A Novel Friendly Jamming Scheme in Industrial Crowdsensing Networks against Eavesdropping Attack

2.1. Network Model

In this paper, we consider a finite disk communication area with radius R, as shown in Figure 1. In this area, a number of legitimate transmitters are distributed according to Poisson point process (PPP) with density $\lambda$. In particular, each transmitter is assumed to follow uniformly independent identical distribution (i.i.d.). We consider a legitimate receiver, located in the center of this network. We assume there is an eavesdropper E with distance D away from the boundary of this communication area, trying to wiretap the confidential communications within the communication area. In order to protect the legitimate transmission, we place multiple friendly jammers at the circular boundary around the protected communication area.

We assume the channel experience Rayleigh fading and path loss. Therefore, the received power of a receiver with distance r from a transmitter is $h{r}^{-\alpha }$, where h is a random variable following an exponential distribution with mean 1 and $\alpha$ is the path loss factor.

2.2. Antennas

There are two types of antennas used in our network: an omni-directional antenna and a directional antenna. Omni-directional antennas radiate/collect radio signals into/from all directions equally. The antenna gain of omni-directional antenna is a constant in all directions, i.e., $G_o=1$. Different from an omni-directional antenna, a directional antenna can concentrate transmitting or receiving capability on some desired directions. Due to the high complexity to approximate a realistic directional antenna, we consider a simplified directional antenna model used in [24,25], as shown in Figure 2. This simplified directional antenna model consists of a main lobe $G_m$ within the beamwidth ${\theta }_m$ and a side lobe $G_s$ for all other directions. When $G_m$ and ${\theta }_m$ is given, $G_s$ can be calculated as follows [25],

In this paper, the receiver, the transmitters and the eavedropper are assumed to be equipped with omni-directional antennas. Then, with respect to jammers, we consider two jammer strategies in this network: (i) OFJ scheme, in which jammers are equipped with omni-directional antennas; (ii) DFJ scheme, in which jammers equipped with directional antennas. For comparison purposes, we also consider a scheme: NFJ scheme, in which no friendly jammers are deployed.

3.1. Impact of NFJ Scheme

In NFJ scheme, there is no friendly jammer at the boundary of the communication area. Therefore, the cumulative interference from friendly jammers $I_j=0$. Then we have ${\mathbb{P}}_T$ in the NFJ scheme as in the following theorem.

Theorem 1.

In the NFJ scheme, the probability of successful transmission is

$${\mathbb{P}}_T={\displaystyle \frac{2^M}{R^{2M}}{\int }_0^{R}exp(-T_p{r_0}^{\alpha }{\sigma }^2){\left({\int }_0^R\frac{r^{1+\alpha }}{r^{\alpha }+T{r_0}^{\alpha }}dr\right)}^{M-1}d{r}_0,}$$

(4)

where $T_p=T/P_t$, and M is the expectation of the number of legitimate transmitters in the communication area.

Proof. 

Let the distance between the receiver and a transmitter be r. Since the receiver is located at the center of the circular area and each transmitter follows uniformly i.i.d., the probability density function of r is as follows,

$$f_r\left(r\right)=\frac{2\pi r}{\pi R^2}=\frac{2r}{R^2},0<r\le R.$$

(5)

After combining Equations (5) and (2), we have ${\mathbb{P}}_T$ as follows,

$$\begin{array}{cc}{\mathbb{P}}_T& ={\int }_0^R\mathbb{P}\left[\frac{P_t{h}_0{r_0}^{-\alpha }}{{\sigma }^2+I_t}\ge T|r_0\right]f_r\left(r_0\right)d{r}_0\hfill \\ & ={\int }_0^R\mathbb{P}[h_0\ge T_p{r_0}^{\alpha }({\sigma }^2+I_t)|r_0]2r_0{R}^{-2}d{r}_0,\hfill \end{array}$$

(6)

where $T_p=T/P_t$.

