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The Long Term Evolution/ System Architecture Evolution (LTE/SAE) of UMTS is one of the latest steps in an advancing series of mobile telecommunication systems. For secure communication, authentication service is one of the most essential services in these networks and guarantee that he/she is authorized for particular services. With evolution mobile networks, authentication and key agreement (AKA) protocol also has been evolved. Evolved Packet System-AKA (EPS-AKA) marks as the enhancement protocol after UMTS-AKA of 3G. This paper first analyses evolved-AKA protocol for LTE/SAE networks and compared its enhancements in contrast with UMTS-AKA, then with respect to different authentication trigger in the integrated LTE networks including: handovers, Tracking Area Update(TAU), registration and service requests introduce a new advancement protocol, which the Serving Network (SN) will be allowed to share with Home Environment (HE) during authentication, whereas in the original one, SN just has the role of broker and it just receives and sends authentication messages between MS and HE. The current 3GPP EPS-AKA has some weaknesses, such as bandwidth consumption and authentication signaling overhead between SN and HN, storage space in serving network and number of complicated hash function which computed during authentication procedure and also incomplete mutual authentication made it vulnerable to some attacks. The new proposed EPS-AKA protocol removes above weaknesses by increasing a little computation in Mobility Management Entity (MME) and generated joined authentication vectors by both MME and Home Subscriber Server/Authentication Center (HSS/AuC). The proposed scheme can satisfy security requirements and its advantages have been verified by simulation which it shows our new proposed protocol caused considerable deduction of authentication traffic load for HSS when density of MS become more.