Homomorphic cryptosystems based on subgroup membership problems

We define an abstract subgroup membership problem, and derive a number of general results for subgroup membership problems. We define an homomorphic public key cryptosystem based essentially on a subgroup membership problem, and show that this abstract construction gives a uniform description of many famous cryptosystems, such as ElGamal, Goldwasser-Micali and Paillier. We show that the abstract theory gives new insights into older results, and allows us to derive new results.