Hardness of Module-LWE with Semiuniform Seeds from Module-NTRU

The module learning with errors (MLWE) problem has attracted significant attention and has been widely used in building a multitude of lattice-based cryptographic primitives. The hardness of the MLWE problem has been established for several variants, but most of the known results require the seed distribution (i.e., the distribution of matrix $\mathbf{A}$ ) to be the uniform distribution. In this paper, we show that under the Module-N-th degree Truncated polynomial Ring Units (NTRU) (MNTRU) assumption, the search MLWE problem can still be hard for some distributions that are not (even computationally indistinguishable from) the uniform distribution. Specifically, we show that if the seed distribution is a semiuniform distribution (namely, the seed distribution can be publicly derived from and has a “small difference” to the uniform distribution), then for appropriate settings of parameters, the search MLWE problem is hard under the MNTRU assumption. Moreover, we also show that under the appropriate settings of parameters, the search learning with errors over rings problem with semiuniform seeds can still be hard under the NTRU assumption due to our results for the search MLWE problem with semiuniform seeds being rank-preserving.