Jamming precoding in AF relay-aided PLC systems with multiple eavessdroppers

Enhancing information security has become increasingly significant in the digital age. This paper investigates the concept of physical layer security (PLS) within a relay-aided power line communication (PLC) system operating over a multiple-input multiple-output (MIMO) channel based on MK model. Specifically, we examine the transmission of confidential signals between a source and a distant destination while accounting for the presence of multiple eavesdroppers, both colluding and non-colluding. We propose a two-phase jamming scheme that leverages a full-duplex (FD) amplify-and-forward (AF) relay to address this challenge. Our primary objective is to maximize the secrecy rate, which necessitates the optimization of the jamming precoding and transmitting precoding matrices at both the source and the relay while adhering to transmit power constraints. We present a formulation of this problem and demonstrate that it can be efficiently solved using an effective block coordinate descent (BCD) algorithm. Simulation results are conducted to validate the convergence and performance of the proposed algorithm. These findings confirm the effectiveness of our approach. Furthermore, the numerical analysis reveals that our proposed algorithm surpasses traditional schemes that lack jamming to achieve higher secrecy rates. As a result, the proposed algorithm offers the benefit of guaranteeing secure communications in a realistic channel model, even in scenarios involving colluding eavesdroppers.

into account the impact of Bernoulli-Gaussian impulsive noise 61 .Finally, the study provides a thorough analysis of secrecy rate, secrecy outage probability, and secrecy capacity of a broadband system 62 .
To the best of our knowledge, there is currently a research gap regarding relay-aided PLS in the presence of multiple eavesdroppers.Motivated by previous studies on cooperative precoding to enhance channel quality for legitimate users 63-65 and cooperative jamming to impair channel quality for unauthorized users in wireless communication 66-68 , we propose a novel cooperative jamming and precoding PLS scheme for IBFD DF relayaided PLC systems.Unlike other studies assuming perfect channel information 69,70 , our proposed approach takes into account imperfect channel information and aims to design the precoding matrices at the legitimate nodes.The objective is to maximize the secrecy rate of AF relay-aided PLC systems with imperfect channel information in the presence of multiple eavesdroppers.To achieve this, we utilize an efficient BCD algorithm to iteratively optimize the precoding matrices.By jointly optimizing these matrices, we aim to enhance the secrecy rate by leveraging the cooperative capabilities of the relay and introducing intentional jamming to disrupt the eavesdroppers' reception.The proposed scheme addresses the challenges posed by multiple eavesdroppers and imperfect channel information, which are critical considerations in practical PLC systems.
To ensure the proposed scheme accurately models real-world power line channels, this paper conducts a characterization of the statistical MIMO PLC channel based on an analysis of a set of experimental field measurements 71 .The analysis takes into account various factors that impact data transfer, including fading effects, multipath propagation, and signal frequency.In addition, the noise in PLC is modeled as Bernoulli-Gaussian impulsive noise 72 .Previous research on PLS has considered different scenarios involving imperfect knowledge of CSI, ranging from passive eavesdroppers with unknown CSI 55-57,61,62 to those with estimation errors 59,73 .We consider a system with globally imperfect channels to provide a more comprehensive and realistic approach.In this scenario, all CSIs are partially known by the legitimate nodes in the PLC system due to channel estimation errors.Furthermore, we extend the study from a single-eavesdropper scenario to a multiple-eavesdropper scenario, considering two types of eavesdropping scenarios: non-colluding and colluding.In the non-colluding scenario, the eavesdroppers operate independently and do not share information, while in the colluding scenario, all eavesdroppers collaborate to intercept the legitimate transmission.Specifically, we investigate the severe colluding case to gain deeper insights into the security performance of the proposed scheme.
The subsequent sections of this paper are organized as follows."System model" presents a detailed description of the system model.In "Simulation and results", the proposed optimization problem is proved to be solvable by a series of transformations.net section showcases the numerical results obtained from the proposed scheme.Finally, "Conclusion" concludes the findings and provides insights for future research directions.
Notations: To simplify the formulation, we denote AA H as A K and the vec(A) denotes the vectorization of a matrix A.

