An investigation of the theory and applications of homomorphic cryptography

Homomorphic cryptography is a unique branch of cryptography concerned with the study of homomorphic cryptosystems. A homomorphic cryptosystem is one which has special mathematical properties that allow an untrusted third party to perform computations on encrypted data without the need to decrypt it. These cryptosystems have promising implications for the design of novel privacy-preserving protocols in a variety of applications such as electronic voting, multiparty computation, private information retrieval and private database queries.;This thesis presents the field of homomorphic cryptography in a pedagogical manner. After introducing mathematical and conceptual preliminaries, we conduct a survey of somewhat-homomorphic cryptosystems—those which are limited in the number of operations that they can perform on ciphertexts. We then present a comparison of these cryptosystems, in both a theoretical and practical setting, using results from benchmarked implementations. We then conduct a survey of recent advances in the field of fully-homomorphic encryption—a relatively new field concerned with cryptosystems which allow an arbitrary number of computational operations to be conducted on ciphertexts. We then present some major applications of homomorphic cryptography, along with select implementations to frame the theory in a practical context. Finally, we conclude with open questions in the field and guidance for further research.