Capturing the vast amount of information encoded in the human genome is a fascinating research problem. The outcomes of this research have significant influences on a number of health-related fields, such as personalized medicine, paternity testing, and disease susceptibility testing. To facilitate these types of large-scale biomedical research projects, it oftentimes requires sharing genomic and clinical data collected by disparate organizations among themselves. In that case, it is of utmost importance to ensure that sharing, managing, and analyzing the data does not reveal the identity of the individuals who contribute their genomic samples. The task of storage and computation on the shared data can be delegated to third-party cloud infrastructures, equipped with large storage and high-performance computation resources. Outsourcing these sensitive genomic data to the third party cloud storage is associated with the challenges of the potential loss, theft, or misuse of the data as the server administrator cannot be completely trusted as well as there is no guarantee that the security of the server will not be breached. In this thesis, I propose methods for secure sharing and computation of three different functions on genomic data.