In this work, the cryptosystems proposed are a slight modification of DSA, Elgamal?s schemes and generalized ?Meta-Elgamal Signature Schemes? of Horster et al. [17]. However, it is not always necessary to consider the generator and its order. We can use a decryption key smaller than that of Elgamal?s scheme.

In general, if we work in a cyclic subgroup of size d (with d as a large prime), then we can keep d secret and we can also use a secret exponent r for decryption of size ?(where ?is some integer which divides ?the size of d). For example, it is possible, for different security levels, to use a 160, 190, or a 256 bit key for decryption. Therefore, the new encryption scheme is faster than the classical Elgamal one?s for decryption process.

Our variants of signatures schemes are more secure in the sense that some known vulnerabilities on the Meta-Elgamal Signatures Schemes do not work with the new modifications proposed. Furthermore, there exist much more variants for our signatures than those of ?Meta-Elgamal Signature Schemes?.

As Elgamal?s encryption scheme, our encryption scheme is based on DDH (Decisional Diffie-Hellman) problem. Moreover, the secrecy of the order ?of the generator g (which is optional), is based on the Integer Factorization Problem.

public key cryptosystem, Meta-Elgamal signatures schemes, provable security, homomorphic, probabilistic, digital signature, DSA, DDH, Elgamal, DLP.