# 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  
QKD schemes are subject to security attacks. In the analysis of quantum security, several types of attacks are normally considered. There, in the most simple level, individual attacks include quantum interactions with a single quantum channel carrying the information to be read as a single register under measurement (in this case the qubit of Bob\({}_{1}\)). Other types of attacks consider collective attacks [48; 49], where measurements are not individual, instead, they are allowed to be performed coherently together. Otherwise, coherent attacks [48; 50] allow us to apply unitary transformations to the whole set of measurements in addition. While collective and coherent attacks are out of the scope of this work, in this subsection, we analyse a type of individual attack showing a certain advantage on the traditional BB84 protocol. For the sifting of the [PERSON]\({}_{1}\) state, [PERSON] achieves it using an alternative pair of \(SWAP\) gates exchanging the states between [PERSON]\({}_{1}\) and [PERSON] to thus steal and return it. In such a case, an intermediate measurement of the stolen state by [PERSON] is then returned to [PERSON]\({}_{1}\) before his measurement (see Figure 6b). The initial state in possession of [PERSON] could be non-meaningful. Otherwise, an alternative \(C^{a}NOT_{b}\) gate arrangement could be performed upon the election of the measurement basis (see Figure 6c,d). In this case, [PERSON] first bets by the basis on which [PERSON] has prepared the state. If she supposes the basis is \(H\), then she should arrange the procedure presented in Figure 6c, just stating a \(C^{a}NOT_{b}\) gate between her qubit in the state \(|0\rangle_{E}\) and controlled by the [PERSON]\({}_{1}\) state; instead, if she bets for the basis \(D\), she should implement the circuit in Figure 6d using complementary \(H\) gates to translate the [PERSON]\({}_{1}\) states to the previous situation, but still returning him his original one. It is immediate to demonstrate that any of those circuits effectively copy the [PERSON]\({}_{1}\) state if [PERSON] hits the correct basis in which [PERSON] has prepared the final state for [PERSON]\({}_{1}\):  
\[|i\rangle_{\text{[PERSON]}_{1}}\in\{|0\rangle,|1\rangle\} \longrightarrow C^{\text{[PERSON]}_{1}}NOT_{\text{[PERSON]}}\cdot|i\rangle_{\text{[PERSON]}_{1} }\otimes|0\rangle_{\text{[PERSON]}}=|i\rangle_{\text{[PERSON]}_{1}}\otimes|i\rangle_{ \text{[PERSON]}} \tag{29}\] \[|i\rangle_{\text{[PERSON]}_{1}}\in\{|+\rangle,|-\rangle\} \longrightarrow H_{\text{[PERSON]}}H_{\text{[PERSON]}_{1}}C^{\text{[PERSON]}_{1}}NOT_{\text{ [PERSON]}}H_{\text{[PERSON]}_{1}}\cdot|i\rangle_{\text{[PERSON]}_{1}}\otimes|0\rangle_{\text{ [PERSON]}}=|i\rangle_{\text{[PERSON]}_{1}}\otimes|i\rangle_{\text{[PERSON]}}\]  
Otherwise, if [PERSON] fails in the basis selection, then her outcomes are non-meaningful so they do not change the overall probabilistic distribution in the BB84 protocol with [PERSON] using the \(SWAP\) gate as before, thus giving the same outcomes previously discussed. As it is well-known for the BB84 protocol, for the individual attacks on the qubits coming from [PERSON], [PERSON] will have success in the 50% (\(\frac{1}{2}/\frac{1}{2}=\frac{1}{2}\) in Table 3) of the useful key where she passes unnoticed. Table 3 classifies the possible cases. There, probability \(P\) is absolute concerning the entire cases, so it should be divided by 1/2 to get the conditional probability for the useful cases. Class 1 in the first row corresponds to the previous situation where the three bases selected meet. In this case, [PERSON] becomes unnoticed and in possession of a valuable key bit. Class 4 corresponds to the cases conducting to the non-useful key because [PERSON] and [PERSON]\({}_{1}\) do not meet in their basis if [PERSON] does or not. Thus, only Classes 1 to 3 correspond with a possible useful key.