# How Secure Are Two-Way Ping-Pong and LM05 QKD Protocols under a Man-in-the-Middle Attack?
## 2 Protocols and Attacks on Them  
Ping-pong (pp) protocol is based on two Bell states [7]. [PERSON] prepares entangled photons in one of the Bell states, sends one of the photons to [PERSON] while keeping the other one in a quantum memory (qm). [PERSON] either returns the photon as is or acts on it so as to put both photons into another Bell states. [PERSON] combines the photon he receives from [PERSON] with the one he kept in qm at a beam splitter (BS) to decode [PERSON]'s messages. The messages are said to be sent in message mode (MM). There is also a control mode (CM) in which [PERSON] measures [PERSON]'s photon and announces her outcomes over a public channel. The Bell basis used in the pp protocol consists of two Bell states  
\[|\Psi^{\mp}\rangle=\frac{1}{\sqrt{2}}(|H\rangle_{1}|V\rangle_{2}\mp|V\rangle_{ 1}|H\rangle_{2}), \tag{1}\]  
where \(|H\rangle_{i}\) (\(|V\rangle_{i}\)), \(i=1,2\), represent horizontal (vertical) polarized photon states. Photon pairs in the state \(|\Psi^{-}\rangle\) are generated by a down-converted entangled photon source. To send \(|\Psi^{-}\rangle\) state [PERSON] just returns her photon to [PERSON]. To send \(|\Psi^{+}\rangle\) she puts a half-wave plate (\(\text{HWP}(0^{\circ})\)) in the path of her photon. The HWP changes the sign of the vertical polarization. At [PERSON]'s BS the photons in state \(|\Psi^{-}\rangle\) will split and those in state \(|\Psi^{+}\rangle\) will bunch together. Eavesdropper [PERSON] carries out an MITM, designed by [PERSON] [23], as follows. She puts [PERSON]'s photon in a qm (delays it in the fiber coil) and makes use of a copy of [PERSON]'s device to send [PERSON] a photon from a down-converted pair in state \(|\Psi^{-}\rangle\) as shown in Figure 1a. When [PERSON] receives the photon from [PERSON] she combines it with the other photon from the pair and determines the Bell state in the same way [PERSON] would. She uses this result to generate the same Bell state for [PERSON] by putting the appropriate HWPs in the path of [PERSON]'s photon. Thus, [PERSON] is able to copy every single message in the MM undetectably and therefore sending messages in the MM is equivalent to sending plain text \"secured\" by the CM. In the LM05 protocol [9; 24] [PERSON] prepares a qubit in one of the four states \(|0\rangle\), \(|1\rangle\) (the Pauli \(\mathbf{Z}\) eigenstates), \(|+\rangle\), or \(|-\rangle\) (Pauli \(\mathbf{X}\) eigenstates) and sends it to his counterpart [PERSON]. In the MM she modifies the qubit state by applying either \(\mathbf{I}\), which leaves the qubit unchanged and encodes the logical \(\mathbf{0}\), or by applying \(i\mathbf{Y}=\mathbf{ZX}\), which flips the qubit state and encodes the logical \(\mathbf{1}\). (\(i\mathbf{Y}|0\rangle=-|1\rangle\), \(i\mathbf{Y}|1\rangle=|0\rangle\), \(i\mathbf{Y}|+\rangle=|-\rangle\), \(i\mathbf{Y}|-\rangle=-|+\rangle\).) [PERSON] now sends the qubit back to [PERSON] who measures it in the same basis in which he prepared it and deterministically infers [PERSON]'s operations, i.e., her messages, without basis reconciliation procedure. Eavesdropper [PERSON] carries out a MITM, designed by [PERSON] ([24], p. 61, Figure 5.5), as follows. As shown in Figure 1b. [PERSON] delays [PERSON]'s photon (qubit) in a fiber spool (a quantum memory) and sends her own decoy photon in one of the four states \(|0\rangle\), \(|1\rangle\), \(|+\rangle\), or \(|-\rangle\) to [PERSON], instead. [PERSON] encodes her message via \(\mathbf{I}\) or \(i\mathbf{Y}\) and sends the photon back. [PERSON] measures it in the same basis in which she prepared it, reads off the message, encodes [PERSON]'s delayed photon via \(\mathbf{I}\), if she read \(\mathbf{0}\), or via \(i\mathbf{Y}\), if she read \(\mathbf{1}\), and sends it back to [PERSON]. [PERSON] never learns the states in which [PERSON] sent his photons but that is irrelevant in MM since only polarization flipping or not flipping encode messages. [PERSON] also need not know [PERSON]'s states [9].