Boneh, Ding and Tsudik presented identity-based mediated RSA encryption and signature systems in which the users are not allowed to decrypt/sign messages without the authorisation of a security mediator.We show that ID-MRSA is not secure and we present a secure modified version of it which is as efficient as the original system. We also propose a generic mediated encryption that translates any identity based encryption to a mediated version of this IBE. It envelops an IBE encrypted message using a user’s identity into an IBE envelope using the identity of the SEM. We present two security models based on the role of the adversary whether it is a revoked user or a hacked SEM. We prove that GME is as secure as the SEM’s IBE against a revoked user and as secure as the user’s IBE against a hacked SEM. We also present two implementations of GME based on Boneh-Franklin FullIBE system which is a pairing-based system and Boneh, Gentry and Hamburg (BGH) system which is pairing-free system.