We introduce a new cryptosystem with trapdoor decryption based on the difficulty of computing discrete logarithms in the class group of the nonmaximal imaginary quadratic order N Δq, where δq = δq2, δ square-free and q prime. The trapdoor information is the conductor q. Knowledge of this trapdoor information enables one to switch to and from the class group of the maximal order N Δ, where the representatives of the ideal classes have smaller coefficients. Thus, the decryption procedure may be performed in the class group of N Δ rather than in the class group of the public N Δq, which is much more efficient. We show that inverting our proposed cryptosystem is computationally equivalent to factoring the non-fundamental discriminant δq, which is intractable for a suitable choice of δ and q. We also describe how signature schemes in N Δq may be set up using this trapdoor information. Furthermore, we illustrate how one may embed key escrow capability into classical imaginary quadratic field cryptosystems.