  We provide a reduction of the Ring-LWE problem to Ring-LWE problems in
subrings, in the presence of samples of a restricted form (i.e. $(a,b)$ such
that $a$ is restricted to a multiplicative coset of the subring). To create and
exploit such restricted samples, we propose Ring-BKW, a version of the
Blum-Kalai-Wasserman algorithm which respects the ring structure. Off-the-shelf
BKW dimension reduction (including coded-BKW and sieving) can be used for the
reduction phase. Its primary advantage is that there is no need for
back-substitution, and the solving/hypothesis-testing phase can be
parallelized. We also present a method to exploit symmetry to reduce table
sizes, samples needed, and runtime during the reduction phase. The results
apply to two-power cyclotomic Ring-LWE with parameters proposed for practical
use (including all splitting types).
