Abstract
We present a new method for computing the scalar multiplication on Koblitz curves. Our method is as fast as the fastest known technique but requires much less memory. We propose two settings for our method. In the first setting, well-suited for hardware implementations, memory requirements are reduced by 85%. In the second setting, well-suited for software implementations, our technique reduces the memory consumption by 70%. Thus, with much smaller memory usage, the proposed method yields the same efficiency as the fastest scalar multiplication schemes on Koblitz curves.
Chapter PDF
Similar content being viewed by others
Keywords
References
Avanzi, R.M., Ciet, M., Sica, F.: Faster scalar multiplication on Koblitz curves combining point halving with the Frobenius endomorphism. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 28–40. Springer, Heidelberg (2004)
Brickell, E.F., Gordon, D.M., McCurley, K.S., Wilson, D.B.: Fast exponentiation with precomputation: algorithms and lower bounds. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 200–207. Springer, Heidelberg (1993)
Chevallier-Mames, B., Ciet, M., Joye, M.: Low-cost solutions for preventing simple side-channel analysis: side-channel atomicity. IEEE Trans. Computers 53(6), 760–768 (2004)
Coron, J.-S., M’Raïhi, D., Tymen, C.: Fast generation of pairs (k, [k]P) for Koblitz elliptic curves. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 151–164. Springer, Heidelberg (2001)
Dahab, R., Hankerson, D., Hu, F., Long, M., López, J., Menezes, A.: Software multiplication using normal bases. Technical report CACR 2004-12. University of Waterloo (2004)
Hankerson, D., López, J., Menezes, A.: Software implementation of elliptic curve cryptography over binary fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–24. Springer, Heidelberg (2000)
IEEE P1363A. Standard specifications for public-key cryptography, annex A, number-theoretic background (2000)
Joye, M., Tymen, C.: Compact encoding of non-adjacent forms with applications to elliptic curve cryptography. In: Kim, K.-c. (ed.) PKC 2001. LNCS, vol. 1992, pp. 353–364. Springer, Heidelberg (2001)
Joye, M., Tymen, C.: Protections against differential analysis for elliptic curve cryptography: An algebraic approach. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 377–390. Springer, Heidelberg (2001)
Kaliski, B.S., Yin, Y.L.: Storage-efficient finite field basis conversion. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 81–93. Springer, Heidelberg (1999)
Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48(177), 203–209 (1987)
Koblitz, N.: CM-curves with good cryptographic properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992)
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)
Massey, J., Omura, J.K.: Computational method and apparatus for finite field arithmetic. US Patent 4587627 (1986)
Möller, B.: Improved techniques for fast exponentiation. In: Lee, P.J., Lim, C.H. (eds.) ICISC 2002. LNCS, vol. 2587, pp. 298–312. Springer, Heidelberg (2003)
Park, D.J., Sim, S.G., Lee, P.J.: Fast scalar multiplication method using change-of-basis matrix to prevent power analysis attacks on Koblitz curves. In: Chae, K.-J., Yung, M. (eds.) WISA 2003. LNCS, vol. 2908, pp. 474–488. Springer, Heidelberg (2004)
Solinas, J.A.: 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.A.: Efficient arithmetic on Koblitz curves. Designs, Codes, and Cryptography 19(2–3), 195–249 (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Okeya, K., Takagi, T., Vuillaume, C. (2005). Short Memory Scalar Multiplication on Koblitz Curves. In: Rao, J.R., Sunar, B. (eds) Cryptographic Hardware and Embedded Systems – CHES 2005. CHES 2005. Lecture Notes in Computer Science, vol 3659. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11545262_7
Download citation
DOI: https://doi.org/10.1007/11545262_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28474-1
Online ISBN: 978-3-540-31940-5
eBook Packages: Computer ScienceComputer Science (R0)