Two Generic Constructions of Probabilistic Cryptosystems and Their Applications

In this paper, we build, in a generic way, two asymmetric cryptosystems with a careful study of their security. We present first an additively homomorphic scheme which generalizes, among others, the Paillier cryptosystem, and then, another scheme, built from a deterministic trapdoor function. Both schemes are proved semantically secure against chosen plaintext attacks in the standard security model and modify versions can be proved secure against adaptive chosen ciphertext attacks. By implementing these constructions with quotients of Z, elliptic curves and quadratic fields quotients we get some cryptosystems yet described in the past few years and provide variants that achieve higher levels of security than the original schemes. In particular, using quadratic fields quotients, we show that it is possible to build a new scheme secure against adaptive chosen ciphertext attacks in the standard security model.