Research On Fully Homomorphic Encryption Schemes Based On SIS And LWE Problems

Fully Homomorphic Encryption has important application value in secure multi-party computing,cloud computing and so on.The first fully homomorphic encryption scheme is constructed by Gentry in 2009,Based on the structural thought of this scheme,there are many improvement schemes have been designed by scholars.However,the research on fully homomorphic encryption is still at the exploratory stage,and there is no big breakthrough.Considering the security of the scheme,the noise is added to the encryption algorithm.However,the noise will increase with homomorphic operation,which results in decryption failure.In order to control the noise growth,the ciphertext must be updated,thus leading the construction method of the scheme is complex and the efficiency is low.Therefore,breaking the structure model of the previous,designing a fully homomorphic encryption scheme that does not need to update the ciphertext and still contains noise term is imminent.In practical applications such as secure multi-party computing,there is no need to compute ciphertext for any times usually,with finite operations in encryption schemes.Thus designing a somewhat homomorphic encryption scheme which is efficient and easy-to-implement also has broad prospects.Given that,the main research work of this thesis is as follows:(1)Based on the small integer solution problem and learning with errors problem,a somewhat homomorphic encryption scheme is constructed,and the plaintext M of the scheme can be selected from the matrix ring of Zm×m·In addition,the correctness,homomorphism and security of the scheme are analyzed as well in this thesis.Compared with Gentry's scheme which is designed in 2010,the number of addition homomorphic operation is extended from finite times to any times,and the message space is extended from Z2m×m to Zqm×m.(2)Based on the small integer solution problem and learning with errors problem,a natural and intuitional fully homomorphic encryption scheme is constructed,which contains noise term and doesn't need the technology of ciphertext update,and the plaintext m of the scheme can be selected fromZmq· In addition,the correctness,homomorphism and security of the scheme are analyzed as well in this thesis.Compared with the typical three kinds of fully homomorphic encryption schemes,the advantages of this scheme are that structure model is natural and intuitional,and it doesn't need the technology of ciphertext update.The noise is obtained by the linear transformation of the key,and it doesn't affect the correctness of decryption after the homomorphic operation.The problems of the scheme are that the key generation algorithm is very complex and the noise term is not randomly selected.