How intractable is the discrete logarithm for a general finite group?

Abstract

GDL is the discrete logarithm problem for a general finitc group G. This paper gives a characterization for the intractability of GDL from the viewpoint of computational complexity theory. It is shown that GDL ∈ NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be exccuted in a polynomial time of the element size. Furthermore, as a natural probabilistic extension, the complexity of GDL is investigated under the assumption that the group law operation is executed in an expected polynomial time of the element size. In this case, it is shown that GDL ∈ MA ∩ co-AM if G ∈ NP ∩ co-NP. Finally, we show that GDL is less intractable than NP-complete problems unless the polynomial time hierarchy collapses to the second level.