Algebraic aspects of solving Ring-LWE, including ring-based improvements in the Blum-Kalai-Wasserman algorithm
This addresses cryptographic security analysis for lattice-based schemes, offering incremental improvements in efficiency for practical Ring-LWE parameters.
The paper tackles the Ring-LWE problem by proposing Ring-BKW, a variant of the Blum-Kalai-Wasserman algorithm that leverages ring structure to reduce it to subring problems, eliminating the need for back-substitution and enabling parallelization, with methods to cut table sizes, samples, and runtime.
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).