A MATLAB-Based Simulation of Quantum Key Distribution Protocols at Telecom Wavelengths Under Various Realistic Conditions

Abstract

We investigate the feasibility of single and entangled photon-based quantum key distribution protocols at telecommunication wavelengths with two types of single photon detectors, namely InGaAs/InP and Silicon-APD, under various realistic conditions. The purpose of the current optical fiber-based simulation is to analyze the various performance parameters. In addition to these, we analyze the effect of possible attacks on the one and two weak decoy state protocols under investigation with the two deployed avalanche photodiodes. The simulation results obtained show that the one and two weak decoy states used in the entangled-based protocol at telecommunication wavelengths with considered attacks and under various industrial parameters outperforms the single photon-based quantum key distribution protocol. In addition, it is also observed that Silicon-APD (the avalanche photodiode) performs better than InGaAs/InP-APD when considering all the conditions.

1. Introduction

At present, quantum technologies are gaining momentum in multidisciplinary fields and are one of the secure technologies based on the unconditional security of quantum postulates, which are different from their classical counterpart, where security is measured in terms of computational complexity. Various security tests have been performed by researchers under different realistic conditions, and it has been further concluded that the performance of quantum cryptography systems also depends on the source characteristics, e.g., single photon sources or entangled [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. In 1992, the first implementation of QKD was proposed, and with time further improvements were accomplished in [14,15,16,17,18,19]. In the current industrial revolution, quantum technologies are being deployed in a number of applications [20,21,22]. Currently, specialty optical fibers are available in the market, which offers very low losses in the range from 0.15 dB/km to 0.17 dB/km at the third telecom window, i.e., 1550 nm [23,24]. This 1550 nm wavelength is the most suitable telecom wavelength as compared to 1310 nm, the reason being that is has less attenuation losses as compared to 1310 nm for real field practical QKD applications. Here, we deployed the frequency up-conversion method [25] to ensure the optimum detection of single photons at 1550 nm. As per the literature, Silicon-APD is the best candidate for such applications at telecom wavelengths, and it offers many significant advantages [18,26,27,28,29,30]. As technological advancements and improvements in optical fiber manufacturing have taken place, specialty fibers (ultra-low loss fibers [23,24,31,32,33,34,35]) are investigated in the current research work. We analyzed the relation between the produced error rate and communication rate with distance for the QKD protocols (Bennett–Brassard 1984 (BB84) and Bennett–Brassard–Mermin 1992 (BBM92)). Further, we investigated the security of BBM92 and BB84 QKD protocols under various attacks.

2. Detectors in Telecom Range

2.1. Single Photon Detectors

The effective approach for detecting single photons at the telecom wavelength of 1550 nm is frequency up-conversion [36]. Conventional InGaAs/InP avalanche photodiodes (APDs), while widely used, exhibit relatively low quantum efficiency and suffer from afterpulsing effects caused by trapped charge carriers. These trapped carriers can be released during subsequent detection windows, resulting in elevated dark count rates that degrade detector performance.

To mitigate this, APDs are often operated in gated mode, where the bias voltage is briefly raised above the breakdown threshold for a few nanoseconds to allow photon detection and is then lowered again to let trapped carriers dissipate. For trapping lifetimes at the order of microseconds, this gating scheme can be applied at megahertz frequencies, with the afterpulse probability reduced by the ratio of the gate width to the interval between gates.

The gate frequency is a critical parameter in many QKD systems as it directly determines the signal pulse repetition rate and, consequently, the achievable secure communication rate. In practice, the dark count rate is strongly influenced by the chosen gate width, which is ultimately limited by the intrinsic response time of the semiconductor material.

In Figure 1 and Figure 2, InGaAs/InP as an avalanche photodiode was deployed to test its performance in many optical fiber-based quantum key distribution applications where its experimental feasibility was investigated to perform as a single photon detector [37,38,39,40,41]. During these experiments, it was observed that this single photon detector has many limitations and performs with poor efficiency due to the existing problems such as severe afterpulse probabilities [42], high background count rates, low value of quantum efficiency, high value of dark count rate. Hence, it is not the right candidate to perform well in gated-mode operation due to the generated trapped charge carriers. It is specifically desired to operate single photon detectors above the breakdown threshold in gated mode for a short duration so that high photon detection events can be achieved where less dark counts exist. Also, in a short span of time, it regains its original state and attains state below breakdown for enough time for trapped charge carriers to leak away. This gated mode operation where single photon detectors work at mega-hertz rates and achievable trapping lifetime exists in microsecond duration. Hence, it is afterpulse probability which is reduced at this point by the amount of gate width to the time separation between gates. It is the gate frequency which decides the pulse repetition rate and affects the final communication rate in most of the quantum key distribution (QKD) protocols. The generation rate of dark counts depends on the semiconductor material’s response time, which further decides the QKD communication distance, and moreover, these are affected by gate width. The value of dark counts in the order of ${10}^{-5}$/gate is achieved with 1–2 ns gate width at the value of ∼1 MHz pulse repetition frequency.

