Boneh et al. introduced mediated RSA (mRSA) in 2001 in an attempt to achieve faster key revocation for medium-sized organizations via the involvement of a security mediator (SEM) as a semi-trusted third party to provide partial ciphertext decryption for the receiver. In this paper, a pairing-free security mediated encryption scheme based on an ElGamal variant is proposed. The scheme features a similar setting as in the mediated RSA but with a different underlying primitive. We show that the proposed security mediated encryption scheme is secure indistinguishably against chosen-ciphertext attack (IND-CCA) in the random oracle via the hardness assumption of the computational Diffie-Hellman (CDH) problem.

Symmetric cryptography allows faster and more secure communication between two entities using the identical pre-established secret key. However, identifying the honest entity with the same secret key before initiating symmetric encryption is vital since the communication may be impersonated. Tea and Ariffin, in 2014, proposed a new identification (ID) scheme based on the Bivariate Function Hard Problem (BFHP) that proved secure against impersonation under passive, active and concurrent attacks via the BFHP-hardness assumption. In this paper, we upgrade the ID scheme and improve some of its settings. Next, we provide the security proof against impersonation under active and concurrent attacks in the random oracle model via the hardness assumption of the One-More BFHP. Finally, we include an additional discussion about the computational efficiency of the upgraded ID scheme based on BFHP and present its comparison with other selected ID schemes.