Secure ElGamal-Type Cryptosystems Without Message Encoding

ElGamal cryptosystem is one of the oldest public-key cryptosystems. It is known to be semantically secure for arbitrary messages in the random oracle model under the decisional Diffie-Hellman assumption. Semantic security also holds in the standard model when messages are encoded as elements in the group for which the decisional Diffie-Hellman assumption is defined. This paper introduces a setting and companion cryptosystem where semantic security can be proved in the standard model without message encoding. Extensions achieving security against chosen-ciphertext attacks are also provided.