System model
We consider a secure transmission system as shown in Figs. 1 and 2, where a source tries to transmit confidential information to legitimate users via an FD relay in the presence of multiple eavesdroppers eavesdropping in different time phases.More specifically, we assume all the eavesdroppers can be divided into two sets by their eavesdropping time.The first one can only eavesdrop on messages in the first time phase and the second one can only eavesdrop messages in the second time phase.Because the relay runs in the full-duplex model, self-interference should be involved.In addition, considering the huge attenuation of signals over long distances in the PLC system, the direct link from the source to the legitimate users can be ignored.In the system, the source(S), the relay(R) and the users(D) are involved N S , N R and N D ports, respectively.Two sets of eavesdroppers are equivalent to two multiple-ports eavesdroppers ( E 1 and E 2 ).Here, both sets of eavesdroppers are equipped with N E ports.
In the system, channels are described by channel transfer function (CTF) H ij,k as the matrix of coefficients, where i, j and k denote the transmitter, receiver and transmission time phases, respectively.Note that H RR,1 refers to the self-interference matrix , and H ij,k stays constant in the transmission process.In this paper, considering the multipath effect, frequency-selective effect, and time delay of power lines, this paper adopts the MK model 71 to model the channel, with channel noise characterized as a Bernoulli-Gaussian pulse noise.Due to the imprecision of channel estimation/feedback and the stealthiness of eavesdroppers, it is challenging for the transmitter to obtain accurate channel state information between the receiver and the transmitter.Therefore, we consider all channels to be imperfect channels.
The imperfect channels are described by the deterministic uncertainty model: where ij,k and H ij,k denote the CTF error and the mean CTF, respectively.The PLC system is operated over two-time phases with the relay in the AF model.In the two time slots, confidential information is transmitted via a source and a relay.While one of the legitimate nodes propagates the information forward, the other sends jamming signal to deal with possible eavesdroppers.By considering the possible worst-case scenario, the model introduces two groups of eavesdroppers who eavesdrop on two time slots respectively and considers the effect of self-interference at the relay.
In the first time phase, the source broadcasts confidential messages to the relay and the messages are inevitably eavesdropped by E 1 .More accurately, the confidential messages are modeled as symbols S ∈ CN (0, 1) mapped by vector W ∈ C N S ×1 : Next, we consider the messages emitted by the relay which only emits jamming: where V ∈ C N R ×1 and Z ∈ CN (0, 1) denote jamming vector and symbol.
With self-interference, the messages received by the relay can be formulated: where n R1 is actually PLC noise based on the Bernoulli-Gaussian noise model at the relay.
Meanwhile, E 1 receives the messages from both the source and the relay: where n E1 is Bernoulli-Gaussian noise at E 1 .
In the second time phase, the source emits the jamming precoded from the relay: Meanwhile, the relay is working in the AF mode so it amplifies and forwards the messages it received to both the users and E 2 : where G ∈ C N R ×N R denotes the amplifying matrix.
Because of the distance, the jamming from the source will not interrupt users but E 2 , i.e.
where n D is Bernouil-Gaussia noise at the users and n E2 is Bernouil-Gaussia noise at E 2 .Above all, we can calculate the signal-to-noise ratio (SNR) at all the receivers. (1) Phase 2 in the PLC system.and σ i is the amplitude of the corresponding Bernouil-Gaussia noise n i .The achievable rate of the legitimate users is as follows: However, the situation for eavesdropping is more complicated, because the eavesdroppers can collude or not.In the colluding case, the eavesdroppers can utilize maximum ratio combining (MRC) to combine their received information.In this typical collusion strategy, the eavesdropping SNR is the sum of all the eavesdroppers.So the achievable rate of the eavesdroppers is as follows:

