In Long Term Evolution (LTE) networks Extensible Authentication Protocol (EAP) is an authentication framework, not a specific authentication mechanism. EAP-AKA is one of the methods of EAP which uses the Authentication and Key Agreement (AKA) mechanism based on challenge-response mechanisms, EAP-AKA is used in the 3rd generation mobile networks then modified and inherited to 4th generation mobile networks (LTE) as Evolved Packet System Authentication and Key Agreement (EPS-AKA) mechanism. EPS-AKA vulnerabilities are disclosure of the user identity, Man in the Middle attack and Denial of Services (DoS) attacks so a robust authentication mechanism must replace EPS-AKA to avoid such attacks. In this paper, Modified Evolved Packet System Authentication and Key Agreement (MEPS-AKA) protocol based on Simple Password Exponential Key Exchange (SPEKE) is proposed to solve these problems. Scyther tool is used to verify the efficiency of the proposed protocol against the mentioned attacks. EPS-AKA and MEPS-AKA are simulated using C programming language to calculate the execution time for both algorithms.