2.2. Photon Detection Using Frequency Up-Conversion

In the sum frequency generation method [25], periodically poled lithium niobate (PPLN) is used; a single photon at 1550 nm interacts with a strong pump at 1320 nm. This is the method which is deployed in 1550 nm up-conversion for single photon detection [28]. A signal of very high conversion efficiency is converted to an output of 700 nm sum-frequency with the help of PPLN waveguide. Due to the presence of guided wave structure in PPLN, this conversion becomes possible, where longer interaction length, tight mode confinement and quasi-phase-matching patterns perform this mechanism, as shown in Figure 1 and Figure 4. Hence, the converted photons are detected by silicon APD using these operations. In most of the QKD industrial applications, out of these two detectors, Si-APD is one of the preferred detectors. It has some important merits over InGaAs/InP APDs, such as a timing resolution as low as 40 ns [30], low dead time (45 ns), low value of dark-count rates, low value of after-pulse at near infrared regions [18]. These characteristics of Si-APD make it a suitable candidate over InGaAs/InP APDs for real field quantum key distribution with high stability count rate of 20 MHz [18,27,30] and better timing accuracy [18,19,45]. In real-field QKD applications, high communication rate is achieved with the use of Si-APD in Geiger mode, which has low afterpulse probability. This Geiger mode is also known as nongated mode of operation of Si-APD. One added advantage of low dead time (45 ns) of Si-APD helps in achieving improved secure key generation rate. During this period, a larger photon flux saturates the set-up; no response is obtained from the photodiode from successive events. As per the standard theory of up-conversion process, the dark count rate $D_{up}$ and quantum efficiency ${\eta }_{up}$ depend on the parameter p (pump power) [28]. When phase-matching and sufficient pump power are available in the said waveguide, maximum (100%) photon conversion is achieved, which results in high quantum efficiency of Si-APD, i.e., 0.46. The fitting curve is drawn with the help of the following equations, based on three-wave interactions coupled mode theory:

$${\eta }_{up}\left(p\right)=a_1{\sin }^2\left(\sqrt{a_{2}p}\right),$$

(1)

where p is in mW, and the values of $a_1$ and $a_2$ are 0.465 and 79.75, respectively.

Here we discuss how the dark counts are controlled by a nonlinear process. First of all, the fiber and phonons of the PPLN disperse the pump photons using spontaneous Raman scattering. Hence, photon generation is achieved at 1550 nm signal wavelength with proper setting of pump power. Further, dark counts are produced due to a combination of pump photons and noise photons in the waveguide through phase-matched sum-frequency generation methods. From Figure 3, we can observe the quadratic dependency of dark counts and pump power. The following equation produces an exact polynomial fitting curve:

$$D_{up}\left(p\right)=b_0+b_{1}p+b_2{p}^2+b_3{p}^3+b_4{p}^4,\left(s^{-1}\right),$$

(2)

where the values of b0, b1, b2, b3, and b4 are 50, 826.4, 110.3, −0.403, and 0.00065, respectively.

The other sources responsible for producing dark counts are up-converted noise signal photons and parametric fluorescence processes [23,46]. According to the described process and spontaneous Raman scattering, extra dark counts are generated. The reason for such an effect in the frequency conversion detector is the absorption of 8.9 μm idler photons, which are absorbed in lithium niobate. Hence, in such fluorescence processes the produced dark counts further limit the QKD performance parameter. To overcome the effects of these unwanted dark counts, the signal wavelengths and pump [39] must be interchanged. All these operations happen within the waveguide and thermal processes of excited vibrational states generate anti-Stokes scattering gain [39].

