Key update and residence management have been investigated as an effective solution to cope with desynchronisation attacks in Mobility Management Entity (MME) handovers. In this paper, we first analyse the impacts of the Key Update Interval (KUI) and MME Residence Interval (MRI) on handover processes and their secrecy performance in terms of the Number of Exposed Packets (NEP), Signaling Overhead Rate (SOR) and Outage Probability of Vulnerability (OPV). Specifically, the bounds of the derived NEP and SOR not only capture their behaviours at the boundary of the KUI and MRI, but also show the trade-off between the NEP and SOR. Additionally, through the analysis of the OPV, it is shown that the handover security can be enhanced by shortening the KUI and the desynchonisation attacks can be avoided with high-mobility users. The above facts accordingly motivate us to propose a Multi-objective Optimisation (MO) problem to find the optimal KUI and MRI that minimise both the NEP and SOR subject to the constraint on the OPV. To this end, two scalarisation techniques are adopted to transform the proposed MO problem into single-objective optimisation problems, i.e., an achievement-function method via Fractional Programming (FP) and a weighted-sum method. Based on the derived bounds on NEP and SOR, the FP approach can be optimally solved via a simple numerical method. For the weighted-sum method, the Firefly Algorithm (FA) is utilised to find the optimal solution. The results show that both techniques can solve the proposed MO problem with a significantly reduced searching complexity compared to the conventional heuristic iterative search technique.