Because of the importance of computing on data with privacy protections, the cryptographic community has developed both theoretical and practical solutions to compute on encrypted data. On the one hand, theoretical schemes, such as fully homomorphic encryption and functional encryption, are secure but extremely inefficient. On the other hand, practical schemes, such as property-preserving encryption, gain efficiency by accepting significant reductions in security. In this thesis, we first study the security of popular property-preserving encryption schemes that are being used by companies such as Microsoft and Google. We show that such schemes are unacceptably insecure for key target applications such as electronic medical records. Second, we propose new models to compute on encrypted data and develop efficient constructions and systems. We propose a new cryptographic primitive called Blind Storage and show how it can be used to realize symmetric searchable encryption, which is much more secure than property-preserving encryption. Finally, we propose a new cryptographic model called Controlled Functional Encryption and develop two efficient schemes in this model