Jamming precoding scheme
In this work, the goal is to maximize the secrecy rate of the system.With the transmit power constraint, the optimization problem can be formulated as follows.
Because of the high non-convexity of the function log |•| , the problem is hard to solve.To make the problem solvable, ( 15) is transformed into an equivalent form through WMMSE algorithm, which can be solved with the BCD algorithm.We first introduce WMMSE algorithm.where However, ( 14) is hard to explicit the Lemma1 directly.As a result, we need to transform (14) in a more compatible form.where ( 12) Vol:.( 1234567890 Then we have (17), where R E are divided as C E1 , C E2 and C E3 for subsequent transformation.Thus, in order to formulate the achievable rate of the eavesdroppers in the colluding case, C E1 and C E2 is equivalent to where Note the decomposition Then to solve C E3 , we also apply Lemma1 : After substituting ( 22)-( 26) into (15), the secrecy rate of the system with the colluding eavesdroppers can be rewritten as where f (W, V, A, G, S i , D i ) is defined in (24).
The max-min problem and constrain tr( can be transformed into an optimization problem by introducing constraints with slack variables β i as follows. The problem (27) can be further transformed as where g(W, V, A, G, S i , D i ) is defined in (25).
However, ( 25) is still convex because of the semi-infinite constraints (28).For i = D , tr(S D M D ) can be rewrit- ten as where S D = F H D F D and the equality tr(A K ) = �vec(A)� 2 is applied.Especially note that for i = E3 , to obtain similiar form as (30), F E3 should be divided as

So we have
Focusing on the uncertain CTF, (30) can be rewritten as follows.16), (26) where the identity vec(ABC) = C T ⊗ A vec(B) is applied.D and D is the linear part and the quadratic part of the CTF uncertainty, respectively.Actually the quadratic part is negligible.Then, we only consider asymptotic form of φ D as where For other situations and for the constraint tr((GH SR,1 , similiar formulas can be obtained through the same method.While the power constraint does not involve any quadratic part of the CTF uncertainty, so the original problem constraints are not relaxed. With (30) and (34) and by exploiting the Schur complement lemma 75 , ( 28) can be rewritten as matrix inequality .
The constraint (36) still contains the uncertainty D .The sign-definiteness lemma is applied to eliminate this uncertainty.
Lemma 2 76 : Given a Hermitian matrix A and arbitrary matrices pair With all components, the problem is equivalent to ( 33) where the function h(W, V, A, G, F i , D i , i , β i ) is defined in (39).Although (43) is still non-convex.However, it is convex with respective to F i or any one in W, V, A, G, D i .As a result, (43) can be solved via BCD algorithm as below.
The variables to be optimized in the optimization problem are divided into the following groups.
By optimizing the problem in order 1 → 2 → 3 → 4 , we denote y (i) k as the optimal value optimized to k in the ith iteration.Since the variables to be optimized always meet the constraints in the optimization process, the optimal value will not decrease : Obviously with constraints ( 16), the secrecy rate is bounded and the objective function value increases in each iteration, which proves the convergence.

