Towards practical non-interactive public-key cryptosystems using non-maximal imaginary quadratic orders
We present a new non-interactive public-key distribution system based on the class group of a non-maximal imaginary quadratic order Cl(Delta(p)). The main advantage of our system over earlier proposals based on (Z/nZ)* [25,27] is that embedding id information into group elements in a cyclic subgroup of the class group is easy (straight-forward embedding into prime ideals suffices) and secure, since the entire class group is cyclic with very high probability. Computational results demonstrate that a key generation center (KGC) with modest computational resources can set up a key distribution system using reasonably secure public system parameters. In order to compute discrete logarithms in the class group, the KGC needs to know the prime factorization of Delta(p) = Delta(1)p(2). We present an algorithm for computing discrete logarithms in Cl(Delta(p)) by reducing the problem to computing discrete logarithms in Cl(Delta(1)) and either F-p* or F-p2*. Our algorithm is a specific case of the more general algorithm used in the setting of ray class groups [5]. We prove - for arbitrary non-maximal orders - that this reduction to discrete logarithms in the maximal order and a small number of finite fields has polynomial complexity if the factorization of the conductor is known.
non-maximal imaginary quadratic order discrete logarithm identity based cryptography non-interactive cryptography