Since h is a random variable following an exponential distribution with mean 1, $\mathbb{P}[h_0\ge T_p{r}_0^{\alpha }({\sigma }^2+I_t)|r_0]$ in Equation (6) can be expressed as

$$\begin{array}{cc}\hfill \mathbb{P}& [h_0\ge T_p{r}_0^{\alpha }({\sigma }^2+I_t)|r_0]\hfill \\ & ={\mathbb{E}}_{I_t}[exp\left((-T_p{r_0}^{\alpha })({\sigma }^2+I_t)\right)|r_0]\hfill \\ & =e^{-T_p{r_0}^{\alpha }{\sigma }^2}{\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)].\hfill \end{array}$$

(7)

Next, we calculate ${\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)]$. If we denote the expected number of legitimate transmitters by M, we can derive the expression of ${\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)]$ as follows,

$$\begin{array}{cc}\hfill {\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)]=& {\mathbb{E}}_{\Phi ,\left\{h_i\right\}}[exp(-T_p{r_0}^{\alpha }\sum _{i\in \Phi /t_0}{P}_t{h}_i{r}_i^{-\alpha })]\hfill \\ \hfill =& {\mathbb{E}}_{\left\{r_i\right\},\left\{h_i\right\}}[exp(-T_p{P}_t{r_0}^{\alpha }\sum _{i=1}^{M-1}{h}_i{r}_i^{-\alpha })]\hfill \\ \hfill =& {\mathbb{E}}_{\left\{r_i\right\},\left\{h_i\right\}}[\prod _{i=1}^{M-1}exp(-T{r_0}^{\alpha }{h}_i{r}_i^{-\alpha })]\hfill \\ \hfill \stackrel{\left(a\right)}{=}& {\left[{\mathbb{E}}_{r,h}(exp(-T{r_0}^{\alpha }h{r}^{-\alpha }))\right]}^{M-1}\hfill \\ \hfill \stackrel{\left(b\right)}{=}& {\left[{\mathbb{E}}_r\left(\frac{1}{1+T{r_0}^{\alpha }{r}^{-\alpha }}\right)\right]}^{M-1}\hfill \\ \hfill =& {\left[{\int }_0^R(\frac{1}{1+T{(r_0/r)}^{\alpha }})f_r\left(r\right)dr\right]}^{M-1}\hfill \\ \hfill =& {\left[\frac{2}{R^2}{\int }_0^R(\frac{r^{1+\alpha }}{r^{\alpha }+T{r_0}^{\alpha }})dr\right]}^{M-1},\hfill \end{array}$$

(8)

where $\left(a\right)$ is derived from the assumption that Rayleigh fading factor of each channel follows exponentially i.i.d., and $\left(b\right)$ can be derived from the property of moment generating function of exponential variable.

Substituting Equation (8) into the corresponding part of Equation (7), we have

$$\mathbb{P}[h_0\ge T_P{r}^{\alpha }({\sigma }^2+I_t)]=exp(-T_p{r_0}^{\alpha }{\sigma }^2)·{\left(\frac{2}{R^2}{\int }_0^R\frac{r^{1+\alpha }}{r^{\alpha }+T·{r_0}^{\alpha }}dr\right)}^{M-1}.$$

(9)

After plugging Equation (9) into Equation (6), we can obtain ${\mathbb{P}}_T$ in Theorem 1. ☐

3.2. Impact of OFJ Scheme

In the OFJ scheme, the friendly jammers placed at the boundary of communication area are equipped with omni-directional antennas, i.e., the antenna gain of jammers is $G_j=G_o=1$. Therefore, $I_j$ in Equation (3) can be expressed as ${\displaystyle \sum _{k=1}^N}{P}_j{h}_k{R}^{-\alpha }$. Then we give ${\mathbb{P}}_T$ in the OFJ scheme by the following theorem.

Theorem 2.

In the OFJ Scheme, the probability of successful transmission is

$${\mathbb{P}}_T=\frac{2^M}{R^{2M}}{\int }_0^{R}exp(-T_p{r_0}^{\alpha }{\sigma }^2){\left[1+T_p{\left(r_{0}R\right)}^{-\alpha }{P}_j\right]}^{-N}{\left({\int }_0^R\frac{r^{1+\alpha }}{r^{\alpha }+T{r_0}^{\alpha }}dr\right)}^{M-1}d{r}_0.$$

(10)

Proof. 

Similar to the proof of Theorem 1, the probability of successful transmission ${\mathbb{P}}_T$ in the OFJ scheme can be expressed as follows,

$${\mathbb{P}}_T={\int }_0^{R}P[h_0\ge T_p{r_0}^{\alpha }({\sigma }^2+I_t+I_j)|r_0]2r_0{R}^{-2}d{r}_0,$$

(11)

where $P[h_0\ge T_p{r_0}^{\alpha }({\sigma }^2+I_t+I_j)|r_0]$ can be derived as follows,

$$\begin{array}{cc}\hfill \mathbb{P}& [h_0\ge T_p{r}^{\alpha }({\sigma }^2+I_t+I_j)|r_0]\hfill \\ & ={\mathbb{E}}_{I_t,I_j}[exp\left((-T_p{r_0}^{\alpha })({\sigma }^2+I_t+I_j)\right)|r_0]\hfill \\ & =e^{-T_p{r_0}^{\alpha }{\sigma }^2}{\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)]{\mathbb{E}}_{I_j}[exp(-T_p{r_0}^{\alpha }{I}_j)],\hfill \end{array}$$

