The security architecture of the mobile networks provides different security features such as authentication, confidentiality, and integrity for both users and network operator. Moreover, the security of mobile communications becomes a hot topic and always need security improvements. GSM cellular system has many weaknesses, especially in the authentication process due to the lack of the physical protection mechanisms in both user and network sides. This paper analyses an existing GSM authentication scheme based on CL-PKC which was proposed in 2014, and it has many weaknesses since it used the basic CL-PKC scheme which has many weaknesses itself. And this paper also proposes an efficient end-to-end scheme for GSM mutual authentication using the concept of a secure CL-PKC, with some modifications in both security and GSM network architectures.