The advancements in wireless technologies have paved the way for offering seamless services to mobile consumers in global mobility networks (GLOMONETs). Nonetheless, since in these networks, the communication channels are open and public, devising proper measures to protect the security and privacy of exchanging messages is critical. Moreover, because of the energy and computational power limitations of the mobile devices, a security mechanism should put the least possible burden on these devices. To do so, plenty authentication schemes have been proposed in recent years to be employed in the context of GLOMONETs. Quite recently, four interesting authentication and key agreement schemes have been put forward, which have improved the efficiency and security of their previous related schemes. Besides the nice contributions of these four, in this paper, it is first indicated that all of them are vulnerable to the known session-specific information attack. Then, a new chaotic map-based authentication scheme is presented that not only is secure against the aforementioned attack but also has less computational complexity and communication overhead. The comprehensive descriptive discussion on security alongside the formal verification of security using the widely accepted ProVerif tool show that our scheme is secure against the known attacks, and the comparative performance analysis demonstrates its betterment for the constrained-resource mobile devices.