(12)

where ${\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)]$ is given by Equation (8), and ${\mathbb{E}}_{I_j}[exp(-T_p{r_0}^{\alpha }{I}_j)]$ can be calculated by

$$\begin{array}{cc}\hfill {\mathbb{E}}_{I_j}[exp(-T_p{r_0}^{\alpha }{I}_j)]=& {\mathbb{E}}_h[exp(-T_p{r_0}^{-\alpha }{\displaystyle \sum _{n=1}^N}{P}_j{h}_n{R}^{-\alpha })]\hfill \\ \hfill =& {\mathbb{E}}_h[{\displaystyle \prod _{n=1}^N}exp(-T_p{r_0}^{-\alpha }{P}_j{R}^{-\alpha }{h}_n)]\hfill \\ \hfill =& {\displaystyle \prod _{n=1}^N}{\mathbb{E}}_h[exp(-T_p{r_0}^{-\alpha }{P}_j{R}^{-\alpha }{h}_n)]\hfill \\ \hfill =& {\left[1+T_p{\left(r_{0}R\right)}^{-\alpha }{P}_j\right]}^{-N}.\hfill \end{array}$$

(13)

Substituting Equations (8) and (13) into the corresponding parts of Equation (12), we have

$$\mathbb{P}[h_0\ge T_P{r}^{\alpha }({\sigma }^2+I_t+I_j)]=exp(-T_p{r_0}^{\alpha }{\sigma }^2)·{\left[1+T_p{\left(r_{0}R\right)}^{-\alpha }{P}_j\right]}^{-N}·{\left(\frac{2}{R^2}{\int }_0^R\frac{r^{1+\alpha }}{r^{\alpha }+T·{r_0}^{\alpha }}dr\right)}^{M-1}.$$

(14)

By plugging Equation (14) into Equation (11), we can obtain ${\mathbb{P}}_T$ of OFJ scheme in Theorem 2. ☐

3.3. Impact of DFJ Scheme

In DFJ scheme, the jammers placed at the boundary are equipped with directional antennas. In particular, we can find that the receiver can be only affected by the side lobe of directional antennas, as shown in Figure 2. Therefore, we have $G_j$ =$G_s$. Thus, $I_j$ in Equation (3) can be expressed as ${\displaystyle \sum _{k=1}^N}{G}_s{P}_j{h}_k{R}^{-\alpha }$. Then, we obtain ${\mathbb{P}}_T$ in the DFJ scheme by the following theorem.

Theorem 3.

In the DFJ scheme, the probability of successful transmission is

$${\mathbb{P}}_T=\frac{2^M}{R^{2M}}{\int }_0^{R}exp(-T_p{r_0}^{\alpha }{\sigma }^2){\left[1+T_p{\left(r_{0}R\right)}^{-\alpha }{P}_j{G}_s\right]}^{-N}{\left({\int }_0^R\frac{r^{1+\alpha }}{r^{\alpha }+T{r_0}^{\alpha }}dr\right)}^{M-1}d{r}_0.$$

(15)

Proof. 

Similar to the proof of Theorems 1 and 2, ${\mathbb{P}}_T$ in DFJ can be expressed as

$$\begin{array}{cc}\hfill {\mathbb{P}}_T=& {\int }_0^{R}P\left[\frac{P_t{h}_0{r_0}^{-\alpha }}{{\sigma }^2+I_t+I_j}\ge T|r_0\right]f_r\left(r_0\right)d{r}_0\hfill \\ \hfill =& {\int }_0^R{\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)]{\mathbb{E}}_{I_j}[exp(-T_p{r_0}^{\alpha }{I}_j)]·2r_0{R}^{-2}{e}^{-T_p{r_0}^{\alpha }{\sigma }^2}d{r}_0,\hfill \end{array}$$

