a key exchange scheme based on lattice

We construct a key exchange scheme with session key security and Alice's privacy, assuming the hardness of "small integer solution" (SIS) problem and "learning with errors" (LWE) problem, which have worst-case to average-case reductions from lattice problems. Our key exchange scheme is the first lattice-based key exchange scheme with provable security. Moreover, our scheme can be modified to be secure under impersonation attack by adding an authentication step, also, it can be modified to be more efficient using a new variant of LWE.