Abstract
No one could deny that Cloud Computing has always been the most amazing innovation over the last decade. It is a platform that hosts computing resources, either hardware or software, offering data storage in remote sites. Due to its numerous advantages, it is doubtless a promising perspective. It is available for all time over the internet, offers a low cost service, easy to maintain, environmentally friendly and has a tremendous performance. However, users and companies relying on this technique have always struggled with the lack of security and privacy in the platform, especially when dealing with critical data storage and transfer. Moreover, even if users encrypt their messages in advance, the cloud service provider ought to decrypt it first in order to carry out different operations which can be a considerable risk, since clouds are not fully trusted. So, to solve this problem, a new encryption manner came along, it is the homomorphic encryption. Using this technique, a cloud service provider can perform different operations on the cipher text without the need to decrypt it. Thus security and confidentiality are guaranteed. The main purpose of this chapter is to discuss the concepts and significance of homomorphic cryptosystems along with the different techniques such as the Fully Homomorphic Encryption (FHE) and the Somewhat Homomorphic Encryption (SHE) and also the related works based on these techniques.