(16)

where ${\mathbb{E}}_{I_t}[exp(-T_p{r_0}^{\alpha }{I}_t)]$ is given by Equation (8), and ${\mathbb{E}}_{I_j}[exp(-T_p{r_0}^{\alpha }{I}_j)]$ can be calculated by

$$\begin{array}{cc}\hfill {\mathbb{E}}_{I_j}[exp(-T_p{r_0}^{\alpha }{I}_j)]=& {\mathbb{E}}_h[exp(-T_p{r_0}^{-\alpha }{\displaystyle \sum _{n=1}^N}{P}_j{G}_s{h}_n{R}^{-\alpha })]\hfill \\ \hfill =& {\displaystyle \prod _{n=1}^N}{\mathbb{E}}_h[exp(-T_p{r_0}^{-\alpha }{P}_j{G}_s{R}^{-\alpha }{h}_n)]\hfill \\ \hfill =& {\left[1+T_p{\left(r_{0}R\right)}^{-\alpha }{P}_j{G}_s\right]}^{-N}.\hfill \end{array}$$

(17)

After plugging Equations (8) and (17) into Equation (16), we can obtain ${\mathbb{P}}_T$ of DFJ scheme in Theorem 3. ☐

4.1. Impact of NFJ Scheme

Firstly we consider the NFJ scheme in which there is no friendly jammer on the boundary of communication area. In this case, the interference from friendly jammers to eavesdropper $I_{je}=0$, then we have the following theorem:

Theorem 4.

In the NFJ scheme, the probability of eavesdropping attack ${\mathbb{P}}_E$ is

$${\mathbb{P}}_E=1-{\left(1-{\left(\frac{2}{\pi R^2}\right)}^M·{\int }_D^{D+2R}exp(-{\beta }_p{l_0}^{\alpha }{\sigma }^2)l_0{Z\left(l_0\right)}^{M-1}·arccos\left[\frac{D}{l_0}+\frac{{l_0}^2-D^2}{2l_0(R+D)}\right]d{l}_0\right)}^M,$$

(20)

where ${\beta }_p=\frac{\beta }{P_t}$, $V\left(l_0\right)={\beta }_p{l_0}^{\alpha }{P}_j$ and

$$Z\left(l_0\right)={\int }_D^{D+2R}(\frac{l^{\alpha +1}}{l^{\alpha }+\beta {l_0}^{\alpha }})arccos\left[\frac{D}{l}+\frac{l^2-D^2}{2l(R+D)}\right]dl.$$

Proof. 

We denote the distance between the eavesdropper and a transmitter by l. The probability density function of l can be expressed as follows [26],

$$f_l\left(l\right)=\frac{2l}{\pi R^2}arccos\left[\frac{D}{l}+\frac{l^2-D^2}{2l(R+D)}\right],D\le l\le D+2R.$$

(21)

Then ${\mathbb{P}}_e$ can be expressed as

$${\mathbb{P}}_e={\int }_D^{D+2R}\frac{2l_0}{\pi R^2}\mathbb{P}[h_0\ge {\beta }_p{l_0}^{\alpha }({\sigma }^2+I_{te})|l_0]·arccos(\frac{d}{l_0}+\frac{{l_0}^2-d^2}{2l_0(R+d)})d{l}_0,$$

(22)

where ${\beta }_p=\frac{\beta }{P_t}$.

Since $h_0$ is a random variable following an exponential distribution with mean 1, $\mathbb{P}[h_0\ge {\beta }_p{l_0}^{\alpha }({\sigma }^2+I_{te})|l_0]$ can be expressed as

$$\begin{array}{cc}\hfill \mathbb{P}& [h_0\ge {\beta }_p{l_0}^{\alpha }({\sigma }^2+I_{te})|l_0]\hfill \\ & ={\mathbb{E}}_{I_{te}}\left[\mathbb{P}(h_0\ge {\beta }_p{l_0}^{\alpha }({\sigma }^2+I_{te})|l_0)\right]\hfill \\ & ={\mathbb{E}}_{I_{te}}[exp\left((-{\beta }_p{l_0}^{\alpha })({\sigma }^2+I_{te})\right)|l_0]\hfill \\ & =e^{-{\beta }_p{l_0}^{\alpha }{\sigma }^2}·{\mathbb{E}}_{I_{te}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{te})].\hfill \end{array}$$

(23)

