# How Secure Are Two-Way Ping-Pong and LM05 QKD Protocols under a Man-in-the-Middle Attack?
## 1 Introduction  
Quantum cryptography, in particular quantum key distribution (QKD) protocols, offers us, in contrast to the classical one, provably unbreakable communication based on the quantum physical properties of the information carriers [1; 2; 3]. So far, implementations were mostly based on the one-way BB84 protocol [4] which is unconditionally secure provided the quantum bit error rate (QBER) is low enough. However, QBER in BB84-like protocols might be high and since we cannot discriminate eavesdropper's ([PERSON]'s) bit flips from bit flips caused by noise in the line, the request of having QBER low enough for processing the bits is often difficult to satisfy. e.g., 4-state BB84 with more than 0.11 [5] and 6-state BB84 [6] with more than 0.126 [5] disturbances (\(D\)) have to be aborted (\(D\) is defined as the amount of polarization-flips caused by [PERSON], the maximum being 0.5). \(D\) includes the inherent QBER as well as possible Eve in the line. If [PERSON] were the only cause of \(D\), the mutual information between the sender ([PERSON]) and [PERSON] (\(I_{AE}\)) wound surpass the one between [PERSON] and the receiver ([PERSON]) (\(I_{AB}\)): \(I_{AE}>I_{AB}\) for \(D>0.11,0.126\), respectively. Protocols using two-way quantum communications have also been proposed. Since they are less efficient versions of BB84 protocols, they have no meaningful advantage. Here we show that the security of some two-way protocols is vulnerable under a man-in-the-middle (MITM) attack. In particular, entangled photon two-way protocols based on two [7] (also called a ping-pong (pp) protocol) and four (\(\Psi^{\mp},\Phi^{\mp}\)) [8] Bell states, on the one hand and a single photon deterministic Lucamarini-Mancini (LM05) protocol, on the other [9; 10]. Several varieties, modifications, and generalisations of the latter protocol are given in [11; 12; 13; 14]. Two varieties were implemented in [15; 16]. The former pp protocol was implemented by [PERSON] and [PERSON] in 2008 [17] while the protocol with four Bell states cannot be implemented with linear optics elements [18; 19]. In the aforementioned references various security estimations have been obtained. In [20] [PERSON], [PERSON], [PERSON], and [PERSON] provide a security proof of an LM05 deterministic QKD for the kind of attack proposed in [9; 10]. Nevertheless, they claim it to be a proof of the unconditional security of LM05. In [21] [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] provide a security proof for a modified pp protocol and prove its security against collective attacks in noisy and lossy channel. All previous elaborations of the security of two-way protocols assume that [PERSON] attacks each signal twice, once on the way from [PERSON] to [PERSON], and later on its way back from [PERSON] to [PERSON], and that in doing so she disturbs the signal in the message mode. However, there is another attack in which an undetectable [PERSON] encodes [PERSON]'s signals by mimicking [PERSON]'s encoding of a decoy signal sent to her which we elaborate on in this paper. We consider the two-way deterministic QKD protocols under a MITM attack where, [PERSON]--undetectable in the message mode (MM)--can acquire all the messages, meaning that there is no disturbance in the MM (\(D_{\text{MM}}\)) at all. In the control mode (CM) there is a disturbance (\(D_{\text{CM}}\)), but there is no critical \(D\) at which [PERSON] and [PERSON] should abort the protocol. The only way to delete bits of the raw key snatched by [PERSON] is via privacy amplification and for disturbances close to \(D_{\text{CM}}=0.5\), when [PERSON] is in the line all time, it seems impossible to distinguish whether [PERSON] has or has not obtained the whole key. In order to verify that conjecture, we prove that the security proof carried out in [20] does not cover a MITM attack and that therefore cannot be called \"unconditional\".