Abstract
There are various expansion methods to accelerate scalar multiplication on special types of elliptic curves. In this paper we present a general expansion method that uses efficient endomorphisms. We first show that the set of all endomorphisms over a non-supersingular elliptic curve E is isomorphic to Z[ ω ] = { a + bω | a,bin Z }, where ω is an algebraic integer with the smallest norm in an imaginary quadratic field, if ω is an endomorphism over E. Then we present a new division algorithm in Z[ ω ], by which an integer k can be expanded by the Frobenius endomorphism and ω. If ω is more efficient than a point doubling, we can use it to improve the performance of scalar multiplication by replacing some point doublings with the ω maps. As an instance of this general method, we give a new expansion method using the efficiently computable endomorphisms used by Ciet et al. [1].
This work was supported by the MOST grant M6-0203-00-0039.
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References
Ciet, M., Lange, T., Sica, F., Quisquater, J.J.: Improved algorithms for efficient arithmetic on elliptic curves using fast endomorphisms. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 388–400. Springer, Heidelberg (2003)
Koblitz, N.: CM-curves with good cryptographic properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992)
Meier, W., Staffelbach, O.: Efficient multiplication on certain non-supersingular elliptic curves. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 333–344. Springer, Heidelberg (1993)
Müller, V.: Fast multiplication on elliptic curves over small fields of characteristic two. Journal of Cryptology 11, 219–234 (1998)
Cheon, J., Park, S., Park, S., Kim, D.: Two efficient algorithms for arithmetic of elliptic curves using Frobenius map. In: Imai, H., Zheng, Y. (eds.) PKC 1998. LNCS, vol. 1431, pp. 195–202. Springer, Heidelberg (1998)
Solinas, J.: An improved algorithm for arithmetic on a family of elliptic curves. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 357–371. Springer, Heidelberg (1997)
Solinas, J.: Efficient arithmetic on Koblitz curves. Designs, Codes and Cryptography 19, 195–249 (2000)
Smart, N.: Elliptic curve cryptosystems over small fields of odd characteristic. Journal of Cryptology 12, 141–151 (1999)
Kobayashi, T., Morita, H., Kobayashi, K., Hoshino, F.: Fast elliptic curve algorithm combining Frobenius map and table reference to adapt to higher characteristic. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 176–189. Springer, Heidelberg (1999)
Kobayashi, T.: Base-φ method for elliptic curves over OEF. IEICE Trans. Fundamentals E83-A, 679–686 (2000)
Lim, C., Hwang, H.: Speeding up elliptic scalar multiplication with precomputation. In: Song, J.S. (ed.) ICISC 1999. LNCS, vol. 1787, pp. 102–119. Springer, Heidelberg (2000)
Gallant, R., Lambert, R., Vanstone, S.: Faster point multiplication on elliptic curves with efficient endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001)
Park, T., Lee, M., Park, K.: New frobenius expansions for elliptic curves with efficient endomorphisms. In: Lee, P.J., Lim, C.H. (eds.) ICISC 2002. LNCS, vol. 2587, pp. 264–282. Springer, Heidelberg (2003)
Silverman, J.: The Arithmetic of Elliptic Curves. Springer, Heidelberg (1986)
Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 3rd edn. Oxford University Press, Oxford (1954)
Gilbert, W.: Radix representations of quadratic fields. J. Math. Anal. Appl. 83, 264–274 (1981)
Cohen, H.: A Course in Computational Algebraic Number Theory, 3rd edn. Springer, Heidelberg (1996)
Cox, D.: Primes of the Form x 2 + ny 2. Fermat, Class Field Theory and Complex Multiplication. Wiley, Chichester (1998)
Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation using mixed coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)
Bailey, D., Paar, C.: Optimal extension fields for fast arithmetic in public key algorithms. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 472–485. Springer, Heidelberg (1998)
Bailey, D., Paar, C.: Efficient arithmetic in finite field extensions with application in elliptic curve cryptography. Journal of Cryptology 14, 153–176 (2001)
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Park, TJ., Lee, MK., Kim, Ey., Park, K. (2004). A General Expansion Method Using Efficient Endomorphisms. In: Lim, JI., Lee, DH. (eds) Information Security and Cryptology - ICISC 2003. ICISC 2003. Lecture Notes in Computer Science, vol 2971. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24691-6_10
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