Following the similar approach in deriving Equation (8), the expression of ${\mathbb{E}}_{I_{te}}[exp(-{\beta }_p{l}^{\alpha }{I}_{te})]$ can be derived by the following equation,

$$\begin{array}{cc}\hfill & {\mathbb{E}}_{I_{te}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{te})]\hfill \\ & ={\mathbb{E}}_{\Phi ,\left\{h_i\right\}}[exp(-{\beta }_p{l_0}^{\alpha }{\displaystyle \sum _{i\in \Phi /b_0}}{P}_t{h}_i{l}_i^{-\alpha })]\hfill \\ & ={\left[{\int }_D^{D+2R}(\frac{1}{1+\beta {(l_0/l)}^{\alpha }})f_l\left(l\right)dl\right]}^{M-1}\hfill \\ & ={\left[\frac{2}{\pi R^2}{\int }_D^{D+2R}(\frac{l^{\alpha +1}}{l^{\alpha }+\beta {l_0}^{\alpha }})arccos\left[\frac{D}{l}+\frac{l^2-D^2}{2l(R+D)}\right]dl\right]}^{M-1}.\hfill \end{array}$$

(24)

If we set $Z\left(l_0\right)={\int }_D^{D+2R}(\frac{l^{\alpha +1}}{l^{\alpha }+\beta {l_0}^{\alpha }})arccos\left[\frac{D}{l}+\frac{l^2-D^2}{2l(R+D)}\right]dl$, Equation (24) can be expressed as

$${\mathbb{E}}_{I_{te}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{te})]={\left[\frac{2Z\left(l_0\right)}{\pi R^2}\right]}^{M-1}.$$

(25)

After plugging Equation (25) into the Equation (23), and substituting the new expression of Equation (23) into Equation (22), we obtain the result of ${\mathbb{P}}_e$ as follows,

$${\mathbb{P}}_e={\left(\frac{2}{\pi R^2}\right)}^M·{\int }_D^{D+2R}exp(-{\beta }_p{l_0}^{\alpha }{\sigma }^2)l_0{Z\left(l_0\right)}^{M-1}·arccos\left[\frac{D}{l_0}+\frac{{l_0}^2-D^2}{2l_0(R+D)}\right]d{l}_0.$$

(26)

Substituting ${\mathbb{P}}_e$ in Equation (26) into Equation (19), we derive the probability of eavesdropping attack ${\mathbb{P}}_E$ of the NFJ scheme as given in Theorem 4. ☐

4.2. Impact of OFJ Scheme

Then we investigate the OFJ scheme, where friendly jammers are equipped with omni-directional antennas. In order to derive ${\mathbb{P}}_e$, we need to calculate the interference from jammers to eavesdropper $I_{je}$ first.

Figure 3 shows the geometrical relationships of the friendly jammers and the eavesdropper. Without loss of generality, we label the jammer which is nearest to the eavesdropper as $J_1$ and the jammer $J_1$ is at the left-hand-side of the eavesdropper. Then we label $J_{2m}$ (where $m=1,2,3,\dots )$ as mth nearest jammer at the right-hand-side of jammer $J_1$ separately. Similarly, we label $J_{2n+1}$ (where $n=1,2,3,\dots )$ as the nth nearest jammer at the left-hand-side of jammer $J_1$ separately.

From the observation point O, $2\phi$ is the relative degree between neighbour jammers and $\gamma$ is the degree between jammer $J_1$ and eavesdropper E. Since the number of jammers is N, we have $2\phi =\frac{2\pi }{N}$. Due to the fact that the eavesdropper is randomly located outside of the protected communication area, we have the probability density function of variable $\gamma$,

Then we can calculate the cumulative interference of the jammers to the eavesdropper based on their geometrical relationships, which is given by the following lemma.

Lemma 1.

When the friendly jammers are equipped with omni-directional antennas, the cumulative interference from the jammers to the eavesdropper is

$$I_{je}=\left\{\begin{array}{c}{P}_j\left[{\displaystyle \sum _{x=-\frac{N-1}{2}}^{\frac{N-1}{2}}}{h}_x{D}_{je}{\left(x\right)}^{-\alpha }\right],N\mathrm{is}\mathrm{odd}\hfill \\ P_j\left[{\displaystyle \sum _{x=-\frac{N}{2}}^{\frac{N-2}{2}}}{h}_x{D}_{je}{\left(x\right)}^{-\alpha }\right],N\mathrm{is}\mathrm{even}\hfill \end{array}\right.,$$

(28)

where $D_{je}\left(x\right)=\sqrt{R^2+L^2-2RLcos(2x\phi +\gamma )}$ .

Proof. 

We present the proof of Lemma 1 in Appendix A.

With the interference from the jammers to the eavesdropper $I_{je}$, we obtain the probability of eavesdropping attack ${\mathbb{P}}_E$ as the following theorem.

Theorem 5.

