of the different elements in this figure.[]{data-label="fig0"}](fig_THA){width="0.95\columnwidth"}

And so the key question is: how do we address covert channels and prove security in QKD with untrusted classical post-processing units? The existence of memory attacks in DI-QKD shows that quantum mechanics alone is not enough. Clearly, we need to include some additional assumptions. To solve this problem, we draw inspiration from the idea of verifiable secret sharing (VSS) [@verifiable1; @verifiable2] and the existence of secure multiparty computations [@book_smc] in conventional cryptography, where it is known that one can achieve information-theoretic security in a $n$-party cryptographic setup if the number of cheaters is less than $n/3$ [@smc1; @smc2; @maurer_smpc].

Standard VSS schemes, however, assume that all channels are classical, so by using error correction and authentication techniques one can basically make these channels perfect. In contrast, in QKD, owing to the noisy and lossy quantum channels controlled by Eve and the quantum no-cloning theorem, to distill a final key Alice and Bob need to apply several classical post-processing steps to the raw data produced by the QKD modules in a setting where [*both*]{} the QKD modules and the classical post-processing units might be corrupted.

![Schematic representation of a MDI-QKD network [@mdiQKD] in 