4. The nack-state protocol

The nack-state protocol is the dual version of the ack state protocol discussed in [5]. Both protocols constitute a generalization of the well-known BB84. The nack state protocol uses couples of parallel and orthogonal states rather than just single non-orthogonal states utilized as a part of BB84. This straightforward distinction makes the nack state strong when facing the IRFS attack, as we will demonstrate later on. We selected the nack prefix to indicate that, provided Alice transmits two quantum states to Bob, the second measurement behaves as the negative acknowledgment (nack) of the one before, since it yields the opposite bit result.

The pair of quantum states is denoted as a biqubit. More specifically, the following biqubits are defined in the nack state protocol: four parallel biqubits ðj0Xi; j0XiÞ,ðj0Zi; j0ZiÞ,ðj1Xi; j1XiÞ,


2, depending on the state number that makes clicks after the basis measurement X or Z is applied to the two consecutive incoming states. This way, the number i will be published

Table 2. Alice sends to Bob the non-orthogonal states ðj0Xi; j0ZiÞ and it shows all the possible measurement results at

basis used Detection event Public disclosure Result

X ∣0Xi, ∣1Xi X,ð Þ 2nM Discard X ∣0Xi, � X, Sð Þ -1 Useful X �, ∣0Xi X, Sð Þ -2 Discard X �, ∣1Xi X, Sð Þ -2 Discard X �, � X,ð Þ 2L Discard Z ∣0Zi, ∣0Zi X,ð Þ 2M Useful Z ∣1Zi, ∣0Zi X,ð Þ 2nM Discard Z �, ∣0Zi Z, Sð Þ -2 Useful Z ∣0Zi, � Z, Sð Þ -1 Discard Z ∣1Zi, � Z, Sð Þ -1 Discard Z �, � Z,ð Þ 2L Discard

∣0Xi, ∣0Zi X ∣0Xi, ∣0Xi X,ð Þ 2M Useful

When applying the ack-state protocol, two consecutive non-orthogonal states are used by Alice and Bob to distill one secret bit. The basis measurement X or Z is declared publicly by Bob along with the sifting instances; he obtained 2ð Þ M , 2ð Þ M , ð Þ S-1 , ð Þ S-2 , and 2ð Þ L . Furthermore, the bits acquired from the single-detection events ð Þ S-1 and ð Þ S-2 are used by Alice to confirm

The nack-state protocol is the dual version of the ack state protocol discussed in [5]. Both protocols constitute a generalization of the well-known BB84. The nack state protocol uses couples of parallel and orthogonal states rather than just single non-orthogonal states utilized as a part of BB84. This straightforward distinction makes the nack state strong when facing the IRFS attack, as we will demonstrate later on. We selected the nack prefix to indicate that, provided Alice transmits two quantum states to Bob, the second measurement behaves as the negative acknowledgment (nack) of the one before, since it yields the opposite bit result.

The pair of quantum states is denoted as a biqubit. More specifically, the following biqubits are defined in the nack state protocol: four parallel biqubits ðj0Xi; j0XiÞ,ðj0Zi; j0ZiÞ,ðj1Xi; j1XiÞ,

iii. The two pulses are lost. This case is denoted as ð Þ �; � or alternatively as 2L.

the single photonic gain of the quantum channel.

4. The nack-state protocol

by Bob.

Bob's side.

Alice's bi-qubit Bob's side

46 Advanced Technologies of Quantum Key Distribution

ðj1Zi; j1ZiÞ and two orthogonal biqubits ðj0Xi; j1XiÞ,ðj0Zi; j1ZiÞ. The parallel and orthogonal biqubits are interleaved at random by Alice. The performance of the protocol is not altered by order of the quantum states within the biqubit (see Figure 5). On the opposite side of the

We expect Alice to send the biqubits ∣0Xi, ∣1Xi and ∣1Zi, ∣1Zi; at that point, every conceivable measurement result at Bob's detector is written. We exhibit the detection event and Bob's corresponding advertisement over the public channel according to Bob's basis selection. Notice that the number of the single detections inside the biqubit, first or second, is openly declared by Bob.

