# How Secure Are Two-Way Ping-Pong and LM05 QKD Protocols under a Man-in-the-Middle Attack?
## 3 Security of the Protocols
The closer \(D_{\text{MM}}\) is to \(0.11\), the more difficult is for [PERSON] and [PERSON], after PA, to extract the secure key from the raw key, since the former becomes smaller and smaller. \"The efficiency of privacy amplification rapidly decreases when [\(D_{\text{MM}}\)] increases\" ([32], p. 165, mid right column). \"At \(D_{\text{MM}}=0.11\) the secure-key length formally vanishes\" ([33], p. 524). See also [31]. For a MITM attack on two-way protocols (which are without any sifting), when [PERSON] is in the line all the time, there is no \(D_{\text{MM}}\) that [PERSON] induces and the mutual information between [PERSON] and [PERSON] as well as between [PERSON] and [PERSON] is equal to unity: \(I_{AB}=1\). In Figure 2b \(D_{\text{CM}}\) indicates the presence of [PERSON], where \(D_{\text{CM}}=0.5\) would mean that [PERSON] is always in the line. For \(D_{\text{CM}}\) slightly below \(D_{\text{CM}}=0.5\) one cannot exclude the possibility that [PERSON] has all the messages. When [PERSON] has all the messages then there is no Alice-Bob \"privacy\" they could amplify. When [PERSON] snatches only a portion of messages, then, for \(D_{\text{CM}}\) close to \(0.5\), there is still a question whether [PERSON] has all messages or not and whether [PERSON] and [PERSON] can erase [PERSON]'s messages with their PA. With that in mind, we can examine the security evaluation for the MITM and verify whether the proofs of unconditional security carried out for other kind of attack on LM05 in [10; 20] might apply to it as well. Figure 2: Mutual information plots for (**a**) one-way protocol BB84; (**b**) two-way protocols with either pp entangled Bell states or with LM05-like single photon states under a MITM attack; (**c**) one-way asymmetric BB84-like protocol, in which one basis serves as MM and the other as CM, under a MITM attack (mcasBB84-MITM); \(I_{AE}\) stands for \(I_{AE}(D_{\text{{\it pd}}-\text{CM}})\). In the aforementioned security proof [20], which is claimed to be unconditional, the authors assume that [PERSON] probes the qubit, entangling it with an ancilla. However, their approach does not cover the MITM attacks. To show this we point to the following steps in their proof of unconditional security vs. two-way-protocol-under-MITM counter-steps:  
* p. 2, 2 nd paragraph from the top: \"[PERSON] announces partial of her key bits in the encoding mode [MM]. They compute the error rate \(e\) in the Alice-Bob channel.\"
* \(I_{AB}=1\) _and [PERSON] does not induce any error in the_ MM, _ever._
* p. 2, Sec. III.A: \"[PERSON] cannot gain any information about [PERSON]'s key bits if she only attacks the qubits after [PERSON]'s encoding operation.\"
* _Since [PERSON] in her MITM sends her own photons to [PERSON] and then reads off_ **I** _or_ _iY from [PERSON]'s encoding of those qubits, [PERSON] gains all information from [PERSON]'s qubits, more precisely, from [PERSON]'s qubits encoded by [PERSON]. Note the neither [PERSON] nor [PERSON] know which states the qubits [PERSON] sends are in. They only control_ **I** _and_ _iY_. * [PERSON]'s most general quantum operation can be described by a unitary operation together with an ancilla. In the Bob-Alice channel, when [PERSON] sends a qubit in state \(|0\rangle\) and [PERSON] measures in the basis \(|0\rangle\),\(|1\rangle\), she will get the measurement outcomes \(|0\rangle\) with probability \(c_{00}^{2}\) or \(|1\rangle\) with probability \(c_{01}^{2}\). * _[PERSON] does not measure qubits. She just applies_ **_I_** _and_ _iY_. * [PERSON]'s most general attack (with ancillas) is \(U_{BE}|0\rangle_{B}|E\rangle=c_{00}|0\rangle_{B}|E_{00}\rangle+c_{01}|1\rangle_ {B}|E_{01}\rangle,U_{BE}|1\rangle_{B}|E\rangle=c_{11}|1\rangle_{B}|E_{11} \rangle+c_{10}|0\rangle_{B}|E_{10}\rangle\), \(U_{BE}|+\rangle_{B}|E\rangle=c_{++}|+\rangle_{B}|E_{++}\rangle+c_{+-}|-\rangle _{B}|E_{+-}\rangle\), \(U_{BE}|-\rangle_{B}|E\rangle=c_{--}|-\rangle_{B}|E_{-+}\rangle+c_{-+}|+\rangle_ {B}|E_{-+}\rangle\).