We discuss the discrete logarithm problem over the class group Cl(Δ) of an imaginary quadratic order OΔ, which was proposed as a public-key cryptosystem by Buchmann and Williams . While in the meantime there has been found a subexponential algorithm for the computation of discrete logarithms in Cl(Δ) , this algorithm only has running time LΔ[1/2, c] and is far less efficient than the number field sieve with Lp[1/3, c] to compute logarithms in IF*p. Thus one can choose smaller parameters to obtain the same level of security. It is an open question whether there is an LΔ[1/3, c] algorithm to compute discrete logarithms in arbitrary Cl(Δ). In this work we focus on the special case of totally non-maximal imaginary quadratic orders OΔpsuch that Δp = Δ1p2 and the class number of the maximal order h(Δ1) = 1, and we will show that there is an LΔp[1/3, c] lgorithm to compute discrete logarithms over the class group Cl(Δp). The logarithm problem in Cl(Δp) can be reduced in (expected) O(log3 p) bit operations to the logarithm problem in IF*p (if (Δ1/ p) = 1) or IF *p2 (if (Δ1/ p) = -1) respectively. This result implies that the recently proposed efficient DSA-analogue in totally non-maximal imaginary quadratic order OΔp  are only as secure as the original DSA scheme based on finite fields and hence loose much of its attractiveness.