The main challenge in building efficient broadcast systems is to encrypt messages with short ciphertexts. In this paper, we present a new construction based on the identity. Our construction contains the desirable features, such as constant size ciphertexts and private keys, short public keys and not fixing the total number of possible users in the setup. In addition, the proposed scheme achieves the full security which is stronger than the selective-identity security. Furthermore we show that the proof of security does not rely on the random oracles. To the best our knowledge, it is the first efficient scheme that is full security and achieves constant size ciphertexts and private keys which solve the trade-off between the ciphertext size and the private key size.