# How Secure Are Two-Way Ping-Pong and LM05 QKD Protocols under a Man-in-the-Middle Attack?
## 4 Conclusions  
To summarize, we considered man-in-middle (MITM) attacks on two kinds of two-way QKD protocols (pp with entangled photons and LM05 with single photons) in which an undetectable [PERSON] can decode all the messages in the message mode (MM) and showed that the mutual information between [PERSON] and [PERSON] is not a function of disturbance in the MM, since there is no disturbance in the MM, but is equal to unity no matter whether [PERSON] is in the line or not. [PERSON] induces a disturbance (\(D_{\text{CM}}\)) only in the control mode (CM). In a way, [PERSON]'s sending of the key is equivalent to sending an unencrypted plain text (via photons obtained by and returned to [PERSON]) secured by an indicator of [PERSON]'s presence. That burdens the protocols with the following drawbacks under a MITM attack:  
* the photons must cover the double distance than in an equivalent one-way BB84-like protocol (mcasBB84) which also has analogous MM and CM modes;
* while the BB84 protocol is unconditionally secure, and its revised version, the macasBB84 protocol proposed recently is only conditionally secure, the proof of unconditional security of the LM05 protocol given in [20] is not valid, as shown in details in Section 3; the mcasBB84 protocol has a predetermined threshold value of the CM disturbance (\(D_{pdCM}\)) at which [PERSON] and [PERSON] must abort the protocol whilst the considered two-way protocols do not have such a critical CM disturbance at which to abort the protocol; \begin{table}
\begin{tabular}{c c c c c} \hline  & **BB84** & **PP** & **LM05** & **mcasBB84-MITM** \\ \hline mode(s) & MM & MM + CM & MM + CM & MM + CM \\ \hline disturbance & \(0\leq D_{\text{MM}}\leq 0.5\) & \(D_{\text{MM}}=0\) & \(D_{\text{MM}}=0\) & \(D_{\text{MM}}=0\) \\  & \(0\leq D_{\text{CM}}\leq 0.5\) & \(0\leq D_{\text{CM}}\leq 0.5\) & \(0\leq D_{\text{CM}}\leq 0.5\) & \(0\leq D_{\text{CM}}\leq D_{pd-\text{CM}}\) \\ maximal disturbance & \(D_{critical-\text{MM}}=0.11\) &? &? & \(D_{pd-\text{CM}}\) \\ \hline secure & for \(D_{\text{MM}}<0.11\) & for \(D_{\text{CM}}<?\) & for \(D_{\text{CM}}<?\) & for \(D_{\text{CM}}<D_{pd-\text{CM}}\) \\ \hline mutual information & \(I_{AB}(D_{\text{MM}})\), \(I_{AE}(D_{\text{MM}})\) & \(\begin{array}{cc}I_{AB}=1\\ 0\leq I_{AE}(D_{\text{CM}})<1\end{array}\) & \(\begin{array}{cc}I_{AB}=1\\ 0\leq I_{AE}(D_{\text{CM}})<1\end{array}\) & \(\begin{array}{cc}I_{AB}=1\\ 0\leq I_{AE}(D_{\text{CM}})<1\end{array}\) & \(\begin{array}{cc}I_{AB}=1\\ 0\leq I_{AE}(D_{\text{CM}})<I_{AE}(D_{pd-\text{CM}})\end{array}\) \\ photon distance & \(L\) & \(4L\) & \(2L\) & \(L\) \\  & trans-mittance & \(\mathcal{T}\) & \(\mathcal{T}^{4}\) & \(\mathcal{T}^{2}\) & \(\mathcal{T}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of a symmetric BB84-like protocol under an arbitrary attack compared with properties of pp, LM05, and asymmetric mcasBB84 protocols under MITM. For the pp and LM05 protocols \(D<0.5\) means that [PERSON] is in the line only a portion of the time and \(D=0.5\) that she is in the line all the time. \(D_{pd-\text{CM}}\) is a predetermined threshold value of \(D<0.5\) for the mcasBB84-MITM [22] protocol. * since there are no errors in the MM mode, the privacy amplification (PA) is the only way to establish the security of the protocols and again in contrast to the mcasBB84 no PA procedures for the two-way protocols have been provided in the literature; Let us elaborate on these point in the reverse order. In the two-way protocols the mutual information between [PERSON] and [PERSON] is always greater than or equal to the one between [PERSON] and [PERSON]. When they are equal, i.e., when [PERSON] is in the line all the time, then [PERSON] and [PERSON] and [PERSON] have identical messages and there is no privacy which can be amplified and PA cannot erase key bits [PERSON] has snatched. For a \(D_{\text{CM}}<0.5\), but close to 0.5, [PERSON] and [PERSON] do not have a procedure and algorithms to obtain the secret key of which [PERSON] possess almost all bits.