The ``learning with errors'' (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications.

Unfortunately, these applications are rather inefficient due to an inherent quadratic overhead in the use of LWE. After a short introduction to the area, we will discuss recent work on making LWE and its applications truly efficient by exploiting extra algebraic structure. Namely, we will define the ring-LWE problem, and prove that it too enjoys very strong hardness guarantees. We will mention some recent cryptographic applications in this line of work.

Based on joint work with Vadim Lyubashevsky and Chris Peikert.