Research On Fully Homomorphic Encryption Schemes

Fully homomorphic encryption (FHE) scheme has widely applications in cryptog-raphy and information security. For example, in secure multiparty computation FHE scheme can be used to project the privacy of each party's data, while the computation involving these data can be interactively done; in privacy-preserving cloud computing, FHE scheme can be used to project users' data from revealing to the cloud server, while users can also enjoy the service supplied by the cloud. The requirements for FHE scheme had been proposed in 1970s, after that some attempts for designing FHE scheme were presented, and some encryption scheme with single homomorphic prop-erty were also proposed. But for a long time, secure FHE scheme which is regarded as an open problem of modern cryptography remained unsolved. Until 2009, Gentry proposed the first FHE scheme with provable security. After that many FHE schemes with provable security have been proposed. But the efficiencies of these FHE schemes do not meet the requirements of practical use. It is strongly required to improve the efficiency of the present FHE scheme and to design new FHE schemes.In this dissertation, we first review the present somewhat homomorphic encryption schemes. We introduce their background knowledge, describe their algorithms, analyze their correctness and security, and we also investigate the capacity of homomorphic evaluation of each somewhat homomorphic encryption scheme by using a multi-layer circuit model.Based on the close looking at these somewhat homomorphic encryption schemes, we find their common point in structure, and we propose an uniform model-dual-noise structure-for representing the present somewhat homomorphic encryption schemes. Under this model we show the difference between perfect FHE and generalized FHE. We also give the explicit condition for the existence of generalized FHE. Furthermore, we propose a universal method to optimize generalized FHE scheme. Our optimization scheme is in circuit level. It is applicable for all present FHE schemes and can work together with an algorithm-level optimization.By the analysis of the present FHE schemes under our dual-noise model, we claim that perfect FHE scheme cannot be achieved by using dual-noise structure. So we try to construct a perfect FHE scheme by using a noise-free structure. We propose a perfect FHE scheme based on multivariate polynomial ring. Our scheme can be viewed as an modified version of Groenber-basis Polly Cracker (GB Polly Cracker) scheme. We use linear/affine transformations instead of adding noise to improve the security of GB Polly Cracker and remain the perfect fully homomorphic property.Finally, we propose a privacy-preserving aggregation scheme for smart grid com-munications. Our aggregation scheme is based on RLWE-FHE scheme. With this aggregation scheme, users' private data can be protect from revealing to the data and control center in smart grid, while the data and control center can obtain some statis-tical quantities about users'data, so that a fine-grained analysis and optimization can be made for the smart grid.