Order-LWE and the Hardness of Ring-LWE with Entropic Secrets

We propose a generalization of the celebrated Ring Learning with Errors (RLWE) problem (Lyubashevsky, Peikert and Regev, Eurocrypt 2010, Eurocrypt 2013), wherein the ambient ring is not the ring of integers of a number field, but rather an order (a full rank subring). We show that our Order-LWE problem enjoys worst-case hardness with respect to short-vector problems in invertible-ideal lattices of the order. The definition allows us to provide a new analysis for the hardness of the abundantly used Polynomial-LWE (PLWE) problem (Stehlé et al., Asiacrypt 2009), different from the one recently proposed by Rosca, Stehlé and Wallet (Eurocrypt 2018). This suggests that Order-LWE may be used to analyze and possibly design useful relaxations of RLWE. We show that Order-LWE can naturally be harnessed to prove security for RLWE instances where the “RLWE secret” (which often corresponds to the secret-key of a cryptosystem) is not sampled uniformly as required for RLWE hardness. We start by showing worst-case hardness even if the secret is sampled from a subring of the sample space. Then, we study the case where the secret is sampled from an ideal of the sample space or a coset thereof (equivalently, some of its CRT coordinates are fixed or leaked). In the latter, we show an interesting threshold phenomenon where the amount of RLWE noise determines whether the problem is tractable. Lastly, we address the long standing question of whether high-entropy secret is sufficient for RLWE to be intractable. Our result on sampling from ideals shows that simply requiring high entropy is insufficient. We therefore propose a broad class of distributions where we conjecture that hardness should hold, and provide evidence via reduction to a concrete lattice problem.