In the OFJ scheme, the probability of an eavesdropping attack ${\mathbb{P}}_E$ of an eavesdropper is:

$${\mathbb{P}}_E=1-{\left(1-{\left(\frac{2}{\pi R^2}\right)}^M{\int }_D^{D+2R}exp(-{\beta }_p{l_0}^{\alpha }{\sigma }^2)l_0{Z\left(l_0\right)}^{M-1}W\left(l_0\right)arccos[\frac{D}{l_0}+\frac{{l_0}^2-D^2}{2l_0(R+D)}]d{l}_0\right)}^M,$$

(29)

where

$$Z\left(l_0\right)={\int }_D^{D+2R}(\frac{l^{\alpha +1}}{l^{\alpha }+\beta {l_0}^{\alpha }})arccos\left[\frac{D}{l}+\frac{l^2-D^2}{2l(R+D)}\right]dl,$$

and

$$\begin{array}{c}W\left(l_0\right)=\left\{\begin{array}{c}{\displaystyle {\displaystyle \prod _{x=-\frac{N-1}{2}}^{\frac{N-1}{2}}}{\int }_0^{\phi }\frac{1}{\phi \left(1+V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }\right)}d\gamma ,N\mathrm{is}\mathrm{odd}}\hfill \\ {\displaystyle {\displaystyle \prod _{x=-\frac{N}{2}}^{\frac{N-2}{2}}}{\int }_0^{\phi }\frac{1}{\phi \left(1+V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }\right)}d\gamma ,N\mathrm{is}\mathrm{even}}\hfill \end{array}\right.,\hfill \end{array}$$

in which $V\left(l_0\right)={\beta }_p{l_0}^{\alpha }{P}_j$ and ${\beta }_p=\frac{\beta }{P_t}$.

Proof. 

Following the similar approach to the proof of Theorem 4, we can get the probability that a certain transmitter can be tapped denoted by ${\mathbb{P}}_e$ as follows,

$$\begin{array}{cc}\hfill {\mathbb{P}}_e=& {\mathbb{E}}_{l_0}\left[\mathbb{P}(SIN{R}_E\ge \beta |l_0)\right]\hfill \\ \hfill =& {\int }_D^{D+2R}\mathbb{P}\left[\frac{P_t{h}_0{l_0}^{-\alpha }}{{\sigma }^2+I_{te}+I_{je}}\ge \beta |l_0\right]f_l\left(l_0\right)d{l}_0\hfill \\ \hfill =& {\int }_D^{D+2R}\frac{2l_0}{\pi R^2}\mathbb{P}[h_0\ge {\beta }_p{l_0}^{\alpha }({\sigma }^2+I_{te}+I_{je})|l_0]·arccos(\frac{d}{l_0}+\frac{{l_0}^2-d^2}{2l_0(R+d)})d{l}_0,\hfill \end{array}$$

(30)

where ${\beta }_p=\frac{\beta }{P_t}$.

Since $h_0$ is a random variable following an exponential distribution with mean 1, $\mathbb{P}[h_0\ge {\beta }_p{l_0}^{\alpha }({\sigma }^2+I_{te}+I_{je})|l_0]$ can be expressed as

$$\mathbb{P}[h_0\ge {\beta }_p{l_0}^{\alpha }({\sigma }^2+I_{te}+I_{je})|l_0]=e^{-{\beta }_p{l_0}^{\alpha }{\sigma }^2}·{\mathbb{E}}_{I_{te}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{te})]·{\mathbb{E}}_{I_{je}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{je})],$$

(31)

where ${\mathbb{E}}_{I_{te}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{te})]={\left[\frac{2Z\left(l_0\right)}{\pi R^2}\right]}^{M-1}$ given by Equation (24).

Then we calculate ${\mathbb{E}}_{I_{je}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{je})]$. For simplicity, we denote $W\left(l_0\right)={\mathbb{E}}_{I_{je}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{je})]$, $V\left(l_0\right)={\beta }_p{l_0}^{\alpha }{P}_j$. With the expression of $I_{je}$ given in Lemma 1, we can obtain $W\left(l_0\right)$ given by the following equation,

