We use the learning with errors (LWE) problem to build a new simple and provably secure key exchange scheme. The basic idea of the construction can be viewed as certain extension of Diffie-Hellman problem with errors. The mathematical structure behind comes from the commutativity of computing a bilinear form in two different ways due to the associativity of the matrix multiplications:
(x^t \times A)\times y = x^t \times (A\times y),
where x,y are column vectors and A is a square matrix.
We show that our new schemes are more efficient in terms of communication and computation complexity compared with key exchange schemes or key transport schemes via encryption schemes based on the LWE problem.
Furthermore, we extend our scheme to the ring learning with errors (RLWE) problem, resulting in small key size and better efficiency.