# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 5 A QKD Protocol Based on a Shared Collaboration among Three Parties
### Action and Impact of a Possible Eavesdropper under an Individual Attack
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Basis Selection and Outcomes Classes**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]\({}_{1}\)**} & \multirow{2}{*}{\(P\)**} \\  & **Basis** & **Out** & **Basis** & **Out** & **Basis** & **Out** \\ \hline
1: Basis selections completely meets & \(B_{A}\) & \(o_{a}\) & \(B_{A}\) & \(o_{a}\) & \(B_{A}\) & \(o_{a}\) & \(\frac{1}{2}\) \\
2: [PERSON] basis fails but [PERSON]\({}_{1}\) output not & \(B_{A}\) & \(o_{a}\) & \(B_{E\
eq A}\) & \(o_{c}\) & \(B_{A}\) & \(o_{a}\) & \(\frac{1}{2}\) \\
3: [PERSON]\({}_{1}\) basis meets but output fails & \(B_{A}\) & \(o_{a}\) & \(B_{E\
eq A}\) & \(o_{c}\) & \(B_{A}\) & \(o_{\beta\
eq a}\) & \(\frac{1}{2}\) \\
4: [PERSON]\({}_{1}\) basis and output fail & \(B_{A}\) & \(o_{a}\) & \(B_{E}\) & \(o_{c}\) & \(B_{\beta\
eq A}\) & \(o_{\beta}\) & \(\frac{1}{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification of basis selection and outcomes considering the presence of an eavesdropper for the traditional BB84 protocol. If [PERSON] and [PERSON]\({}_{1}\) spend some part of the key, they could detect the [PERSON] presence [11] by comparing their outcomes in a reconciliation procedure. Considering just the useful key (or part of it), 25% (\(\frac{1}{8}/\frac{1}{2}=\frac{1}{4}\) in Table 3) of it corresponds to the QBER in the protocol (Class 3). QBER is due to the presence of an eavesdropper, or otherwise to the presence of noisy communication in the quantum channel. Finally, Class 2 corresponds to the cases where the presence of [PERSON] is undetectable for [PERSON] and [PERSON]\({}_{1}\). Despite this, [PERSON] has no certainty if their key is correct because her basis does not meet with that of them (assuming she has access to that information published in a public channel). In the following, we will assume that such a class is not successful for [PERSON]. For the scheme presented here, the situation runs identical if such attack is performed just before [PERSON]\({}_{1}\) measures his state, but after to be assembled by [PERSON]/[PERSON]\({}_{0}\). In any case, this intervention assumes the possibility to steal the system and then be reintegrated ready for the Bob\({}_{1}\) measurement (otherwise gaining complete access to it, still being classically unnoticed, but no quantumly). In any case, it requires non-trivial interventions on the Bob\({}_{1}\) state as it was shown in the circuits in Figure 6b,c. Despite their technical complexity, note those procedures should be performed during the state transmission in the BB84 protocol, but in the current procedure, they should be performed after [PERSON] measures their control state. Despite, if [PERSON] pretends to perform the sifting of the Bob\({}_{1}\) state under the current protocol, then the probability of Class 1 (\(P_{E}\) in the following) and the only successful for [PERSON], changes dramatically. Then, we perform our analysis around this quantity and upon such assumptions. Figure 7a exhibits the eavesdropper temporal action through the events in the protocol. Instead of remaining the quantum channel opened all time as in the BB84 original eavesdropping, in our case, [PERSON] needs to sift the [PERSON]\({}_{1}\) state during the entire process consisting of the stages shown in black. There, the action performed by [PERSON] to define \(K\) (orange) determines the beginning of the assembling of the transmitted state to [PERSON]\({}_{1}\). Such selection could be made since the beginning when she prepares the control state (particularly if she wants to maximize \(C_{0}(p_{0},K)\) by selecting (\(p_{0}=p_{0_{max}}\)), or otherwise just before [PERSON]\({}_{0}\) should perform his processing \(\mathcal{U}_{0_{2}}\) if milder \(P_{j}\) values are allowed. Thus, [PERSON] can perform several types of individual attacks during such a period. Type A, after [PERSON]'s control measurement, has been already discussed as equivalent to the traditional individual attack in the BB84 protocol.