# Quantum Key Distribution in the Presence of the Intercept-Resend with Faked States Attack
## 3 The _Nack State_ Protocol  
The _nack state_ protocol is the dual version of the _ack state_ protocol discussed in [19]. Both of them constitute a generalization of the well-known \(BB84\). The _nack state_ protocol uses pairs of parallel and orthogonal states instead of only single non-orthogonal state used in the \(BB84\). This simple difference makes the _nack state_ resilient to the \(IRFS\) attack, as we will show later on. We chose the _nack_ prefix  
Figure 2: In the quantum blinding Intercept Resend with Faked States (\(IRFS\)) attack, [PERSON] uses the same optical receiver unit as [PERSON] to detect [PERSON]’s states in a random basis. Then, she prepares those quantum states, but she sends her results to [PERSON] as bright light pulses instead of quantum pulses. (**a**) [PERSON] uses the same basis as [PERSON]; (**b**) [PERSON] and [PERSON]’s basis are opposite. to denote that provided [PERSON] sends two quantum states to [PERSON], the second measurement acts as the negative acknowledgment (_nack_) of the previous one, because it yields the opposite bit result. In the rest of this section, we describe the _nack state_ protocol, and in Section 4, we discuss how the _nack state_ protocol is capable of detecting the \(IRFS\) attack. We refer to the pair of quantum states as a biqubit. To be more specific, the following biqubits are defined in the _nack state_ protocol: four parallel biqubits \((\ket{0_{X}},\ket{0_{X}}),(\ket{0_{Z}},\ket{0_{Z}}),(\ket{1_{X}},\ket{1_{X}})\), \((\ket{1_{Z}},\ket{1_{Z}})\) and two orthogonal biqubits \((\ket{0_{X}},\ket{1_{X}}),(\ket{0_{Z}},\ket{1_{Z}})\). The parallel and orthogonal biqubits are randomly interleaved by [PERSON]. The order of the quantum states within the biqubit does not affect the behavior of the protocol (see Figure 3). On the other side of the quantum channel, [PERSON] measures two incoming states of a biqubit using the same measurement basis (\(X\) or \(Z\)). The _nack state_ protocol can be described according to the following steps:  
1. [PERSON] is equipped with a photon source with an expected photon number \(\mu\) that exhibits a Poisson distribution. [PERSON] randomly chooses between a parallel or an orthogonal biqubit, and she prepares the biqubit to send it to [PERSON] through the quantum channel;
2. [PERSON] measures the biqubit (two incoming pulses) using the same measurement basis \(X\) (or \(Z\)) that he chooses randomly (in Section 4.2, we discuss that the consecutiveness of states can be avoided if [PERSON] sends a burst of the first states of each pair, followed by a burst of the second states of each pair);
3. [PERSON] announces publicly his measurement basis choices;
4. To share secret bits, [PERSON] and [PERSON] perform sifting using single compatible events and double compatible matching detection events (from parallel states). Similarly, they apply sifting to the double detection events that contain a single compatible detection event. For this purpose, [PERSON] indicates if the single detection is the first or the second inside the biqubit;
5. Finally, they use an error correction algorithm and a privacy amplification method usually used in \(BB84\)-based protocols. Figure 3: We represent the quantum states as black dots in a simplified Bloch sphere over two dimensions. The quantum measurement bases \(X\) and \(Z\) are illustrated here as a horizontal and a vertical line, respectively. In this representation, two concentric circles define the order in which the states are prepared and sent. Accordingly, the inner circle state contains the state that is first sent, and the outer circle state is prepared and transmitted. As discussed in [19], the non-orthogonal states are useful to detect the Photon Number Splitting (\(PNS\)) attack; an example is shown in **(a)**. In the _nack state_ protocol, [PERSON] chooses randomly two consecutive parallel states as the case depicted in **(c)**\((\ket{1_{Z}},\ket{1_{Z}})\).