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Collisions in Fast Generation of Ideal Classes and Points on Hyperelliptic and Elliptic Curves

Abstract.

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.

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Correspondence to Tanja Lange.

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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.

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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

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Keywords

  • Public Key Cryptography
  • Discrete Logarithm
  • Hyperelliptic Curves
  • Koblitz Curves
  • Collisions