On the Hardness of Module Learning with Errors with Short Distributions

The Module Learning With Errors (M-LWE) problem is a core computational assumption of lattice-based cryptography which offers an interesting trade-off between guaranteed security and concrete efficiency. The problem is parameterized by a secret distribution as well as an error distribution. There is a gap between the choices of those distributions for theoretical hardness results (standard formulation of M-LWE , i.e., uniform secret modulo q and Gaussian error) and practical schemes (small bounded secret and error). In this work, we make progress toward narrowing this gap. More precisely, we prove that M-LWE with uniform η-bounded secret for any 1 ≤ η≪ q and Gaussian error, in both its search and decision variants, is at least as hard as the standard formulation of M-LWE , provided that the module rank d is at least logarithmic in the ring degree n. We also prove that the search version of M-LWE with large uniform secret and uniform η-bounded error is at least as hard as the standard M-LWE problem, if the number of samples m is close to the module rank d and with further restrictions on η. The latter result can be extended to provide the hardness of search M-LWE with uniform η-bounded secret and error under specific parameter conditions. Overall, the results apply to all cyclotomic fields, but most of the intermediate results are proven in more general number fields.