Abstract
In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a modified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as \(\sum^n_{i=1}2^{b_i}3^{t_i}\) where (b i ) and (t i ) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we propose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups.
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Bernstein, D.J., Birkner, P., Lange, T., Peters, C.: Optimizing Double-Base Elliptic-Curve Single-Scalar Multiplication. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 167–182. Springer, Heidelberg (2007)
Bernstein, D.J., Lange, T.: Explicit-formulas database, http://hyperelliptic.org/EFD
Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29–50. Springer, Heidelberg (2007)
Bernstein, D.J., Lange, T.: Inverted Edwards coordinates. In: Boztaş, S., Lu, H.-F(F.) (eds.) AAECC 2007. LNCS, vol. 4851, pp. 20–27. Springer, Heidelberg (2007)
Bernstein, D.J., Lange, T.: Analysis and optimization of elliptic-curve single-scalar multiplication. In: Finite fields and applications: proceedings of Fq8, pp. 1–19 (2008)
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), 385–434 (1986)
Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Cryptography. Chapman and Hall, Boca Raton (2006)
Dimitrov, V., Cooklev, T.: Two algorithms for modular exponentiation using nonstandard arithmetics. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 78(1), 82–87 (1995)
Dimitrov, V., Imbert, L., Mishra, P.K.: Efficient and secure elliptic curve point multiplication using double-base chains. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 59–78. Springer, Heidelberg (2005)
Doche, C., Icart, T., Kohel, D.R.: Efficient scalar multiplication by isogeny decompositions. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 191–206. Springer, Heidelberg (2006)
Doche, C., Imbert, L.: Extended Double-Base Number System with Applications to Elliptic Curve Cryptography. In: Barua, R., Lange, T. (eds.) INDOCRYPT 2006. LNCS, vol. 4329, pp. 335–348. Springer, Heidelberg (2006)
Duquesne, S.: Improving the arithmetic of elliptic curves in the Jacobi model. Inf. Process. Lett. 104(3), 101–105 (2007)
Edwards, H.M.: A normal norm for elliptic curves. Bulletin of the American Mathematical Society 44, 393–422 (2007)
Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)
Hisil, H., Carter, G., Dawson, E.: New formulae for efficient elliptic curve arithmetic. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 138–151. Springer, Heidelberg (2007)
Hisil, H., Koon-Ho Wong, K., Carter, G., Dawson, E.: An intersection form for jacobi-quartic curves. Personal communication (2008)
Liardet, P., Smart, N.P.: Preventing SPA/DPA in ECC systems using the jacobi form. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 391–401. Springer, Heidelberg (2001)
Longa, P., Gebotys, C.: Setting speed records with the (fractional) multibase non-adjacent form method for efficient elliptic curve scalar multiplication. Technical report, Department of Electrical and Computer Engineering University of Waterloo, Canada (2009)
Longa, P., Miri, A.: New composite operations and precomputation scheme for elliptic curve cryptosystems over prime fields. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 229–247. Springer, Heidelberg (2008)
Meloni, N.: New point addition formulae for ECC applications. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 189–201. Springer, Heidelberg (2007)
Mishra, P.K., Dimitrov, V.S.: Efficient quintuple formulas for elliptic curves and efficient scalar multiplication using multibase number representation. In: Garay, J.A., Lenstra, A.K., Mambo, M., Peralta, R. (eds.) ISC 2007. LNCS, vol. 4779, pp. 390–406. Springer, Heidelberg (2007)
Yao, A.C.: On the evaluation of powers. SIAM Journal on Computing 5(1), 100–103 (1976)
Zeckendorf, E.: Représentations des nombres naturels par une somme de nombre de Fibonacci ou de nombres de Lucas. Bulletin de la Soci. Royale des Sciences de Liège, pp. 179–182 (1972)
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Méloni, N., Hasan, M.A. (2009). Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases. In: Clavier, C., Gaj, K. (eds) Cryptographic Hardware and Embedded Systems - CHES 2009. CHES 2009. Lecture Notes in Computer Science, vol 5747. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04138-9_22
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DOI: https://doi.org/10.1007/978-3-642-04138-9_22
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