We analyze several mutual authentication and key exchange protocols (MAKEPs), and present a number of essential properties of the protocols for secure mobile communications. To address the weaknesses of existing protocols, we propose an improved version of MAKEP known as EC-MAKEP. Besides supporting the essential features present in the existing protocols, the proposed protocol also provides the user anonymity and forward secrecy properties that many of the existing protocols do not support. Further, the proposed protocol compares favorably with ES-MAKEP, an improved version of the early MAKEPs, in terms of computation cost and communication bandwidth. In addition, EC-MAKEP supports an implicit authentication of server and a dual authentication of client.