It is the waveguide bandwidth which has to be tuned properly to adjust both the generated noise photons and the number of dark counts. For a detector, the term $D_{\mathit{up\; Hz}}$ is written as $D_{\mathit{up\; Hz}}$ $=D_{up}/B_d{s}^{-1}H{z}^{-1}$, where $B_d$ is the dark count per mode. From the principle of communication systems, B is the bit rate defined for an ideal communication system, having bandwidth B with a matched filter, where the term $1/B$ is defined for an ideal communication system. The term $d_{up}$ is the dark count per time window, which is equal to $D_{up}$ Hz and is one of the deciding factors of QKD performance. Here the dark count per time window does not depend on the bit rate B which is the optimum filtering case. In the gated mode of InGaAs/ InP APD, the term $D_{APD}\left(s^{-1}\right)$ is the dark-count rate of the InGaAs/ InP APD, $1/B$ is gate width, $d_{APD}$ is the dark counts per gate calculated by the expression $D_{APD}/B$. All the required parameters like dark-counts are mentioned in

Table 1. Parameters for dark counts.

	$\mathit{InGaAs}/\mathit{InP}$ APD	Up-Converter
Dark count rate ($s^{-1}$)	$D_{APD}$	$D_{up}$
Dark counts per mode $\left(s^{-1}H{z}^{-1}\right)$	-	$D_{up}Hz={\displaystyle \frac{D_{up}}{B_d}}$
Dark counts per time window/gate	$d_{APD}=D_{APD}{\displaystyle \frac{1}{B}}$	$d_{up}=D_{up}Hz$

. The bit rate is B and $B_d$ is the waveguide bandwidth. The value of $D_{APD}={10}^4,s^{-1}$ is deployed in InGaAs/InP APD.

The expression for the normalized noise equivalent power (NEP) is written as ${\displaystyle \frac{\sqrt{2D_{up}}}{{\eta }_{up}}}$, where $D_{up}=6.4\times {10}^3{s}^{-1}$ and ${\eta }_{up}=0.075$. All these mathematical expressions are used in the up-conversion process. The term $D_{up}$ Hz is calculated based on the detector’s operating point. In the same context, for an up-converter detector, and for the value of bandwidth $B_d=50$ GHz, the optimum value of $d_{up}$ is $1.3\times {10}^{-7}$. All these are the important exercises prior to performing any implementation of QKD in a real field scenario.

The current research work deals with the effects of the waveguide bandwidth on the produced dark counts in Si-APD and also pump power, which affects its characteristics in frequency up-converted quantum communication systems at telecommunication wavelength and decides the detection performance.

3. BB84 Quantum Key Distribution Protocol

The randomly modulated single photons in two non-orthogonal bases are transmitted from Alice to Bob in a BB84 (Bennett–Brassard 1984) protocol. The received photon polarization states, at Bob’s end, are measured in a random polarization basis. Further, it is required to analyze the effects of the hybrid attacks such as intercept-resend, individual attack and photon-number splitting (PNS) attacks [3]. Under these attacks the expression for the secure communication rate is written as

$$R_{BB84}={\displaystyle \frac{\nu p_{click}}{2}}\{\tau (e,\beta )+f\left(e\right)[elo{g}_{2}e+(1-e)lo{g}_2(1-e)]\},$$

(3)

Here, the sifting parameter is ${\displaystyle \frac{1}{2}}$ and the term $\nu$ is defined as the repetition rate of the transmission.

Now, for the analysis of security of BB84 protocol, we consider hybrid attacks, e.g., intercept-resend attack and beam-splitter attack.

To calculate the key generation rate, we need to compute the privacy amplification shrinking factor ($\tau$), with respect to average collision probability ($p_c$).

The transmitted photons after reaching Bob’s side are detected with detection probability expressed as

$$p_{click}=p_{signal}+p_{dark},$$

(4)

where

$$p_{signal}=\mu \eta {10}^{-(\alpha L+L_r)/10},$$

(5)

$$p_{dark}=4d,$$

(6)

where the mean photon number per pulse is denoted by $\mu$, optical fiber loss coefficient is written as $\alpha$ with unit dB/km, detector quantum efficiency is written as $\eta$, losses in the receiver unit are represented by $L_r$, communication link by L with unit in km, and dark counts per measurement time window by d. Here four detectors are deployed in the detection unit of Bob. The mean photon number, $\mu$, is the most contributing parameter that has to be optimized, and therefore, for an ideal case $\mu =1$, but in the case of a Poisson photon source, this parameter has to be tuned to obtain the optimum results [47].

$$e={\displaystyle \frac{(\frac{1}{2}{p}_{dark}+b{p}_{signal})}{p_{click}}},$$

(7)

where baseline system error rates and error rate are represented by b and e, respectively.

