In view of the fact that the rapid development of quantum computer technology poses serious threat to the security of the traditional public-key cryptosystem, a new authenticated key exchange protocol scheme based on Ring Learning With Errors (RLWE) problem was proposed. By using Peikert error reconciliation mechanism, both parties of communication can directly obtain the shared bit value of the uniform distribution and get the same session key. The encoding bases of lattice was used to analyze the error tolerance, and reasonable parameters were selected to ensure that both parties can get the same session key with significant probability. The security of the protocol was proved in the BR (Bellare-Rogaway) model with weak perfect forward secrecy. The security of the protocol was attributed to the difficult RLWE problem of lattice, so that the protocol can resist quantum attacks. Compared with the existing authenticated key exchange protocols based on RLWE, the size of the parameter value modulus decreases from sub-exponential to polynomial magnitude, thus the corresponding amount of computation and communication are also significantly reduced. The results show that the proposed scheme is a more concise and efficient post quantum authenticated key exchange protocol.

A lattice signature scheme was proposed and some parameter choosing rules were illustrated concerning Small Integer Solution (SIS) problem and random oracle model of lattice. Then the results of the length of the keys that were generated under different parameter circumstances were compared. Afterwards the security and efficiency with the signature scheme were verified. At last, for the purpose of fairness, and reliability in multipartite authentication, the signature scheme was combined with key distribution and escrow, a new authentication scheme with the Singular Value Decomposition (SVD) algorithm based on mathematical matrix decomposition theory was proposed.