Research On Zero-knowledge Proofs Based On Lattice And Its Application In Blockchains

Zero-knowledge proof is an important part of cryptography,which means that the prover can convince the verifier that a statement is true without revealing valid knowledge.Most of non-interactive zero-knowledge proof schemes are based on trapdoor permutations,pairing problem,indistinguishability obfuscation and so on,but less on lattice difficulty problem.So it is still an unsolved problem to construct a noninteractive zero knowledge scheme based on lattice difficult problems.With the development of the Blockchain technology,privacy protection in Blockchain has become a key issue for the implementation of Blockchain technology.Zcash based on succinct non-interactive arguments of knowledge is one of digital currencies with the highest anonymous,but the zero knowledge proof schemes in Zash can't resist quantum attack.Therefore,it is important to design a zero-knowledge proof scheme against quantum attack to improve the security of Blockchain.Based on the Analysis,this thesis makes the following work:(1)Combined with the full homomorphic hybrid encryption framework proposed by Gentry et al,a interactive zero-knowledge proofs scheme based on fully homomorphic encryption FHE-ZKP was proposed,and converts it into a noninteractive zero-knowledge proof scheme in the random Oracle model by the “Fiatshamir heuristic” method.In addition,the completeness,soundness and zeroknowledge of the scheme FHE-ZKP were proved.We used the software Python and Magma to do a confirmatory experiment on this scheme,and carried out a usability and practicability analysis.The security of fully homomorphic encryption scheme is based on the LWE problem,and the security of the interactive zero-knowledge proof scheme is based on the SIS problem.(2)Based on the fully homomorphic hybrid encryption framework proposed by Gentry et al.,a designated-verifier non-interactive zero-knowledge proofs based on the fully homomorphic hybrid encryption in the common reference string model FHHEDVNIZKP was proposed.Its security depends on the SIS problems.Based on the fully homomorphic hybrid encryption and the sampling algorithm on the lattice about gadgets matrix,the scheme FHHE-DVNIZKP was proposed.In order to analyze its security,we proved its completeness,soundness and zero knowledge.