Table 2. Error-correction algorithm given in [48].

e	$\mathit{f}\left(\mathit{e}\right)$
0.01	1.16
0.05	1.16
0.1	1.22
0.15	1.35

[48] mentions values of $f\left(e\right)$, which are based on an error-correction algorithm.

The main shrinking factor $\tau (e,\beta )$ in privacy amplification is written as

$$\tau =-lo{g}_2{p}_c$$

(8)

where the average collision probability is represented by $p_c$ and is used to account for the amount of Eve’s mutual information shared between Bob and Alice.

Further, $\tau$ is expressed as

$$\tau (e,\beta )=-\beta lo{g}_2\left[{\displaystyle \frac{1}{2}}+2\left({\displaystyle \frac{e}{\beta }}\right)-2{\left({\displaystyle \frac{e}{\beta }}\right)}^2\right]$$

(9)

The fraction of single-photon states emitted from the source can be written as

$$\beta ={\displaystyle \frac{p_{click}-p_m}{p_{click}}},$$

(10)

where the probability of the multi-photon quantum states are denoted by the term $p_m$. The values of $\beta =1$ or $p_m$ = 0 are written for an ideal single photon source. In addition, photon emission probability in a Poisson source, $p_m$ is written as

$$p_m=1-(1+\mu )e^{-\mu }$$

(11)

Here the photon number splitting (PNS) attack is performed by Eve to extract the information and computed by the term $\beta$. To hide her presence, Eve performs quantum nondemolition measurements of the photon number in each pulse without any errors. As the source emits more than one photon, due to these multiple photons, Eve copies one photon in a quantum memory, and after Bob performs a basis announcement, Eve performs a delayed quantum measurement on the photon. Here in the BB84 protocol, the PNS attack limits the performance of the laser source. The expression ${10}^{{\displaystyle \frac{-\alpha L}{10}}}$ is used to denote the loss in the secure communication rate with the distance, for low error rate and $p_{dark}\ll p_{signal}\ll 1$. In a different scenario, under similar constraints, when the used source is an ideal single-photon source, it is analyzed that $R_{BB84}\approx {\displaystyle \frac{1}{2}}\nu p_{signal}$, i.e., the secure key rate starts decreasing only linearly with the increasing fiber distance.

Assume that Eve has a quantum memory with infinitely long coherence time which is used for storing the intermediate information (basis measurements) transferred between Alice and Bob. In the case when Eve does not possess a quantum memory, she has to apply a polarization measurement with arbitrary random basis selection. Under such a situation, Equation (9) can be written as

$$\tau (e,\beta )=-{\displaystyle \frac{1+\beta }{2}}lo{g}_2\left[{\displaystyle \frac{1}{2}}+4\left({\displaystyle \frac{e}{1+\beta }}\right)-8{\left({\displaystyle \frac{e}{1+\beta }}\right)}^2\right]$$

(12)

There are other alternatives to make BB84 more robust against PNS attack, such as deploying a sifting procedure [49] or introducing decoy states [50,51,52]. A Poisson source is deployed in the BB84 protocol to compute the secure communication distance achieved. In addition to this, vacuum plus weak decoy state protocol [51] will be used in the current research work.

4. BBM92 Quantum Key Distribution Protocol

The two-photon variant of BB84 is the Bennett–Brassard-Mermin 1992 (BBM92) protocol. Alice and Bob each provide a photon of an entangled photon pair, as shown in Figure 2, for which they perform the polarization state measurement in a randomly selected basis out of the two considered non-orthogonal bases. In [4], it was mentioned that the BB84 with a single-photon source, i.e., with $\beta =1$, and BBM92 both have the same average collision probability $p_c$. Now, $\tau$, the shrinking factor, is written as

$$\tau \left(e\right)=-lo{g}_2\left[{\displaystyle \frac{1}{2}}+2e-2e^2\right]$$

(13)

The above relation highlights how there is no analogy to a photon-number splitting attack in BBM92. As compared to BB84, the BBM92 protocol is more robust against the attacks, and less prone to errors generated by the dark counts, as highlighted by the simulated results in Figures 14–23. One dark count cannot contribute to the errors in the BBM92 protocol. Further, the secure communication rate equation under the effect of an individual attack is expressed as [4]:

$$R_{BBM92}={\displaystyle \frac{\nu p_{coin}}{2}}\{\tau \left(e\right)+f\left(e\right)[elo{g}_{2}e+(1-e)lo{g}_2(1-e)]\}$$

(14)

Now the expression for the probability of a coincidence between Bob and Alice is written as

$$p_{coin}=p_{true}+p_{false}$$

(15)

Assume that the photon source is located in between the two authentic communicating parties [4]. Further, the expressions $p_{false}$ and $p_{true}$ represent false coincidence probability and true coincidence probability, respectively, and these are different expressions for a Poissonian entangled-photon source and a deterministic entangled-photon source.

