On the Cryptographic Value of the q^th Root Problem

Abstract

We show that, for a prime q and a group G, if ord(G) = q ^k r, k>1, and r is smooth, then finding a q^th root in G, is equivalent to the discrete logarithm problem over G (note that the discrete logarithm problem over the group G reduces to the discrete logarithm problem over a subgroup of order q – see reference [5]). Several publications describe techniques for computing q^th roots (see [3] and [1]). All have the stated or implied requirement of computing discrete logarithm in a subgroup of order q.

The emphasis here will be on demonstrating that with a fairly general q ^th root oracle, discrete logarithms in a subgroup of order q may be found, describing the cryptographic significance of this problem, and in introducing two new public key signature schemes based on it.