In mobile network, ensuring privacy to the subscriber is an obligation of the mobile operator. It is the responsibility of the operator to ensure that the private information about the subscriber like his/her identity does not get compromised.While adopting technologies with regards to issues like this, operators mostly follow specifications standardized by the third Generation Partnership Project (3GPP). In a recent technical specification published by 3GPP for 5G mobile network, an Elliptic Curve Integrated Encryption Scheme (ECIES) is adopted for identity privacy. However, this scheme does not emphasize on protecting the identity of the home network with which the subscriber is registered. This may make the subscriber vulnerable to tracking by an adversary in a visited network through his/her home network identity. In this paper, we propose an enhanced scheme called scheme for home network identity privacy. Through a formal and an informal security analysis, the proposed scheme is found to be secured. Through a performance analysis, it is found that the scheme achieves its objective of securing the home network identity of the subscriber at the expense of 47% additional computation at the UE in comparison with the execution time that is incurred by ECIES.