(1) Expression for $p_{true}$ for a deterministic entangled-photon source:

$$p_{true}={\eta }^2{10}^{-(\alpha L+2L_r)/10}$$

(16)

$$p_{false}=8d\eta {10}^{-(\alpha L+2L_r)/20}+16d^2$$

(17)

(2) Expression for $p_{false}$ for a Poissonian entangled-photon source:

$$p_{true}=c_1,$$

(18)

$$p_{false}=16d^2{c}_2+8d{c}_3+c_4,$$

(19)

$$c_1={\displaystyle \frac{1}{cos{h}^4\chi }}{\displaystyle \frac{2t_L^{2}tan{h}^2\chi }{{[1-tan{h}^2\chi {(1-t_L)}^2]}^4}}$$

(20)

$$c_2={\displaystyle \frac{1}{cos{h}^4\chi }}{\displaystyle \frac{1}{{[1-tan{h}^2\chi {(1-t_L)}^2]}^2}}$$

(21)

$$c_3={\displaystyle \frac{1}{cos{h}^4\chi }}{\displaystyle \frac{2t_L(1-t_L)tan{h}^2\chi }{{[1-tan{h}^2\chi {(1-t_L)}^2]}^3}}$$

(22)

$$c_4={\displaystyle \frac{1}{cos{h}^4\chi }}{\displaystyle \frac{4t_L^2{(1-t_L)}^{2}tan{h}^4\chi }{{[1-tan{h}^2\chi {(1-t_L)}^2]}^4}}$$

(23)

and

$$t_L=\eta {10}^{-(\alpha L+2L_r)/20}$$

(24)

The description of the said parameters has been mentioned in the previous section. The term $\chi$ depends on the average photon-pair number per pulse, i.e., the nonlinear coefficient, the interaction time of the down-conversion process and the pump energy. In a Poissonian entangled-photon source, the said term is a free variable which needs to be optimized. Further, the error rate expression can be written as follows:

$$e={\displaystyle \frac{\frac{1}{2}{p}_{false}+b{p}_{true}}{p_{coin}}}$$

(25)

In a single photon source based on BB84 and BBM92 protocols with the conditions $p_{false}\ll p_{true}$ and small error rates, the secure key generation rate decreases linearly with the optical fiber transmission used as a quantum channel. For a BBM92 QKD protocol, no quantum memory is required to attack, and hence, Equation (14) is alone computed by IR (intercept and resend) attack.

The probability to receive n photon pairs in a particular pump pulse is expressed as

$$P_n={\displaystyle \frac{(n+1){\gamma }^n}{{(1+\gamma )}^{n+2}}},$$

where $\gamma =sin{h}^2\left(\chi \right)$ belongs to pump power of the laser used. $2\gamma =\mu$ is the mean number of pairs per pump pulse

$$\gamma =sin{h}^2\left(\chi \right),$$

$${\displaystyle \frac{\mu }{2}}=sin{h}^2\left(\chi \right).$$

The parameter $\chi$ depends on the average photon-pair number per pulse, i.e., the nonlinear coefficient, the pump energy, and the interaction time of the down-conversion process. There are some important considerations which affect the performance parameters. The measurement time window is affected by the timing jitter of the detectors. The dark counts are generated from either the thermal generation of a carrier in the sensitive part or the release of a carrier trapped by defects in the junction in the course of a previous avalanche. The latter type of dark count is known as an afterpulse. In addition to this, to reduce the effects of afterpulse probability, self-differencing circuits are used which are made of coaxial cables of precise length, so that they match with the laser clock frequency. Afterpulse probability is an important factor to consider while improving the performance of any quantum communication system. Afterpulse probability is defined as the ratio of total afterpulse counts to the photon counts. To overcome the effects of an afterpulse in gated mode operation, set the interval of the time windows longer than the lifetime of the trapped carrier. To reduce the dark counts in the gated mode operation, which was generated due to afterpulse effects, and to eliminate those dark counts, we need to set the gate-off time longer than the lifetime of trapped carriers. The said lifetime is related to the gate-off time beyond which the dark count probability does not vary. It is also essential to set the gate repetition frequency below 1 MHz, so that the afterpulse effects can be eliminated. Here we observe from the simulation results shown in the plots that the entanglement-based BBM92 protocol performs better than the single photon-based BB84 protocol, as highlighted by the simulated results in Figures 14–23. The reason is that the BBM92 is less vulnerable to the errors generated from dark counts, because a dark count of only one cannot contribute to errors in the said BBM92 protocol. To overcome the effects of the afterpulse, we can employ long dead time. Also, the effect of saturation present in Si-APD is related to its dead time. This effect is apparent even for small losses in the fiber and also a concern for high bit rates.

