Résumé
We prove that the Module Learning With Errors $\mathrm {M\text {-}LWE}$ problem with binary secrets and rank $d$ is at least as hard as the standard version of $\mathrm {M\text {-}LWE}$ with uniform secret and rank $k$, where the rank increases from $d \ge (k+1)\log _2 q + \omega (\log _2 n)$, and the Gaussian noise from $\alpha$ to $\beta = \alpha \cdot \varTheta (n^2\sqrt{d})$, where $n$ is the ring degree and $q$ the modulus. Our work improves on the recent work by Boudgoust et al. in 2020 by a factor of $\sqrt{md}$ in the Gaussian noise, where $m$ is the number of given $\mathrm {M\text {-}LWE}$ samples, when $q$ fulfills some number-theoretic requirements. We use a different approach than Boudgoust et al. to achieve this hardness result by adapting the previous work from Brakerski et al. in 2013 for the Learning With Errors problem to the module setting. Theproof applies to cyclotomic fields, but most results hold for a larger classof number fields, and may be of independent interest.