Security of some cryptosystems on lattices

Since its introduction in 2005, the Learning With Errors (LWE) problem has had a profound impact in both theoretical and applied cryptography. The reason for the increased interest in the LWE problem is its hardness with respect to the lattice problems Decisional Approximate Shortest Vector Problem and Approximate Shortest Independent Vector Problem in the worst case. Thus, as a result of the LWE hardness assumption, LWE-based cryptographic systems are conjectured to be post-quantum secure. This paper provides complete proofs of the security of some lattice-based cryptosystems, which are based on the LWE assumption.