4.1. Beam-Splitter Attack

In the multi-photon transmission process from Alice to Bob, Eve takes advantage of these multi-photons by intercepting and further making a copy of these coherent quantum states. In this process, a beam splitter with transmission ${\eta }_{BS}$ is used by Eavesdropper. In addition to this, in place of lossy fiber, Eve deploys a lossless fiber. To hide her Eavesdropping attempts, at Bob’s end, she also replaces inefficient detectors by the ideal detectors. Bob’s signal photon detection probability is represented by $p_{signal}$, which is the same as written in Equation (5). At this stage, Eve makes efforts just to hide her presence by altering the beam-splitter transmission ${\eta }_{BS}$ to

$${\eta }_{BS}=\eta {10}^{-(\alpha L+L_r)/10},$$

(26)

To extract the meaningful information shared by Bob, Eve can use an interferometer of delay time $M\tau$. To compute the amount of measured information, we have to find out some important parameters. The detection probabilities in a given time frame are $\mu {\eta }_{BS}$ and $\mu (1-{\eta }_{BS})$, which are the values at Eve’s and Bob’s ends, respectively. At the same time, the expression for the detection probability can be written as ${\mu }^2{\eta }_{BS}(1-{\eta }_{BS})$. As per the conditional probability, at a particular time, the probability value of the obtained information bits by an Eavesdropper when Bob has already detected that photon during that time frame can be expressed as ${\mu }^2{\eta }_{BS}(1-{\eta }_{BS})/\mu {\eta }_{BS}=\mu (1-{\eta }_{BS})$. Here the probability value achieved by Eve is $\mu (1-{\eta }_{BS})/N$. This is the received probability by Eve in case she does not have a quantum memory with infinitely long coherence time. On the other side, the two authentic communicating parties, Alice and Bob, can disclose their outcomes with random delay; hence, with a quantum memory with a long coherence time, Eve can steal the information. At this stage, instead of a beam splitter, an optical switch with an interferometer is deployed by Eve to extract the pulses for which differential phase information was already received by Bob. By introducing this strategy, Eve achieves enough information, which is equal to $2\mu (1-{\eta }_{BS})$. Following this strategy Eve obtains an amount of information which is equal to $p_c=1$, which is further equal to $2\mu (1-{\eta }_{BS})$ or $\mu (1-{\eta }_{BS})/N$. No errors are introduced in the Beam-Splitter attack and remaining bit fractions can be expressed as follows:

(i) In case there is no quantum memory:

$${\gamma }_1=1-\mu {\displaystyle \frac{(1-{\eta }_{BS})}{N}}=1-{\displaystyle \frac{\mu }{N}}+{\displaystyle \frac{p_{signal}}{N}},$$

(27)

(ii) When there is a quantum memory:

$${\gamma }_2=1-2\mu (1-{\eta }_{BS})=1-2\mu +2p_{signal},$$

(28)

4.2. Intercept-Resend Attack

In an another Eavesdropping strategy, the information carrying photons from Alice to Bob is attacked by the Eavesdropper, known as an intercept-resend attack.

Under intercept-resend and beam-splitter attacks, Eavesdropper is not familiar with $p_c={\displaystyle \frac{1}{2}}$ number of bits; further, it is equal to $\gamma -{\displaystyle \frac{e}{N(1-1/2N)}}$. Here, the expression for the privacy amplification shrinking factor is written as

$$\tau (e,\gamma )=\gamma -{\displaystyle \frac{e}{N(1-1/2N)}}$$

(29)

The expression for the BB84-QKD secure key generation rate under intercept-resend and beam-splitter attacks is written as

$$R_{BB84}={\displaystyle \frac{\nu p_{click}}{2}}\{\tau (e,\beta )+f\left(e\right)[elo{g}_{2}e+(1-e)lo{g}_2(1-e)]\}$$

