Lower Bounds for Discrete Logarithms and Related Problems

This paper considers the computational complexity of the discrete logarithm and related problems in the context of “generic algorithms”—that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded as a unique binary string. Lower bounds on the complexity of these problems are proved that match the known upper bounds: any generic algorithm must perform Ω(p1/2) group operations, where p is the largest prime dividing the order of the group. Also, a new method for correcting a faulty Diffie-Hellman oracle is presented.