Generalization of a Variant of

We propose a generalization of a variant of Paillier Cryptosystem that is an additive homomor-phic Cryptosystem, meaning that one can combine cipher texts into a new cipher text that is the encryption of the sum of the messages of the original cipher texts. We propose a threshold version of this generalized system, allowing a number of servers to share the knowledge of the secret key, such that any large enough subset of them can decrypt a cipher text, while smaller subsets have no useful information. Using the generalized system some inter-active proofs are presented for proving various claims. We combine our generalized scheme with Elgamal Cryptosystem; this leads to a new cryptosystem. Also we present a threshold version for this new Cryptosystem