$$\begin{gathered} \begin{array}{c}\begin{array}{c}W\left(l_0\right)={\mathbb{E}}_{I_{je}}[exp(-{\beta }_p{l_0}^{\alpha }{I}_{je})]\hfill \\ =\left\{\begin{array}{c}{\mathbb{E}}_{h,\gamma }\left[exp\left(-{\displaystyle \sum _{x=-\frac{N-1}{2}}^{\frac{N-1}{2}}}V\left(l_0\right)h_k{D}_{je}{\left(x\right)}^{-\alpha }\right)\right],N\mathrm{is}\mathrm{odd};\hfill \\ {\mathbb{E}}_{h,\gamma }\left[exp\left(-{\displaystyle \sum _{x=-\frac{N}{2}}^{\frac{N-2}{2}}}V\left(l_0\right)h_k{D}_{je}{\left(x\right)}^{-\alpha }\right)\right],N\mathrm{is}\mathrm{even}.\hfill \end{array}\right.\hfill \end{array} \\ \stackrel{\left(c\right)}{=}\left\{\begin{array}{c}{\mathbb{E}}_{\gamma }\left[{\displaystyle \prod _{x=-\frac{N-1}{2}}^{\frac{N-1}{2}}}{\mathbb{E}}_h\left[exp\left(-V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }{h}_k\right)\right]\right],N\mathrm{is}\mathrm{odd};\hfill \\ {\mathbb{E}}_{\gamma }\left[{\displaystyle \prod _{x=-\frac{N}{2}}^{\frac{N-2}{2}}}{\mathbb{E}}_h\left[exp\left(-V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }{h}_k\right)\right]\right],N\mathrm{is}\mathrm{even}.\hfill \end{array}\right.\hfill \end{array} \\ \stackrel{\left(d\right)}{=}\left\{\begin{array}{c}{\displaystyle \prod _{x=-\frac{N-1}{2}}^{\frac{N-1}{2}}}{\mathbb{E}}_{\gamma }\left[\frac{1}{1+V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }}\right],N\mathrm{is}\mathrm{odd};\hfill \\ {\displaystyle \prod _{x=-\frac{N}{2}}^{\frac{N-2}{2}}}{\mathbb{E}}_{\gamma }\left[\frac{1}{1+V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }}\right],N\mathrm{is}\mathrm{even}.\hfill \end{array}\right. \\ \stackrel{\left(e\right)}{=}\left\{\begin{array}{c}{\displaystyle {\displaystyle \prod _{x=-\frac{N-1}{2}}^{\frac{N-1}{2}}}{\int }_0^{\phi }\frac{1}{\phi \left(1+V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }\right)}d\gamma ,N\mathrm{is}\mathrm{odd};}\hfill \\ {\displaystyle {\displaystyle \prod _{x=-\frac{N}{2}}^{\frac{N-2}{2}}}{\int }_0^{\phi }\frac{1}{\phi \left(1+V\left(l_0\right)D_{je}{\left(x\right)}^{-\alpha }\right)}d\gamma ,N\mathrm{is}\mathrm{even}.}\hfill \end{array}\right.. \end{gathered}$$

(32)

where $\left(c\right)$ is derived from the independence between an eavesdropper’s location and the distribution of fading channel, $\left(d\right)$ follows from the property of moment generating function of exponential variable, $\left(e\right)$ is derived with the probability density function of $\gamma$ as given in Equation (27).

After plugging Equation (32) into Equation (31), and substituting the new expression of Equation (31) into Equation (30), we obtain the result of ${\mathbb{P}}_e$ as given in the following expression,

$$\begin{array}{cc}\hfill {\mathbb{P}}_e=& {\left(\frac{2}{\pi R^2}\right)}^M·{\int }_D^{D+2R}exp(-{\beta }_p{l_0}^{\alpha }{\sigma }^2)l_0{Z\left(l_0\right)}^{M-1}W\left(l_0\right)arccos[\frac{D}{l_0}+\frac{{l_0}^2-D^2}{2l_0(R+D)}]d{l}_0,\hfill \end{array}$$

(33)

Substituting the ${\mathbb{P}}_e$ in Equation (33) into Equation (19), we obtain the result of ${\mathbb{P}}_E$ given in Theorem 5. ☐

4.3. Impact of DFJ Scheme

In order to derive the probability of an eavesdropping attack ${\mathbb{P}}_E$ in the DJF scheme, we need to evaluate the interference from the friendly jammers equipped with directional antenna to the eavesdropper.

However, when the jammers are deployed densely or the distance between the eavesdropper and the commmunication area D is large, there may be more than one jammer that interferes with the eavesdropper via their main lobes simultaneously (as shown in Figure 4). Therefore, we first investigate the number of friendly jammers which interfere with the eavesdropper via main lobes.

In Figure 4, we show the main lobes of 3 jammers. Due to the fact that the eavesdropper is nearest to the jammer $J_1$, the eavesdropper locates in the area between line a and line b, where line a is the extended line of $O{J}_1$ and line b is the perpendicular bisector of segment $J_1{J}_2$.

