A cryptosystem based on non - maximal imaginary quadraticorders with fast

We introduce a new cryptosystem with trapdoor decryption based on the diiculty of computing discrete logarithms in the class group of the non-maximal imaginary quadratic order O q , where q = q 2 , 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 ideal class group of the maximal order O, where the representatives of the ideal classes have smaller coeecients. Thus, the decryption procedure may be performed in the class group of O rather than in the class group of the public O q , which is much more eecient. We show that inverting our proposed cryptosystem is computationally equivalent to factoring the non-fundamental discriminant q; which is intractable for a suitable choice of and q. We also describe how signature schemes in O q may be set up using this trapdoor information. Furthermore, we illustrate how one may embed key escrow capability into classical imaginary quadratic eld cryptosystems.