Rare diseases severely affect many people across the world at the present time. Researchers conduct studies to understand the reasons behind rare diseases and as a result of this research, diagnosis, and treatment methods are developed. Rare disease analysis is performed to specify the disease-causing variants on the genome data of patients. The researchers need access to as much genome data as possible to find causing variants of rare diseases. On the other hand, the genome data of patients should be protected because it can be used to detect the identity of individuals. The researchers are not able to share the genome data of patients easily because of regulations such as General Data Protection Regulation (GDPR). For this reason, rare disease analysis should be performed in a secure way that protects the privacy of patients while enabling the collaboration of multiple medical institutions. In this context, a privacy-preserving collaborative system for rare disease analysis should be provided. This thesis study focuses on the utilization of fully homomorphic encryption, a method that enables unlimited number of operations to be performed on encrypted data, for privacy-preserving collaborative rare disease analysis. Two different methods, the boolean circuit method, and the integer arithmetic method, are implemented to perform rare disease analysis on the encrypted genome data to find disease-causing variants, and various experiments are performed to assess the efficiency of the proposed methods.