One-time-commutative public key encryption

Commutative encryption denotes an encryption scheme where an encrypted message could be repeatedly encrypted, which is especially useful in the public key encryption system. For instance, a ciphertext which is already encrypted using Alice's public key could be `commutatively encrypted' using Bob's public key without decryption. The architecture can be realized as presented in Fig. 1. Then, it becomes a ciphertext which could be decrypted only when both Alice and Bob participant in. The commutative feature is suitable when applied to some co-operation between groups or teamworks. In this work, we propose a one-time commutative encryption scheme based on ElGamal encryption and Ateniese et al.'s re-encryption schemes. Besides the original algorithms of public key encryption such as key generation, encryption and decryption, there are two extra algorithms called commutative encryption and commutative decryption in the new proposed scheme. An encrypted message can be `commutatively encrypted' only one time, and the `commutatively encrypted' message should be decrypted using `commutative decryption'. On the other hand, when considering the notion of security, our commutative encryption is proved IND-CPA secure based on Ateniese et al.'s assumption in the standard model.