# 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
Despite this, such a possibility is not meaningful here because the information is not properly travelling through a quantum channel. Anyway, [PERSON]\({}_{1}\) should avoid this possibility by using this resource rapidly after the assembling. In addition, if [PERSON] has not to control the time assembling of the transmitted state, she can perform her attack just before [PERSON]'s control measurement (type C) or still before the \(\mathcal{U}_{0_{2}}\) codification by [PERSON]\({}_{0}\) (type B). Both attacks conduct to the same [PERSON]'s success probability outcome as it is seen in the Appendix C. Note this procedure is not equivalent to the BB84 one, so the success probability changes, despite it still requires the exchange methods depicted in Figure 6.  
Figure 7: (**a**) Eavesdropping temporal action through the protocol; (**b**) contour lines of \(C_{0}(p_{0},K)\) as function of \(p_{0}\) and \(K\) with their maximal values as blue dots; and contour lines for the conditional probability for the [PERSON] success, \(P_{E}\), as function of \(\omega_{1}\) and \(K\) in color for (**c**) \(k=0,\theta=0\); (**d**) \(k=0,\theta=\frac{\pi}{2};(\textbf{e})\,k=1,\theta=0\); and (**f**) \(k=1,\theta=\frac{\pi}{2}\). \(P_{j}\geq 0.9\) corresponds to the gray region. In addition, the difference does not depend on the open selection of \(C_{0}(p_{0},K)\) (becoming better choosing \(C_{0_{max}}\) to maximize \(P_{j}\)) as will be seen. For both cases (Type B and C), \(P_{E}\) is given by (see Appendix C):  
\[P_{E} = \frac{P}{P_{j}}=\frac{C_{0}(p_{0},K)}{P_{j}}(f_{k\oplus 1}(\frac{ \theta}{2})\sqrt{1-K^{2}f_{K}^{2}(\omega_{1})}-Kf_{k\oplus 1}(\omega_{1}))^{2} \tag{30}\]  
then, an attempt to maximize \(P_{j}\) (the rate for the successful key) directly should raise the value of \(P\), but not \(P_{E}\). Figure 7b shows such dependence through contour lines. As function of \(p_{0}\), \(K\) could be selected almost openly, despite only one value maximizes the value of \(C_{0}(p_{0},K)\) with \(p_{0}\) fixed (blue dots in Figure 7b). The color of each point on the curves reflects the \(C_{0}(p_{0},K)\) value (darkest for the lowest values and brightest for the highest ones). Blue dots mark the value where the maximum of \(C_{0}(p_{0},K)\) is reached for a given \(p_{0}\) or \(K\). Nevertheless, many other values of \(K\) could still keep \(P_{j}\) in a higher value to maintain the performance of the key generation, but still to conveniently reduce \(P_{E}\). In Figure 7c-f, the involved regions marked in gray with \(P_{j}\geq 0.9\) (just for the \(j+m=1\) cases), remark certain criteria for the random election of \(K\). In any case, the meaningful quantity to evaluate the performance of [PERSON] is the conditional probability \(P_{E}\) which is independent of \(C_{0}(p_{0},K)\). It just accounts for the useful key cases when [PERSON] reaches correctly her selection \(|b_{j}\rangle\), as similarly the conditional probabilities for the Table 3 were calculated concerning the useful key outcomes. It is plotted as a function of \(\omega_{1}\) and \(K\), and coloured in agreement with the color bar inside. Cases correspond to (c) \(k=0,\theta=0\); (d) \(k=0,\theta=\frac{\pi}{2}\); (e) \(k=1,\theta=0\); and (f) \(k=1,\theta=\frac{\pi}{2}\) (remembering that the selected state for [PERSON]\({}_{1}\) is \(k\oplus 1\)). In any case, clearly, the strategy lets a notable lowering for the probabilities for the success of [PERSON] to great extent. The reason is now evident, \(P\) has a wide distribution in the region where \(P_{j}\) is high, then it still lets the selection of low values for \(P\). The last behavior is due to the election of basis by [PERSON] before [PERSON] did, thus modifying the global state and lowering the success notably in most cases.