Research On Shamir’s Homomorphic Secret Sharing Scheme And Its Applications

Homomorphic secret sharing schemes allow participants to reconstruct the addition or multiplication of multiple secrets by operating the received sub-shares. And the single secret is private. Shamir’s secret sharing scheme which is widely used has the property of additive homomorphism. But to our knowledge, the research on multiplicative homomorphism is limited to calculate the multiplication of two secrets once in the existing literature. In this thesis, the multiplication of multiple secrets can be reconstructed once. The known electronic voting schemes based on homomorphic secret sharing require the voter to distribute shares for each candidate, and the vote recorder is also need to calculate every candidate’s votes. To improve the computational efficiency and transmission efficiency, the electronic voting protocol with many candidates is designed based on the above-mentioned multiplicative homomorphism. The voters and the vote recorders can realize the voting process through distributing shares and reconstructing secret for only one time. The known large-scale key management schemes have some problems such as performance bottleneck, security, fault-tolerance performance and so on. So the group key management protocol with multiple servers is designed, which combines the decentralized and distributed multicast key management protocols. The additive homomorphism and Chinese Remainder Theorem are used to solve the above-mentioned problems and improve the efficiency of message transmission and the memory property.With the solution of multiplicative homomorphic secret sharing scheme, it is useful to calculate the multiplication of multiple secrets in secure multi-party computation, and it is equivalent to realize homomorphic encryption. The problem of low efficiency when existed multiple candidates is resolved in the designed electronic voting protocol. The problems of performance bottleneck, security and the efficiency of message transmission are resolved in the group key management protocol which combines the decentralized and distributed multicast key management protocols.Firstly, the concept of the easy-computing discrete logarithm is proposed based on the discrete logarithm, which makes the scheme have multiplicative homomorphism. In addition, it can achieve the multiplication of multiple secrets once. Secondly, the electronic voting protocol is appropriate for the case of large-scale candidates. The Godel coding is used to code multiple voting results for multiple candidates into one result. Then multiplicative homomorphism is used to share this result, and the votes for each candidate are calculated by decoding the reconstructed result. Each voter just distributes shares once and the vote recorder just reconstructs once for all candidates. The efficiency of protocol is improved. Lastly, the additive homomorphism of Shamir secret sharing scheme is applied to design the group key management protocol with multiple servers which combines the decentralized and distributed multicast key management protocols. The problems of performance bottleneck, security and other issues are resolved in this protocol. The Chinese Remainder Theorem is introduced to distribute keys, which reduces the communication complexity and realizes the key transmission in open channels.