Towards Practical Non-interactive Public Key Cryptosystems Using Non-maximal Imaginary Quadratic Orders (Extended Abstract)

We present a new non-interactive public key distribution system based on the class group of a non-maximal imaginary quadratic order Cl(Δ_p). The main advantage of our system over earlier proposals based on (ℤ/nℤ). [19],[21] 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. In order to compute discrete logarithms in the class group, the KGC needs to know the prime factorization of Δ_p = Δ1p2. We present an algorithm for computing discrete logarithms in Cl(Δ_p) by reducing the problem to computing discrete logarithms in Cl(Δ₁) and either Fp or Fp2. We prove that a similar reduction works for arbitrary non-maximal orders, and that it has polynomial complexity if the factorization of the conductor is known.