(Bob) has $s$ ($r$) classical post-processing units at their disposal, each of them ideally purchased from a different provider. Alice’s modules QKD$_{{\rm A}i}$, with $i=1,\ldots,n$, are connected to the classical post-processing units CP$_{{\rm A}i'}$, with $i'=1,\ldots,s$, via secure channels ([*i.e.*]{}, channels that provide both secrecy and authentication). Also, all the units CP$_{{\rm A}i'}$ are connected to each other via secure channels. The same applies to Bob. Importantly, since all these secure channels are located only within Alice and Bob’s labs, in practice they could be implemented, for instance, by using physically protected paths ([*e.g.*]{}, physical wires that are mechanically and electrically protected against damage and intrusion) which connect only the prescribed devices. Furthermore, each QKD$_{{\rm A}i}$ is connected to its partner QKD$_{{\rm B}i}$ via a quantum channel, and each CP$_{{\rm A}i'}$ is connected to all CP$_{{\rm B}i''}$, with $i'=1,\ldots,s$ and $i''=1,\ldots,r$, via authenticated classical channels [@wegman; @wegman2].

Moreover, for simplicity, we shall consider a so-called threshold active adversary structure. That is, we will assume that up to $t<n$ pairs of QKD modules, up to $t'<s/3$ units CP$_{{\rm A}i'}$ and up to $t''<r/3$ independent units CP$_{{\rm B}i''}$ could be corrupted. We say that a pair of QKD modules is corrupted when 