The term of $\omega$ in Figure 4 is the degree between line a and $J_{1}E$. When $\omega \ge \frac{{\theta }_m}{2}$, the eavesdropper locates in area $A_0$, there is no jammers interfering it with its main lobe. When $\omega \le \frac{{\theta }_m}{2}$, the area that the eavesdropper locates depends on the distance D. When $\omega \le \frac{{\theta }_m}{2}$, the eavesdropper locates in $A_1$ if $D\le d$, there will be one jammer interfering with the eavesdropper via its main lobe; the eavesdropper locates in $A_2$ if $D\ge d$, there will be two jammers interfering with the eavesdropper via their main lobes.

Similarly, we denote $A_k$ to be the intersection area of k jammers’ main lobe directions between line a and line b. When the eavesdropper locates in area $A_k$, there will be k jammers interfering with the eavesdropper via their main lobes. We denote the number of friendly jammers (interfering the eavesdropper via their main lobes) by $N_d$. When $N_d$ is fixed, the interference of directional jammers to eavesdroppers is shown in the following lemma.

Lemma 2.

When $N_d=k$, the interference of jammers on eavesdropper in the DFJ scheme is

$$I_{je}=\left\{\begin{array}{c}{P}_j{G}_n\left[{\displaystyle \sum _{x=-\frac{k-1}{2}}^{\frac{k-1}{2}}}{h}_x{D}_{je}{\left(x\right)}^{-\alpha }\right]+I_{js},k\mathrm{is}\mathrm{odd}\hfill \\ P_j{G}_n\left[{\displaystyle \sum _{x=-\frac{k}{2}}^{\frac{k-2}{2}}}{h}_x{D}_{je}{\left(x\right)}^{-\alpha }\right]+I_{js},k\mathrm{is}\mathrm{even}andk\ne 0\hfill \\ I_{js},k=0\hfill \end{array}\right.,$$

(34)

where $G_n=G_m-G_s$, $D_{je}\left(x\right)=\sqrt{R^2+L^2-2RLcos(2x\phi +\gamma )}$ , and

$$I_{js}=\left\{\begin{array}{c}{P}_j{G}_s\left[{\displaystyle \sum _{x=-\frac{N-1}{2}}^{\frac{N-1}{2}}}{h}_x{D}_{je}{\left(x\right)}^{-\alpha }\right],N\mathrm{is}\mathrm{odd};\hfill \\ P_j{G}_s\left[{\displaystyle \sum _{x=-\frac{N}{2}}^{\frac{N-2}{2}}}{h}_x{D}_{je}{\left(x\right)}^{-\alpha }\right],N\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$

Proof. 

We present the proof of Lemma 2 in Appendix B. ☐

With the interference from the jammers to the eavesdropper, we obtain the probability of eavesdropping attack ${\mathbb{P}}_E$ as the following theorem,

Theorem 6.

In DFJ scheme, the upper bound of probability of eavesdropping attack ${\mathbb{P}}_E$ is given by

$$\begin{array}{c}\hfill {\mathbb{P}}_E=\frac{\phi -{\gamma }_0}{\phi }·\left[1-{\left(1-{\mathbb{P}}_e(N_d=0)\right)}^M\right]+\frac{{\gamma }_0}{\phi }·\left[1-{\left(1-{\mathbb{P}}_e(N_d=1)\right)}^M\right],\end{array}$$

where ${\mathbb{P}}_e(N_d=k)={\left(\frac{2}{\pi R^2}\right)}^M{\int }_D^{D+2R}exp(-{\beta }_p{l_0}^{\alpha }{\sigma }^2)l_0{Z\left(l_0\right)}^{M-1}{W}^{\prime }\left(l_0,k\right)arccos[\frac{D}{l_0}+\frac{{l_0}^2-D^2}{2l_0(R+D)}]d{l}_0$,

${\beta }_p=\frac{\beta }{P_t}$, $V_1\left(l_0\right)={\beta }_p{l_0}^{\alpha }{P}_j{G}_s$, $V_2\left(l_0\right)={\beta }_p{l_0}^{\alpha }{P}_j{G}_n$, $Z\left(l_0\right)={\int }_D^{D+2R}(\frac{l^{\alpha +1}}{l^{\alpha }+\beta {l_0}^{\alpha }})arccos\left[\frac{D}{l}+\frac{l^2-D^2}{2l(R+D)}\right]dl$, ${\gamma }_0=\frac{{\theta }_m}{2}-arcsin\left(Rsin\left(\frac{{\theta }_m}{2}\right)/L\right)$,