Non-Commutative Ring Learning With Errors From Cyclic Algebras

The Learning with Errors (LWE) problem is the fundamental backbone of modern lattice based cryptography, allowing one to establish cryptography on the hardness of well-studied computational problems. However, schemes based on LWE are often impractical, so Ring LWE was introduced as a form of `structured LWE, trading off a hard to quantify loss of security for an increase in efficiency by working over a well chosen ring. Another popular variant, Module LWE, generalizes this exchange by implementing a module structure over a ring. In this work, we introduce a novel variant of LWE over cyclic algebras (CLWE) to replicate the addition of the ring structure taking LWE to Ring LWE by adding cyclic structure to Module LWE. The proposed construction is both more efficient than Module LWE and conjecturally more secure than Ring LWE, the best of both worlds. We show that the security reductions expected for an LWE problem hold, namely a reduction from certain structured lattice problems to the hardness of the decision variant of the CLWE problem. As a contribution of theoretic interest, we view CLWE as the first variant of Ring LWE which supports non-commutative multiplication operations. This ring structure compares favorably with Module LWE, and naturally allows a larger message space for error correction coding.