Weak discrete logarithms in non-abelian groups

The intractability of the traditional discrete logarithm problem (DLP) forms the basis for the design of numerous cryptographic primitives. In [2] M. Sramka et al. generalize the DLP to arbitrary finite groups. One of the reasons mentioned for this generalization is P. Shor’s quantum algorithm [4] which solves efficiently the traditional DLP. The DLP for a nonabelian group is based on a particular representation of the group and a choice of generators. In this paper we show that care must be taken to ensure that the representation and generators indeed yield an intractable DLP. We show that in PSL(2, p) = 〈α, β〉 the generalized discrete logarithm problem with respect to (α, β) is easy to solve for a specific representation and choice of generators α and β. As a consequence, such representation of PSL(2, p) and generators should not be used to design cryptographic primitives. 2000 Mathematics Subject Classification: 68P25, 94A60.