Table 3. The nack-state protocol running without blunders in the quantum channel is shown with each of the possible measurement results at Bob's detectors.

quantum channel, Bob measures two incoming states of a biqubit utilizing the same measurement basis (X or Z). The following steps depict the nack state protocol:

same measurements on her stored states. In Bob's side, a distribution over the following sifting events is achieved 2ð Þ M , 2ð Þ nM , ð Þ S-1 , ð Þ S-2 and 2ð Þ L , where every one may originate in parallel

After Bob declares both the measurement bases (X or Z) and the sifting occurrences, Eve executes the measurements utilizing the same measurement bases and she gets the same bits from the multi-photonic single sifting instances: ð Þ S-1 and ð Þ S-2 , parallel and non-orthogonal. Moreover, the same outcomes from the 2ð Þ M measurements of the parallel and (a half of the) non-orthogonal multi-photonic states are acquired by the eavesdropper. However, she cannot acquire the secret bits from the 1-state ð Þ S-i and 2ð Þ M sifting occurrences, given that the

In order to get the secret bits, Eve obstructs the 1-photon states which incorporate single and double-detection events from parallel and non-orthogonal states. In doing that, an error gain in the photonic gain of the single and double-detection events is introduced by Eve. At that point, Eve executes a channel substitution expanding the transmittance of the channel. The fiber

coefficient measured in dB=km and the length l is measured in km. Moreover, the local transmittance at Bob's side, ηB, is defined as tBη<sup>D</sup> where tB is the internal transmittance of optical components and η<sup>D</sup> is the quantum efficiency of Bob's detectors. Then, the general transmission and detection efficiency at Bob's side ηBT is computed as ηBT ¼ tBηDTAB [18]. A mathematical description of the gain of detection events will be presented in the following section.

In Table 4 (upper part), the gain of the single-detection events is depicted with the Qð Þ <sup>þ</sup> symbol. According to Ma et al. [18], the gain of detection events is acquired from two origins: the photon source and the quantum channel. The photon source presents an expected photon number μ, and it adopts Poisson distribution. Contrastively, the quantum channel exhibits a distribution that is computed for every i photons' state (where i is the quantity of photons in each pulse) that is named yield. The gain Qi of i photons' state is the product of the probability of Alice sending an i photons' state (that adopts Poisson distribution) and the yield of i photons' state (and background states). It will generate a gain at Bob's side provoked by the

1. The fiber channel transmittance among Alice and Bob is denoted as TAB <sup>¼</sup> <sup>10</sup>�α<sup>l</sup>

the loss coefficient measured in dB/km, and the length l is measured in km. Moreover, the local transmittance at Bob's side, ηB, is written as tB � η<sup>D</sup> where tB is the internal transmittance of optical components and η<sup>D</sup> is the quantum efficiency of Bob's detectors. Then, the overall transmission and detection efficiency at Bob's side, ηBT, is computed as

μi

<sup>i</sup>! e�<sup>μ</sup> where Yi is the yield of i

<sup>10</sup> where α is

<sup>10</sup> where α is the loss

Quantum Flows for Secret Key Distribution http://dx.doi.org/10.5772/intechopen.75964 49

or non-orthogonal states; however, just Alice knows those outcomes.

eavesdropped cannot discriminate parallel and non-orthogonal states.

channel transmittance among Alice and Bob is written as TAB <sup>¼</sup> <sup>10</sup>�α<sup>l</sup>

detection of events corresponding to the relation Qi ¼ Yi

<sup>η</sup>BT <sup>¼</sup> tB � <sup>η</sup><sup>D</sup> � TAB and typically <sup>η</sup>BT ranges to 10�<sup>3</sup> [18];

The yield Yi is computed across the following steps:

5.1. The gain of detection events

photons' state.


Table 3 exhibits a case of the nack state protocol. Here, two biqubits are transmitted to Bob from Alice. The first biqubit is the orthogonal pair ðj0Xi; j1XiÞ, and the second biqubit is the parallel pair ðj1Zi; j1ZiÞ. In case the two states sent by Alice reach Bob's detection system with no failure, a double-detection event is generated. In the situation that just one of the two states of the biqubit reaches Bob's station, he gets a single-detection event.

The nack-state protocol has been conceived of to use the same optical hardware of the BB84 protocol; thus, it can be configured in most QKD systems as a software module application. However, two additional tasks must be implemented: the random computation of biqubits before preparing and sending the quantum states and the sifting stage of the protocol, which must include (1) sifting of single matching (compatible or non-compatible), where Bob announces the number of the single-detections inside the biqubit and (2) sifting of double detection, matching or non-matching, from parallel or orthogonal states. The error correction and privacy amplification stages of the QKD protocol do not require changes.
