Study Of Fully Homomorphic Cryptography Theory

In the past thirty-six years’ development of public-key cryptography, a series of public-key encryption schemes have been presented, most of which are homomorphic encryption schemes. But building up a perfect fully homomorphic encryption scheme remains an open problem that puzzled cryptologists for 30 or 40 years. The core technology of cloud computing, ciphertexts retrieval, secure multi-party computation and integrity check of remote data objects put forward new requirements for the development of cryptography. In addition, it is also very necesssary to design a new secure and efficient fully homomorphic encryption scheme. Thus, the attention of this thesis has been focused on the study of the theory of fully homomorphic encryption and the design of practical fully homomorphic encryption scheme. Our work is centred on the very problem and include these aspects next:At first, in order to comprehensively and deeply analysing the fully homomorphic encryption,this thesis researches the representative homomorphic encryption schemes. There are a great many homomorphic encryption schemes in literature such as RSA Cryptosystem, Paillier Cryptosystem, ElGamal Cryptosystem and so on. In this thesis, we mainly analyse the essence of homomorphic, classify homomorphic cryptosystems and obtain several homomorphic properties. Under the gudience of these theory,a new semantically secure targeted fully homomorphic encryption scheme has been presented. Based on whose homomorphic property subtlely, a new secure two-party computation protocol has been designed.Second,in order to grasp the approach of designing a practical fully homomorphic cryptosystem,the two fully homomorphic encryption schemes are studied-DGHV cryptosystem which is short for fully homomorphic encryption over the integer and fully homomorphic encryption using ideal lattice and which belongs to the Gentry series. The Cryptography features and limitations of ideal lattice are provided in this paper. Under the guidence of Gentry’s idea, an improved somewhat homomorphic encryption scheme whose public-key size is shorter and is more efficient is presented on the basis of DGHV scheme.What is more, in order to design a fully homomorphic Encryption on lattice theory, a new fully homomorphic encryption scheme which is based on secure assumption in post quantum environment is designed. In this thesis, taking LWE problem as an example, the key techniques of fully homomorphic encryption scheme are provided. Under the direction of these key technique, a novel fully homomorphic encryption scheme based on ciphertext flattening is designed according to the characteristics of the matrix. This scheme keeps away from introducing a new secure assumption in classic fully homomorphic encryption schemes, which makes its security more definite. In this new scheme, the size of public-key is O(n2l) and the expansion between plaintext and ciphertext is O(n2l),both of which are greatly improved compared with the previous fully homomorphic encryption schemes.At last, the research fully homomorphic encryption scheme brings new idea for the design of homomorphic signature and homomorphic message authentication. On the basis of researching on fully homomorphic encryption schemes, the process of homomorphic message authentication is analysed and an improved homomorphic message authentication algorithm whose security depends on psuedo-random function is presented, which avoids the leakage of plaintext and guarantees the two kinds of unforgeability and provides a new idea for the integrality check of partial data.In the ending, the whole work is summarized and the ideas and plan of work next is also provided.