(30)

The term $\nu$ is referred to as the transmission repetition rate. The expression for dark count probability $p_{dark}$ is written as

$$p_{dark}=4d$$

(31)

5. Results and Discussion

In the current research, we introduced a frequency up-conversion process and analyzed two QKD protocols, namely BBM92 and BB84. The description of the two single photon detectors with the detailed simulation parameters is already mentioned in the previous section. In the current simulation at 1550 nm, we have used 0.17dB/km ((ultra low loss fibers [23]) and 0.2 dB/km) as an attenuation coefficient. The other values deployed in the simulation are baseline system error rate, $b=0.01$, and the extra loss at the receiver end is $L_r$ = 1 dB. The remaining values used in the simulations have been written in the figure caption. In the weak-laser-pulse of BB84 QKD, the mean photon number $\mu$ affects the secure key generation rate and is one of the important parameter to be optimized; too low of a value of $\mu$ leads to dark counts and a higher value gives rise to a PNS attack. The parameter $\chi$ is another important term to be tuned to achieve improved secure key rate in the BBM92 protocol with a Poissonian entangled-photon source.

Figure 1 and Figure 3 represent the related experiment setup for the current research work, and the detailed description is mentioned in [28,43]. The simulation results are highlighted in the figures (Figure 2, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21) and produced from Equations (1)–(31). Inspecting all these, it is observed that the shown results for the considered QKD protocols under the two types of detectors described earlier depend on various parameters, such as dark counts, d, detector quantum efficiency, $\eta$, afterpulse probability, and transmission repetition rate, $\nu$. Here, out of the two considered detectors, Si-APD with non-gated mode operation and improved timing jitter values provides better results at 1 GHz and 10 GHz, as shown in the simulated results. The considered parameter, dead time, $t_d$, is one of the major obstacles in achieving desired optimum results. In the considered Poisson photon source, the term $e^{-\delta \nu p_{click}{t}_d}$ represents the probability value of the two events occurring in increased timing, $t_d$, where the deployed number of detectors decides the value of $\delta$. The dead time, $t_d$, for the used Si-APD detector is 45 ns. The variations in achieved secure communication distance are properly simulated with the variations in pump power, p. This is achieved by the proper curve fitting method and the used parameter values are shown in the figure captions. From Figure 2, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21, it is observed that the improved results are achieved for secure key generation rates ranging from ${10}^4$ bits/s to ${10}^9$ bits/s, in the range of 100 km to 600 km communication distance. All these improved results satisfy the acceptable practical quantum bit error rates as shown in Figure 20 and Figure 21. The upper bound of the error rate for secure key generation is computed to be 11.4% if the ideal value of the efficiency of the error correction algorithm $f\left(e\right)=1$. In addition to these, from Figure 22 and Figure 23, it is observed that using Si-APD, BBM92 QKD protocol with deterministic entangled photon source (Figure 22) attains more than 500 km communication distance as compared to a Poissonian entangled photon source (Figure 23), within the acceptable quantum bit error rate (QBER). Hence, under the considered scenario, we achieve improved results for the BBM92 QKD protocol, as highlighted by the simulated results in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23.

Analyzing all the results, we claim that the BBM92 QKD protocol under PNS attack outperforms the BB84 protocol even if Eve has a quantum memory, the reason being that it is independent of the value of N, which is nothing but a delay term. In addition to this, from simulation results, in the absence of the quantum memory with an infinitely long coherence time, the communication distance achieved and the secure key generation rate are much affected. Finally, we can claim that the two QKD protocols under investigation with Si-APD under frequency up-conversion show their practical feasibility under the mentioned realistic conditions, where Si-APD provides improved results as compared to the InGaAs/InP APD. Also, under all such practical and realistic scenarios, the entangled-based BBM92 QKD protocol outperforms the BB84 QKD protocol, as highlighted by the simulated results in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23.

The source of the improved results obtained includes operation in the said frequency range where Si-APDs offer lower noise and higher efficiency, reduced afterpulsing and better timing resolution, and up-conversion enables the use of these advantages at telecom wavelength. To validate the effectiveness of the up-conversion detection technique employed in this work, we conduct a comparative simulation study using conventional InGaAs/InP-APDs operated directly at 1550 nm, without frequency conversion. These detectors, although widely used, are known to suffer from higher dark count rates, limited detection efficiency, and elevated afterpulse probabilities, particularly in gated mode operations.

