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One of the recently proposed authentication schemes for 4G/5G mobile networks is a secure anonymity key of authentication and key agreement scheme (SAK-AKA), wherein the home subscriber server (HSS), mobility management entity (MME), and user equipment (UE) represent authentication entities. In the SAK-AKA scheme, UE authentication is performed through an initial authentication processes session (IAPS) and subsequent authentication processes session (SAPS) phases to ensure that the UE is authorized to use the network services. In IAPS, the MME accesses the HSS to obtain the authentication parameters and takes the delegation from the HSS to execute the authentication process with the UE mutually. Since accessing the HSS is expensive, the HSS generates a number <inline-formula> <tex-math notation="LaTeX">$\left ({k }\right)$ </tex-math></inline-formula> of authentication vectors (AVs) and then sends them back to the MME as one unit at a time. SAPS is then executed <inline-formula> <tex-math notation="LaTeX">$\left ({k }\right)$ </tex-math></inline-formula> times to authenticate the UE without going back to the HSS. Therefore, it is useful to choose a specific value of <inline-formula> <tex-math notation="LaTeX">$\left ({k }\right)$ </tex-math></inline-formula> that would reduce the traffic cost. In this paper, we proposed an analytical cost function model to examine the effect of the <inline-formula> <tex-math notation="LaTeX">$\left ({k }\right)$ </tex-math></inline-formula> value on the transmission cost of authentication processes. The proposed model assumes that the signaling traffic of the SAK-AKA scheme is represented by a Poisson process where the authentication request time has an exponential distribution. This model was analyzed numerically to identify appropriate values of the (<inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-AVs) that can minimize the overhead messages cost of the SAK-AKA scheme.

Reducing the Overhead Messages Cost of the SAK-AKA Authentication Scheme for 4G/5G Mobile Networks