Research And Design Of Provable Security Public-key Cryptosystem Based On Lattice

The theory of lattice is hot topic for its quantum-resistant. The public-keycryptosystem on lattice is not only resisting quantum attack, but also working efficiently.The NTRU is the unique effective public-key cryptosystem on lattice, but the proof ofthe security for NTRU is absent. So how to design the security and efficient public-keycryptosystem is the research contents in this paper. Our research is as follows:At first, in order to design the public-key cryptosystem based on lattice, we must beclear about the hard problem on lattice. The lattice-reduction is presented to analyze theproblem on lattice. So it becomes the first research point of our paper. A new scheme oflattice-reduction is put forward and realized. The vector get from our scheme is shorterthan the vector from other scheme. And also the method to generate the lattice when weuse the lattice-reduction to analyze the public-key cryptosystem is raised. It is applied tosolve the Regev’s public-key cryptosystem based on learing with errors (LWE) in ourpaper.Secondly, we research the previous public-key cryptosystem based on lattice andimproved two of them. The Regev’s cryptosystem is improved to bits encrypted. Thesecurity of it is proved equal to the LWE. And also according to the first cryptosystembased on learning with errors over the ring, we present another public-key cryptosystembased on lattice, which is satisfied chosen plaintext security under the static model.What’s more, two signature scheme based on lattice is represented. The first one isbased on LWE. We also use the signature scheme to construct a group signature scheme.The analysis of this scheme shows it is high security. The second one is an identitybased signature scheme. The core technology of this scheme is trapdoor function andbonsai trees. We prove it is chosen message attack security under the random oracle.The comparison between the new scheme and the old one shows it has high speed.In the end, we constructed two public-key encryption schemes. The framework ofthe first one is similar as NTRU. But its hard problem is relying on the R-LWE. At thesame time, we point out it is more security than NTRU. In the second scheme we leadinto the hash function. So the scheme can satisfy chosen cipertext attack security. Thetechnology of this scheme is new. And it is simpler than the past scheme, which satisfieschosen cipertext attack security, based on lattice.