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Lower bounds on generic algorithms in groups

  • Ueli Maurer
  • Stefan Wolf
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1403)

Abstract

In this paper we consider generic algorithms for computational problems in cyclic groups. The model of a generic algorithm was proposed by Shoup at Eurocrypt '97. A generic algorithm is a generalpurpose algorithm that does not make use of any particular property of the representation of the group elements. Shoup proved the hardness of the discrete logarithm problem and the Diffie-Hellman problem with respect to such algorithms for groups whose order contains a large prime factor. By extending Shoup's technique we prove lower bounds on the complexity of generic algorithms solving different problems in cyclic groups, and in particular of a generic reduction of the discrete logarithm problem to the Diffie-Hellman problem. It is shown that the two problems are not computationally equivalent in a generic sense for groups whose orders contain a multiple large prime factor. This complements earlier results which stated this equivalence for all other groups. Furthermore, it is shown that no generic algorithm exists that computes p-th roots efficiently in a group whose order is divisible by p2 if p is a large prime.

Keywords

Diffie-Hellman protocol discrete logarithms generic algorithms roots in finite groups complexity lower bounds 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Ueli Maurer
    • 1
  • Stefan Wolf
    • 1
  1. 1.Computer Science DepartmentSwiss Federal Institute of Technology (ETH Zürich)ZürichSwitzerland

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