Research On Multiplic Ative Homomorphic Linear Secret Sharing Scheme And Its Application

The main idea of homomorphic secret sharing is that the sum or product of multiple secrets can be restored using the reconstruction algorithm with the input for the result of adding or multiplying to secret shares under the constraint of protecting each secret.For the widely used linear secret sharing scheme(LSS),it has the additive homomorphism.Although multiplication can be realized by using addition of LSS,the computation efficiency of the multiplication of many secrets is lower.Therefore,it is necessary to realize the multiplicative homomorphism of LSS.From the known documents,the problems of LSS multiplicative homomorphism are as follows:Firstly,the schemes which the method based on discrete logarithm is scalable have problems that the study focus on the Shamir secret share scheme(SSS)and the secrets and their product must be less than modulus of discrete logarithm.Secondly,it is only used in electronic voting at present,thus it is necessary to expand its application field.It is known that the electronic auction protocol based on the additive homomorphism of secret sharing(SSS as an example)has the efficiency problem as follows:The computational overhead of tender and bid opening are respectively(8)9)6))and(6)7)2)~2),the communicational overhead are respectively(8)9))and(6)).In above,6),,8),9)are respectively the number of prices,the threshold value,the number of bidders and the number of servers.To resolve above problems,the research work of this thesis are as follows:Firstly,the multiplicative homomorphism is realized by introducing the easy-to-calculate discrete logarithm into Brickell secret sharing scheme(BSS),Massey secret sharing scheme(MSS)and Asmuth-Bloom secret sharing scheme(ABSS).Compared with the multiplicative homomorphism of SSS,their computational efficiency is impro-ved.Secondly,factorization and godel coding are introduced for the first time in the scheme of realizing the multiplicative homomorphism of the SSS,BSS,MSS,ABSS.The experiment shows that based on the scheme of easy-to-calculate discrete logarithm,only2 of random integers of 8 bit can be multiplied in 1800s.This scheme can multiply at least 50 of random integers of 64 bit.Thirdly,based on LSS additive and multiplicative homomorphism,secure scalar product algorithm and secure evaluation of multivariate polynomials algorithm are designed.The experiment shows that algorithms based on LSS additive and multiplicative(factorization)homomorphism are more practical(security scalar product algorithm:it is required only 3s that 64-bit 50-dimensional vector does scalar product;secure evaluation of multivariate polynomials algorithm:64-bit polynomial of degree 2is required Only 0.49s).Fourly,the distributed electronic auction protocol for Multiple items is designed by combining the multiplicative homomorphism of ABSS.Compared with the electronic auction protocol based on the additive homomorphism of secret sharing(SSS as an example)toitems,the communication cost is its 1?6)when bidding and the communication cost is its 1?6)when bid opening.