Abstract
Scalar multiplication on Koblitz curves can be very efficient due to the elimination of point doublings. Modular reduction of scalars is commonly performed to reduce the length of expansions, and τ-adic Non-Adjacent Form (NAF) can be used to reduce the density. However, such modular reduction can be costly. An alternative to this approach is to use a random τ-adic NAF, but some cryptosystems (e.g. ECDSA) require both the integer and the scalar multiple. This paper presents an efficient method for computing integer equivalents of random τ-adic expansions. The hardware implications are explored, and an efficient hardware implementation is presented. The results suggest significant computational efficiency gains over previously documented methods.
This work was supported by the project “Packet Level Authentication” funded by TEKES.
Chapter PDF
Similar content being viewed by others
References
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 nonsupersingular elliptic curves. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 333–344. Springer, Heidelberg (1993)
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. Des. Codes Cryptogr. 19(2-3), 195–249 (2000)
Lange, T., Shparlinski, I.: Collisions in fast generation of ideal classes and points on hyperelliptic and elliptic curves. Appl. Algebra Engrg. Comm. Comput. 15(5), 329–337 (2005)
Lange, T., Shparlinski, I.E.: Certain exponential sums and random walks on elliptic curves. Canad. J. Math. 57(2), 338–350 (2005)
Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Information Theory IT-22(6), 644–654 (1976)
ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985)
National Institute of Standards and Technology (NIST): Digital signature standard (DSS). Federal Information Processing Standard, FIPS PUB 186-2 (2000)
Lange, T.: Koblitz curve cryptosystems. Finite Fields Appl. 11(2), 200–229 (2005)
Järvinen, K., Forsten, J., Skyttä, J.: Efficient circuitry for computing τ-adic non-adjacent form. In: ICECS 2006. Proc. of the IEEE Int’l. Conf. on Electronics, Circuits and Systems, Nice, France, pp. 232–235 (2006)
Dimitrov, V.S., Järvinen, K.U., Jacobson, J.M.J., Chan, W.F., Huang, Z.: FPGA implementation of point multiplication on Koblitz curves using Kleinian integers. In: Goubin, L., Matsui, M. (eds.) CHES 2006. LNCS, vol. 4249, pp. 445–459. Springer, Heidelberg (2006)
Lutz, J., Hasan, M.A.: High performance FPGA based elliptic curve cryptographic co-processor. In: ITCC 2004. International Conference on Information Technology: Coding and Computing, vol. 02, pp. 486–492. IEEE Computer Society Press, Los Alamitos (2004)
Standaert, F.X., Peeters, E., Rouvroy, G., Quisquater, J.J.: An overview of power analysis attacks against field programmable gate arrays. Proc. IEEE 94(2), 383–394 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brumley, B.B., Järvinen, K. (2007). Koblitz Curves and Integer Equivalents of Frobenius Expansions. In: Adams, C., Miri, A., Wiener, M. (eds) Selected Areas in Cryptography. SAC 2007. Lecture Notes in Computer Science, vol 4876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77360-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-77360-3_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77359-7
Online ISBN: 978-3-540-77360-3
eBook Packages: Computer ScienceComputer Science (R0)