Koblitz curves have been proposed to quickly generate random ideal classes and points on hyperelliptic and elliptic curves. To obtain a further speed-up a different way of generating these random elements has recently been proposed. In this paper we give an upper bound on the number of collisions for this alternative approach. For elliptic Koblitz curves we additionally use the same methods to derive a bound for a modified algorithm. These bounds are tight for cyclic subgroups of prime order, which is the case of most practical interest for cryptography.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Behlen, P.: Algebraic geometry and coding theory, PhD thesis, Eindhoven University of Technology, 2001
Bosma, W.: Signed bits and fast exponentiation. J. Théorie des Nombres Bordeaux 13, 27–41 (2001)
Doumen, J.: Some applications of coding theory in cryptography, PhD thesis, Eindhoven University of Technology, 2003
Everest, G., van der Poorten, A.J., Shparlinski, I.E., Ward, T.B.: Recurrence sequences. Amer. Math. Soc. 2003
Galbraith, S.D., McKee, J.: The probability that the number of points on an elliptic curve over a finite field is prime. J. London Math. Soc. 62, 671–684 (2000)
Günther, C., Lange, T., Stein, A.: Speeding up the arithmetic on Koblitz curves of genus two. Proc. SAC’00, Lect. Notes in Comp. Sci. Springer-Verlag, Berlin, 2012, 106–117 (2001)
Koblitz, N.: CM curves with good cryptographic properties. Proc. Crypto’91, Lect. Notes in Comp. Sci. Springer-Verlag, Berlin, 576, 279–287 (1992)
Koblitz, N.: Almost primality of group orders of elliptic curves defined over small finite fields. Experiment. Math. 10, 553–558 (2001)
Lange, T.: Efficient arithmetic on hyperelliptic curves. PhD thesis, Universität Gesamthochschule Essen, 2001
Lange, T.: Koblitz Curve Cryptosystems. To appear in Finite Fields and Their Applications, online 24 August 2004
Müller, V.: Fast multiplication on elliptic curves over small fields of characteristic two. J. Cryptol. 11, 219–234 (1998)
Mumford, D.: Tata lectures on Theta II. Birkhäuser, 1984
Pohlig, S., Hellman, M.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Transactions on Information Theory IT-24, 106–110 (1978)
Smart, N.P.: Elliptic curve cryptosystems over small fields of odd characteristic. J. Cryptol. 12, 141–151 (1999)
Solinas, J.: Efficient arithmetic on Koblitz curves. Designs, Codes and Cryptography 19, 195–249 (2000)
Acknowledgement The authors would like to thank Bernd Sturmfels whose suggestion has led to a substantial improvement of our preliminary result. This paper was written during a visit of the second author to the Ruhr-Universität Bochum whose generous support and hospitality are gratefully acknowledged.
About this article
Cite this article
Lange, T., Shparlinski, I. Collisions in Fast Generation of Ideal Classes and Points on Hyperelliptic and Elliptic Curves. AAECC 15, 329–337 (2005). https://doi.org/10.1007/s00200-004-0161-9
- Public Key Cryptography
- Discrete Logarithm
- Hyperelliptic Curves
- Koblitz Curves