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The Learning with Errors (LWE) problem is the fundamental backbone of modernlattice based cryptography, allowing one to establish cryptography on thehardness of well-studied computational problems. However, schemes based on LWEare 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 inefficiency by working over a well chosen ring. Another popular variant, ModuleLWE, 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 byadding cyclic structure to Module LWE. The proposed construction is both moreefficient than Module LWE and conjecturally more secure than Ring LWE, the bestof both worlds. We show that the security reductions expected for an LWEproblem hold, namely a reduction from certain structured lattice problems tothe hardness of the decision variant of the CLWE problem. As a contribution oftheoretic interest, we view CLWE as the first variant of Ring LWE whichsupports non-commutative multiplication operations. This ring structurecompares favorably with Module LWE, and naturally allows a larger message spacefor error correction coding.

Non-Commutative Ring Learning With Errors From Cyclic Algebras