Design Of Homomorphic Encryption Schemes Based On Common Divisor Matrix Assumption And LWE Problem

Homomorphic encryption means that the results of addition or multiplication operation for ciphertext are consistent with the corresponding operation for plaintext.On the other hand,based on the computational problem of matrix ring,quantum-resist encryption schemes can be constructed.The n x n invertible matrix is generally used as the key in existing matrix homomorphic encryption schemes,which results that the form of keys' length can only be fixed to n2,once the key is generated,only n x n matrix can be encrypted.In view of this,the generalized inverse matrix is first combined with homomorphic encryption in this thesis.First,a symmetrical fully homomorphic encryption scheme is designed;second,the generalized inverse matrix is applied to the first matrix homomorphic encryption scheme designed by Gentry and others in 2010,and a new(asymmetric)somewhat homomorphic encryption scheme is constructed.The research work is as follows:(1)Based on the relationship between the general m x n column full rank matrix and its generalized inverse,a symmetrical fully homomorphic encryption scheme is designed,which has the IND-CPA(indistinguishability under chosen-plaintext attacks)security and based on(untexted)common divisor matrix assumption.Compared with similar schemes,the key's length is variable in the form of mn.Second,the generalized inverse matrix is not unique,the secret key pair generated by the scheme is not unique as well,thus improving the security of the key to a certain extent.(2)The key factor from symmetry to asymmetry is to look for a one-way trapdoor function in the design of cryptosystems.Used check matrix as a trapdoor,combined with the related properties of generalized inverse matrix,a new somewhat homomorphic encryption scheme is constructed in this thesis,which has the IND-CPA security and based on the hardness of the learning with errors(LWE)problem.Compared with GHV10 scheme,the random[n,k]linear code can be generated or stored in advance,and then the corresponding elementery transformation of generator matrix and check matrix are used,thus leading the key generation algorithm is simple.In addition,the generalized inverse has been acted on plaintext,so it can encrypt the matrix of fixed number of rows and any number of columns after the key pair is generated.(3)Mathematical software Magma is used as the platform to verify the above two matrix homomorphic encryption schemes.The results show that our schemes can be verified by experiments on the basis of correct theory.