In contrast, the up-conversion scheme shifts the incoming telecom-band photons to the visible spectrum using a PPLN waveguide, thereby enabling the use of high-performance Si-APDs. These detectors operate in non-gated mode, offer significantly lower noise, superior timing resolution, and improved quantum efficiency.

The simulation results highlighted in the said figures clearly indicate that up-conversion detection achieves lower QBER, higher secure key rates, and longer communication distances. For example, at 10 GHz repetition rate, the Si-APD-based system achieves a secure distance improvement of up to a significant communication distance compared to InGaAs-based systems. This improvement is directly attributable to the enhanced detector characteristics enabled by frequency up-conversion.

Our revised study includes a direct comparison between up-conversion-based detection and standard InGaAs/InP-APD-based detection at 1550 nm. The observed improvements in secure key rate and communication distance confirm that frequency up-conversion provides a substantial advantage. These improvements are primarily due to reduced noise, higher detector efficiency, and lower timing jitter of visible-wavelength Si-APDs, which become accessible via the up-conversion technique. This comparative analysis rigorously substantiates the claimed benefits.

For baseline comparison, we reference prior work employing direct InGaAs/InP-APD detection at 1550 nm. Yuan et al. [54] demonstrated GHz rate BB84 with key rates up to ≈2.37 Mb/s at 5.6 km and ≈2.9 kb/s at 101 km, using detectors with ≈${10}^{-6}-{10}^{-5}$ dark counts per gate and ≈10% efficiency. Stucki et al. [39] reported error rates of ≈10% at ≈54 km with a dark count of ≈2.8 × ${10}^{-5}$ per gate at 10% efficiency. Namekata et al. [55] achieved 21.3% detection efficiency and ≈2.1 × ${10}^{-6}$ dark counts with ≈ 3.7% afterpulse probability over ≈73 km fiber. Comandar et al. [36] demonstrated room-temperature telecom APDs yielding ≈1.26 Mb/s secure key rate over 50 km despite ≈ 6% afterpulse noise. Comparing all these with our simulated results, we claim that Si-APD with the frequency up-conversion method provides improved results, as shown in the simulated figures. These benchmarks define the performance limits of conventional detector approaches, against which our up-conversion with Si-APD system demonstrates quantifiable gains in key rate, QBER reduction, and transmission distance under comparable parameters.

To rigorously evaluate the performance benefits of the proposed up-conversion detection scheme, we conducted a comparative simulation study using conventional InGaAs/InP avalanche photodiodes (APDs) operating directly at 1550 nm, and our up-conversion with Si-APD detection setup. Identical system parameters—fiber loss, repetition rate, detector gate width, and protocol configuration—were used to isolate the impact of detection technology.

The results demonstrate a clear advantage of the up-conversion scheme in terms of quantum bit error rate (QBER), secure key rate, and achievable communication distance. Specifically, from the simulation results shown, the secure key rate for the up-conversion detector is much higher than that of the conventional InGaAs setup, primarily due to improved detection efficiency and significantly reduced dark counts. Furthermore, due to reduced afterpulsing and improved timing jitter, the up-conversion system supports higher repetition rates and more stable synchronization.

6. Conclusions

The two QKD protocols with the two considered APDs perform much better while deploying PPLN waveguide with the frequency conversion method, which is reflected in the simulated results. The improved results obtained provide enhanced secure communication distance and secure key generation rates with Si-APD, as highlighted in various plots, which proves its practical feasibility under the considered scenario. Superconducting single photon detectors have low dark count value but require a cryogenic environment to perform, as well as being costly to set up under realistic conditions, which makes them overall complex.

Here, we have simulated the two QKD protocols using the two single photon detectors at telecommunication wavelength. In addition to this, individual and hybrid attacks have been taken into account. The generated simulated results show that under the said attacks and considered simulation parameters with the frequency up-conversion method, the entanglement-based BBM92 QKD protocol outperforms the BB84 QKD protocol with the deployment of Si-APD as a single photon detector. In addition to this, to evaluate the performance of the quantum communication system, we tested the effects of quantum memory, which Eve uses to extract the information. Under all these scenarios and at high frequencies, i.e., at 1 GHz and 10 GHz, it attains a longer secure communication distance in the range of 150 km to 610 km with an improved higher secure key rate in the range of ${10}^3$ to ${10}^9$ bits/s. To compensate for other fiber losses such as chromatic dispersion and birefringence in optical fibers, it is highly required to use dispersion compensation techniques [56] and phase-encoding protocols [57].