## 4 The Photon Number Splitting Attack  
In the PNS attack, the eavesdropper captures at least one photon from each of the multi-photon states with the aim of storing them in quantum memory, while she blocks the single photon states in the quantum channel. Once [PERSON] has revealed over the public channel the basis measurements he has used, the eavesdropper performs the same measurements on the quantum states she has stored [9]. When the PNS attack is applied to the ack-QKD protocol, the eavesdropper captures at least one photon of the multi-photon states (parallel and non-orthogonal), and she waits for [PERSON]'s announcements about the measurement bases he has used in order to apply the same measurements on her stored states. At [PERSON]'s side, a distribution over the following sifting instances is obtained: (\(2M\)), (\(2 nM\)), (\(S\)-1), (\(S\)-2) and (\(2L\)); where each one may come from parallel or non-orthogonal states, but only [PERSON] knows those results. After [PERSON] announces both the measurement bases (\(X\) or \(Z\)) and the sifting instances, [PERSON] performs the measurements using the same measurement bases, and she gets the same bits from the multi-photonic single sifting instances: (\(S\)-1) and (\(S\)-2), parallel and non-orthogonal. Furthermore, the eavesdropper obtains the same results from the (\(2M\)) measurements of the parallel and (a half of the) non-orthogonal multi-photonic states. However, she cannot obtain the secret bits from the one-state (\(S\)-\(i\)) and (\(2M\)) sifting instances, because the eavesdropper cannot distinguish between parallel and non-orthogonal states. To get the secret bits, [PERSON] blocks the one-photon states, which include single- and double-detection events from parallel and non-orthogonal states. In doing that, [PERSON] introduces an error gain in the photonic gain of the single- and double-detection events. Then, [PERSON] performs a channel substitution increasing the transmittance of the channel. The fiber channel transmittance between [PERSON] and [PERSON] is written as \(T_{AB}=10^{-\frac{\eta l}{10}}\), where \(\alpha\) is the loss coefficient measured in \(dB/km\) and the length, \(l\), is measured in \(km\). Furthermore, the local transmittance at [PERSON]'s side, \(\eta_{B}\), is represented as \(t_{B}\eta_{D}\), where \(t_{B}\) is the internal transmittance of optical components and \(\eta_{D}\) is the quantum efficiency of [PERSON]'s detectors. Then, the overall transmission and detection efficiency at [PERSON]'s side, \(\eta_{BT}\), is computed as \(\eta_{BT}=t_{B}\eta_{D}T_{AB}\)[8]. However, in replacing \(T_{AB}\), [PERSON] can adjust the photonic gain of the single detection events or the double detection events, but not both simultaneously. In contrast, [PERSON] uses the double matching detection events (\(2M\)) and the (\(S\)-\(i\)) sifting instances, which are compatible with the states she prepared, to verify corresponding photonic gains, parallel and non-orthogonal. We define the quantum photonic error gain (QPEG or, simply, \(\Delta\)Q) as the deviation gain from the reference photonic gain that is caused by [PERSON]'s apparatus at [PERSON]'s receiver station (see Table 3). We will denote the QPEG of double \((+,+)\) detection events as \(\Delta Q_{(+,+)}\) and the QPEG of single \((\pm,\mp)\) detection events as \(\Delta Q_{(\pm,\mp)}\). We calculate \(\Delta Q_{(+,+)}\) as the difference \(Q_{(+,+)_{AB}}-Q_{(+,+)_{EB}}\), where the symbol, \((+,+)_{AB}\), defines the reference photonic gain at calibration time of the double \((+,+)\) detection events and \((+,+)_{EB}\) denotes the gain of the double detection events at [PERSON]'s side, but in the presence of [PERSON] at running time). Similarly, we calculate \(\Delta Q_{(\pm,\mp)}\) as \(Q_{(\pm,\mp)_{AB}}-Q_{(\pm,\mp)_{EB}}\), where we use the subindex of \(\left(\pm,\mp\right)_{AB}\) and \(\left(\pm,\mp\right)_{EB}\) with the same purpose.