Use of mobile personal computers in open networked environment is revolutionalising the way we use computer. Mobile networked computing is raising important information security and privacy issues. This paper is concerned with the design of authentication protocols for a mobile computing environment. The paper first analyses the authentication initiator protocols proposed by Belier, Chang and Yacobi (BCY) and the modifications considered by Carlsen and points out some weaknesses. The paper then suggests improvements to these protocols. The paper proposes secure end-to-end protocols between mobile users using both symmetric and public key based systems. These protocols enable mutual authentication and establish a shared secret key between mobile users. Furthermore, these protocols provide a certain degree of anonymity of the communicating users to be achieved vis-a-vis other system users.