Simulation and results
In the simulation parts, the statistical MIMO PLC channels are generated by formula (1) in reference 71 , and the specific parameters are shown in Table 2.And the noises in PLC are also modeled as a Bernoulli-Gaussian impulsive noise.
In this section, numerical results are presented to prove the effectiveness of precoding jamming scheme in terms of average secrecy rate.In this part, without specific definition, we consider Besides, for the simplicity, the CTF uncertaninty bound δ ij,k are related to corresponding determinant of mean CTF with one certain coefficient, or δ ij,k = µ H ij,k .Apparently, it accords with the natural assumption that CTF with larger determinant tends to be more uncertain.
Figures 3 and 4 illustrate the relationship between the average secrecy rate and the number of iterations, assuming power constraints P S = P R = P = 10 dB .Notably, the average secrecy rate consistently stabilizes after approximately 40 iterations.This indicates that heightened CTF uncertainty detrimentally impacts the secrecy rate.Moreover, the proposed approach exhibits superior performance with an increased number of legitimate user ports and a reduced number of eavesdropper ports.This distinction becomes particularly pronounced in scenarios characterized by larger CTF uncertainty.In essence, the number of ports directly correlates with the capacity for receiving or intercepting information.
We examine the characteristics of the proposed scheme under varying transmit power levels in Figs. 5 and 6.The analysis reveals an increase in the average secrecy rate as the transmit power is raised.However, beyond a (43) max 16), ( 40), ( 41) Parameters of the PLC channels 71 .The average secrecy rate(bit/s/Hz) The average secrecy rate(bit/s/Hz) The average secrecy rate(bit/s/Hz)

5.
Average secrecy rate versus power constraint comparison of different ports number and CTF uncertainty.transmission power of 10 dB, especially in scenarios with more eavesdropper ports and increased CTF uncertainty, the secrecy rate experiences only marginal improvement.This is attributed to the fact that elevating transmit power enhances not only the capacity of legitimate users but also that of eavesdroppers colluding to boost their eavesdropping rate.Consequently, this simultaneous enhancement leads to only a slight alteration in the overall secrecy rate.Additionally, when the numbers of legitimate user ports and eavesdropper ports both increase from 2 to 3, there is a notable decrease in the average secrecy rate.This observation suggests that in collusion scenarios, the expansion of the eavesdroppers' port number has a more substantial impact on the PLC system than the growth in the number of legitimate users' ports.
We assess the influence of jamming by showcasing the numerical outcomes of our proposed schemes alongside a comparable one lacking jamming signals in Fig. 7.This figure shows improvements achieved by the proposed algorithm compared to the traditional one.As shown in Fig. 7, the proposed algorithm achieves higher secure rates than the traditional algorithm when µ = 0.05 or µ = 0.075 , and the advantage increases with increasing power.Specifically, when µ = 0.05 and the transmission power is 20 dB, the secure rate of the proposed algorithm can reach 0.85 bit/s/Hz, while the traditional algorithm achieves 0.61 bit/s/Hz.In contrast to the jamming-free scheme, our proposed approach demonstrates superior performance, particularly regarding the average secrecy rate, especially under conditions of lower CTF uncertainty and increased transmit power.This suggests that, to a certain degree, jamming has the capability to disrupt the interception efforts of eavesdroppers, even in scenarios characterized by higher CTF uncertainty.
Figure 8 depicts how the ports of eavesdroppers affect the PLC system, where N S = N R = N D = 2 .It suggests the ability of eavesdroppers increases with the growth of their ports, especially in few ports.

Conclusion
The paper introduces a precoding jamming scheme aimed at bolstering the security of AF relay-aided PLC systems when faced with the challenge of multiple colluding eavesdroppers, while also considering CTF uncertainty.
The numerical results unequivocally establish the superiority of our proposed scheme compared to a jammingfree alternative.Notably, the effectiveness of the proposed scheme is underscored, especially in scenarios characterized by elevated CTF uncertainty.

Lemma1 74 :
Define the MSE matrix where R ≻ 0 .Then we have Furthermore, auxiliary matrices S i , M i , D i are introduced to reformulate the part of log |•| in the objective function in (15) as follows.

Figure 3 .
Figure 3. Average secrecy rate versus numbers of iterations comparison of different ports number and CTF uncertainty.

Figure 4 .
Figure 4. Average secrecy rate versus numbers of iterations comparison of different ports number and CTF uncertainty.

Figure 6 .Figure 7 .
Figure 6.Average secrecy rate versus power constraint comparison of different ports number and CTF uncertainty.

Figure 8 .
Figure 8.Average secrecy rate versus different eavesdropping ports number.