On the Hardness of Module Learning With Errors with Short Distributions

Abstract

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 $\eta$-bounded secret for any $1\leq \eta \ll q$ 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 $\eta$-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 $\eta$. The latter result can be extended to provide the hardness of search M-LWE with uniform $\eta$-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.