# _photonics_
## 2 QKD Protocol and Architecture Assumptions in the LEO
### Weak + Vacuum Decoy-State BB84 Protocol  
In our simulation, we used the most well-known variation of the BB84 protocol [6], the weak + vacuum decoy-state BB84 protocol [35]. The concept behind the BB84 protocol is that [PERSON] encodes bit sequences into different polarization states of single photons. Its security generally relies on the fact that [PERSON]'s presence cannot remain undetected due to the quantum nature of single photons and will therefore inevitably introduce errors in the bit sequences. The only thing [PERSON] and [PERSON] have to do is to sacrifice a part of the exchanged bits to estimate the error rate. If the bit error rate is lower than a specific threshold, they proceed to the procedure of the secure key distillation. Despite that the unconditional security of the BB84 protocol has been proven [5; 6; 36], security loopholes may arise when moving on its practical implementations. For example, due to the hardness of engineering on-demand single-photon sources, most QKD implementations and experiments rely on the use of highly attenuated laser sources which emit probabilistically photons, including also multi-photon pulses. In order not to compromise security by the presence of these multi-photon pulses, the selected intensity level leads to an average photon number much smaller than one, thereby leading to lower detection rates. Even in that case, there exist a non-zero probability of emitting multi-photon states, which can open the possibility of photon-number-splitting (PNS) attacks. The need to counter the PNS attack [37] triggers the invention of the decoy-state protocol [35; 38; 39], which allows the efficient distillation of secure keys using weak coherent pulse-based QKD systems that were once vulnerable [8]. In the weak + vacuum state modification, this kind of attacks is encountered using decoy and vacuum states in order to precisely measure the attenuation of the channel, the  
Figure 1: Positions of the three optical ground stations (OGSs) of Skinakas, Helmos, and Cholomon-das in Greece. The black line corresponds to a random passage of one of the 10 satellites of the constellation over Greece.  
background noise, and detector's dark counts. It has been shown that, with the decoy modification, higher key rates and longer communication distances can be achieved [40; 41].  
For the reasons mentioned above, we adopted the weak + vacuum decoy-state BB84 protocol [35]. The introduction of the decoy states enhances the detection of eavesdropping via PNS attacks, thus allowing Alice to transmit a higher mean photon number per pulse, improving key rates and reaching longer distances. It is important to mention here that this protocol is suitable for free-space transmission, since experimental results have already shown very good polarization stability in the satellite-to-ground transmission [42; 43]. The normalized SKR (bps/pulse) is lower bounded by the following inequation according to [35], as described in Appendix A:  
\[\frac{\text{SKR}}{f_{rep}}\geq\ q\big{\{}Q_{1}[1-H_{2}(e_{1})]-Q_{\mu}f\big{(} E_{\mu}\big{)}H_{2}\big{(}E_{\mu}\big{)}\big{\}}, \tag{1}\]  
where \(f_{rep}\) is the transmitters pulse repetition rate, \(q\) is the protocol efficiency, subscript \(\mu\) is the average photon number per signal in signal states, \(Q_{\mu}\) and \(E_{\mu}\) are the gain and the quantum bit error rate (QBER) of signal states, respectively, \(Q_{1}\) and \(e_{1}\) are the gain and the error rate of the single-photon state in signal states, respectively, \(f(x)\) is the bi-directional error correction rate, and \(H_{2}(x)\) is the binary entropy function.