Subscription privacy of a user has been a historical concern with all the previous generation mobile networks, namely, GSM, UMTS,and LTE. While a little improvement have been achieved in securing the privacy of the long-term identity of a subscriber, the so called IMSI catchers are still in existence even in the LTE and advanced LTE networks. Proposals have been published to tackle this problem in 5G based on pseudonyms, and different public-key technologies. This paper looks into the problem of concealing long-term identity of a subscriber and presents a technique based on identity based encryption (IBE) to tackle it. The proposed solution can be extended to a mutual authentication and key agreement protocol between a serving network (SN) and a user equipment (UE). This mutual authentication and key agreement protocol does not need to connect with the home network (HN) on every run. A qualitative comparison of the advantages and disadvantages of different techniques show that our solution is competitive for securing the long-term identity privacy of a user in the 5G network.