On Module Lattices with Galois-Symmetries: What You See Is Not What You Get

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This paper deals with the hardness of finding short vectors in module lattices. Let $K$ be a number field of degree $d$ and $\mathcal{O}_K$ its ring of integers. We show that if a module lattice $M$ of rank $n$ in $\mathcal{O}_K^n$ has some Galois-symmetries, namely if it is fixed coordinate-wise (as a set) by a group $G$ of automorphisms of $K$, then $M$ can actually be seen as a module of rank~$n$ over a subfield~$K'$ of $K$ ($K'$ is the fixed-field of $G$), whose degree is $|G|$ times smal