to achieve our goal, which uses as main ingredients VSS schemes [@verifiable1; @verifiable2; @maurer_smpc] and privacy amplification techniques (see Appendix \[tool\]). The former is employed to defeat corrupted classical post-processing units. Indeed, given that $t'<s/3$ and $t''<r/3$, the use of VSS schemes allows to post-process the raw keys generated by the QKD modules in a distributed setting by acting only on raw key shares. More importantly, this post-processing of raw key shares can be performed such that no set of corrupted classical post-processing units can reconstruct $k_{\rm A}$ and $k_{\rm B}$ and, moreover, it is also guaranteed that $k_{\rm A}$ and $k_{\rm B}$ is a correct key independently of the misbehaviour of the corrupted units which might wish to purposely introduce errors. In this regard, a key insight of our paper is to show that, since all the classical post-processing techniques that are typically applied in QKD are “linear” in nature ([*i.e.*]{}, they involve simple functions in linear algebra such as bit-wise XOR and multiplications of matrices), they are easily implementable in a distributed setting.

![[**(a)**]{} Schematic representation of a QKD setup with multiple QKD modules and classical post-processing units. We assume that up to $t<n$ pairs of QKD modules, 