which each user has multiple QKD transmitter modules and classical post-processing units. Note that the measurement devices are often the most expensive components of an entire QKD system because single photon detection is highly non-trivial. MDI-QKD allows measurement modules to be totally untrusted, which means that there is no need for redundant measurement modules if our proposal is employed with MDI-QKD. The users just need to have multiple transmitters and classical post-processing units, which thanks to the development of cheap chip-based QKD systems [@chip1; @chip2; @chip3], we believe, could render our proposal cost effective in the future. We remark that our approach is also fully compatible with quantum relays and quantum repeaters.[]{data-label="fig_mdi"}](fig_mdi){width="0.96\columnwidth"}

A key contribution of this paper is thus to show how, despite these obstacles, such VSS approach could be adapted to QKD to re-establish its security. The price that we pay is that now Alice and Bob have to use a redundant number of QKD modules and classical post-processing units. Fortunately, however, with the recent development of measurement-device-independent QKD (MDI-QKD) [@mdiQKD; @mdiQKDe1; @mdiQKDe2; @mdiQKDe3; @mdiQKDe4] and chip-based QKD [@chip1; @chip2; @chip3], the cost of QKD modules might decrease dramatically over time, see Fig. \[fig\_mdi\]. So, it is not 