# Secrecy at the Physical Layer in TWC with Two Half-Duplex DF Relays in the Presence of an External Eavesdropper

## System Model of Two-Way Communication with DF Relays

In this model, two sources share their information signal with each other through two half-duplex DF relays as in Sharma et al. (2019). The complete system model of this TWC network has been shown in Figure 12.7. In this communication, an external eavesdropper efforts to eavesdrop the message of both the sources, but not simultaneously at a time. Each of the relays has two directional antennas, one is used for receiving the information signal, while the other is for transmitting the information signal. The directional antennas at each relay assist the relay for two-way communication. The rest of the communication nodes has omnidirectional antennas. The EAV keep its position in such a way that it presents at almost equal distance from both the sources, and this position provides the proper receiving of the information signal from both the sources. We have found this particular position of EAV on a line that is perpendicular to another line that connects to both the sources. Due to directional antennas, the EAV is unable to receive the relayed signal. The communication completes in two phases; first is a broadcasting phase, and another is the relaying phase as shown in Figure 12.8. In the broadcast phase, both the sources transmit the signals. In broadcasting phase, the information signal of source one (SI) is received by the

FIGURE 12.7 System model of two-way communication with two half-duplex DF relay.

FIGURE 12.8 The time frame of two-way communication with two half-duplex DF relay.

relay one (R1), and the information signal of source two (S2) is received by the relay two (R2). In the relaying phase, the S2 receives the signal from Rl, and SI receives the signal from R2.

All channels are Rayleigh fading. The channel has Gaussian noise with zero mean and variance *N _{0}.* The channel coefficients are denoted by

*h*the channel gain is denoted by

_{>r}*g*and mean channel gains are denoted by Q

_{tj}_{1;}, where / = (51. R1,52, R2, E), here subscript E indicates for EAV, and

*j =*(51,R1,52.R2,£). All the channels are independent and nonidentically distributed random variables.

### Signal strength

The signal strength at any receiving node is represented in signal-to-noise ratio (SNR) or signal-to-interference plus noise ratio (SINR) parameter. In the broadcasting phase, the information signal of the SI is received by the Rl, and the information signal of the S2 is received by the (R2). The received signal at the Rl is

where *P _{sl}* is the 51 ’s transmit power, x

_{sl}message of 51, and

*n*is the Gaussian noise. From Eq. (12.36), SNR at the Rl is

_{0}In the same way, the SNR at the *R2* is

The SINR of information signal S1 at the EAV is

The SINR of *S2* at the EAV is

In the relaying phase, the information signal strength of 51 at the 52 via *R*1 is

where *a* is the fraction of relay power, i.e., *aP _{R}* has assigned to

*R*1 and (1 -

*a)P*has assigned to

_{R}*R2.*The information signal strength of 52 at the 51 via

*R2*is

The end-to-end information signal strength of 51 at the 52 is and end-to-end information signal strength of 52 at 51 is

### Global secrecy capacity

Using the Shannon channel capacity formula, the channel capacity of information signal one is

and channel capacity of information signal two is

The channel capacity of the information signal one at the EAV is The channel capacity of the information signal two at the EAV is

Now, the secrecy capacity of information signal one is defined as

The secrecy capacity of information signal two is defined as (Pan et al. 2015; Kalamkar and Banerjee 2017)

The global secrecy capacity is expressed as (Zhang et al. 2017; Sharma et al. 2019)

## SOP Formulation with DF Relays and Optimality of Source Power and Fraction of Relay Power

### SOP formulation

We have defined the SOP in the previous Section 12.2.2. Here, we express the SOP as

Equation (12.52) is further simplified as

The term P(c|ff_{2} < Cth^{SEC}) (let P(Cjf£ < Cth^{SEC}) = *h )* in closed form is expressed as

where all constants are given as

The term *I _{2}* in closed form is expressed as

where all constants are given as

We put /, from Eq. (12.54) and *U* from Eq. (12.56) in Eq. (12.53); we get the SOP in final closed-form expression as

### Optimality of source power and fraction of relay power

The optimal value of source power and fraction of relay power is obtained by differentiating the SOP with respect to source power and fraction of relay power, respectively Sharma et al. 2019). First, the SOP is differentiated with respect to *P _{SI }*keeping the fixed value of

*^SI*and

*a.*For finding the optimal value, we equate this differentiation to zero. The expression is given as

The expression of can be expressed as

and the expression of can be expressed as

*3P*n

*&I*

We can find the final expression of Eq. (12.59) by putting —— from Eq. (12.60)

*61**, dPs]*

and —— from Eq. (12.61) in Eq. (12.59). The obtained expression is a transcenden- *dP*n

tal equation, which the bisection method of the numerical method is used to solve. The solution of the optimal value is shown in the next subsection.

Next, the optimality of « can be found following the same procedure as we have followed to find the optimality of *P _{S1}.* The derivatives of the SOP with respect to «is expressed as

where — can be expressed as

*3a*

and —can be expressed as *3a*

From the expression in Eqs. (12.63) and (12.64), we obtain the final expression of Eq. (12.62) as a transcendental equation, which the bisection method of the numerical method is used to solve. The solution of the optimal value is shown in the next subsection.