. 324–344. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13051-4_20
Oliveira, T., LĂłpez, J., Aranha, D.F., RodrĂguez-HenrĂquez, F.: Lambda coordinates for binary elliptic curves. In: Bertoni, G., Coron, J.-S. (eds.) CHES 2013. LNCS, vol. 8086, pp. 311–330. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40349-1_18
Solinas, J.A.: Efficient arithmetic on Koblitz curves. Des. Codes Crypt. 19(2/3), 195–249 (2000)
Taverne, J., Faz-Hernndez, A., Aranha, D.F., RodrĂguez-HenrĂquez, F., Hankerson, D., LĂłpez, J.: Speeding scalar multiplication over binary elliptic curves using the new carry-less multiplication instruction. J. Crypt. Eng. 1(3), 187 (2011)
Trost, W.R., Guangwu, X.: On the optimal pre-computation of window \(\tau \) NAF for Koblitz curves. IEEE Trans. Comput. 65(9), 2918–2924 (2016)
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Li, W., Yu, W., Li, B., Fan, X. (2019). Speeding up Scalar Multiplication on Koblitz Curves Using \(\mu _4\) Coordinates. In: Jang-Jaccard, J., Guo, F. (eds) Information Security and Privacy. ACISP 2019. Lecture Notes in Computer Science(), vol 11547. Springer, Cham. https://doi.org/10.1007/978-3-030-21548-4_34
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