Multi-Key Fully Homomorphic Encryption Schemes And Applications In Regression Algorithm

The rapid development of the emerging Internet industries in an open environment makes the daily life of people full of various kinds of data.Meanwhile the data are related with their information and privacy.On the one hand,the information leakage and blackmail attack affect the privacy of data,threatening the vital interests of the individual.On the other hand,independent data and zombie data restrict the availability,impeding the values of data.Then the homomorphic encryption keeping the privacy and availability of data is the key technique for the secure usage of data,since it can not only encrypt the data to protect the information,but also support the exchange,sharing,processing and usage on encrypted data.Therefore,facing with different centers possessing large number of data,multi-key fully homomorphic encryption(MKFHE)makes it possible to securely using the encrypted data from different involved parties.On the basis of this,for the goal of collaborative and intelligent computations on data from multi parties,taking the regression models in machine learning as the scenarios,this thesis focuses on the design of MKFHE and the applications on the regression algorithms.There are some challenges surrounding with MKFHE,such as the efficiency and the ciphertext size.In this thesis,we propose two MKFHE schemes from the perspective of noise and without noise to improve the performance.One is to decrease the noise and modulus,the other is to skip the complicated noise managements and reduce the ciphertexts size.In the aspects of applications,we design privacy-preserving linear regression algorithm and cox proportional hazards algorithm,estimating the efficiency from computational complexity and communication overheads.The details are as follows.1.For the representation MKFHE with underlying GSW13,i.e.,MW16,there is a common random string(CRS)with which every involved party must share each other,before they generate their own public and private key.Hence we propose a modified MKFHE in the plain model aiming to solve the drawback.In our scheme,every involved party can generate his own key independent on a CRS.Considering the linear combination procedure(LCP)in MW16,a main step transferring GSW13 ciphertexts to multi-key ciphertexts,leads to more noise and smaller efficiency,we improve the LCP and generate a different matrix with less noise.As a result,the modulus of MKFHE is reduced and the efficiency is enhanced.2.For the problem of the ciphertext size of MKFHE dependent on the number of involved parties,we propose an MKFHE scheme from additively homomorphic encryption algorithm.It is pointed out that the ciphertexts in the proposed scheme have no relation with the number of parties.Concretely,from the perspective of additively homomorphic encryption,this thesis turns to zero-knowledge encryption switching protocols,consisting two-party decryption protocol and ciphertext multiplication protocol.Then the ciphertexts from different parties can achieve collaborative and secure computations.Moreover,an instantiation with the El Gamal variant scheme is presented.Compared with the mainstream MKFHE schemes,the instantiation is noiseless,without expanded procedure and evaluation key,supports new parties to be involved at any time.That is,the scheme we propose is relatively efficient.3.Giacomelli et al.proposed a privacy-preserving linear regression algorithm with the following system model: the pair of public and secret key is generated by a third authority,and every involved party shares the same pair.Aiming to the drawback of system model,a practical privacy-preserving linear regression algorithm is proposed to deal with the problem.First,the system model is modified.Every party generates his own public key and secret key,without depending on a third authority.Second,the data are encrypted by the MKCKKS17 scheme which can directly encrypt rational numbers.Then the transforms between integers and rational numbers are unnecessary.Third,a preprocessing phase is presented to generate a column full-rank encrypted matrix,instead of an assumption,since the correctness of the linear regression model largely depends on the column full-rank matrix composed of data.Finally,the security proof and the performance evaluations from theoretical and experimental analyses are shown to demonstrate the feasibility.4.Lu et al.proposed a cox proportional hazards algorithm where two encryptions are needed and the relinearization is generated by interactions.In order to solve the problems,especially the interactive relinearization,this thesis presents an improved cox regression algorithm.First,the system model is improved.The entity cloud service provider(CSP)is assumed to be semi-malicious rather than semi-honest.Then CSP may perform dishonest computations,nevertheless all his behaviors are recorded on a witness tape.Second,the succinct multi-key CKKS17 scheme is improved,surrounding with the relinearization.Third,a succinct multi-key homomorphic message authenticator is designed to make sure the results returned by CSP are correct without forgery.Moreover,the performance analyses demonstrate that our cox regression algorithm is relatively efficient.