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User Cooperation for IRS-aided Secure SWIPT MIMO Systems

In this paper, intelligent reflecting surface (IRS) is proposed to enhance the physical layer security in the Rician fading channel where the angular direction of the eavesdropper is aligned with a legitimate user. In this scenario, we consider a two-phase communication system under the active attacks and passive eavesdropping. Particularly, in the first phase, the base station avoids direct transmission to the attacked user. While, in the second phase, other users cooperate to forward signals to the attacked user with the help of IRS and energy harvesting technology. Under the active attacks, we investigate an outage constrained beamforming design problem under the statistical cascaded channel error model, which is solved by using the Bernstein-type inequality. As for the passive eavesdropping, an average secrecy rate maximization problem is formulated, which is addressed by a low complexity algorithm. Numerical results show that the negative effect of the eavesdropper's channel error is greater than that of the legitimate user.


I. INTRODUCTION
Communication security is widely regarded as one of the important issues in wireless communications. Traditionally, security is enforced by imposing cryptographic protocols in the application layer [1]. However, this upper layer solution is not flexible as it requires complex key exchange protocols.
Fortunately, it is shown by Wyner [2] that secure communication can be guaranteed by adopting advanced signal processing techniques developed in the physical layer. In specific, these techniques exploit differences in channel conditions and interference environment to enhance the received signal of legitime users (LUs) and suppress the signal received by the eavesdropper (ED).
In order to enhance physical layer security, intelligent reflecting surface (IRS), a kind of passive metasurface, has emerged as a promising technique [3]- [5]. By changing the reflection direction of the incident signal, the IRS is capable of reconfiguring the wireless channels, enhancing the signal from the base station (BS) to LU and suppressing the radio frequency (RF) power leaked to the ED [6]- [10].
Furthermore, the IRS can be readily coated on existing buildings, such as the walls and ceilings, which reduces the cost and complexity of deployment operations. Hence, IRS holds great promise for security enhancement as it provides a cost-effective and energy-efficient approach.
In general, ED works in two modes: active attacks and passive eavesdropping [11], [12]. In an active attack, in order to mislead the BS to send signals to the ED, the ED pretends to be a LU sending pilot signals to the BS during the channel estimation procedure. Nonetheless, a passive attack is more challenging to tackle since the passive ED can hide itself and its channel state information (CSI) is not available at the BS.
precoders and the passive reflection beamforming are updated by using the proposed semidefinite programming (SDP) and penalty convex-concave procedure (CCP) technique respectively in an iterative manner.
• For the passive ED case with only partial CSI, we maximize an average secrecy rate subject to the unit-modulus constraint of the reflection beamforming and the energy harvesting constraint.
To address the numerical integration in the objective function, an angular secrecy model, which is analytically non-convex, is proposed. A low-complexity algorithm is proposed based on the MM-based alternate optimization (AO) framework, where the precoders are updated by solving a convex optimization problem and the reflection beamforming is updated in a closed-form solution which is globally optimal.
• The numerical results demonstrate that the level of the cascaded CSI error plays a vital role in the IRS-aided secure communication systems. In particular, at low error of cascaded CSI, the secrecy rate increases with the number of elements at the IRS due to the increased beamforming gain. However, at large level of cascaded CSI error, the secrecy rate decreases with the number of elements at the IRS due to the increased channel estimation error. Hence, whether to enable the IRS for enhancing the security capacity in the communication systems depends on the level of the cascaded CSI error. In addition, the IRS can enhance the average secrecy rate under the passive eavesdropping.
The remainder of this paper is organized as follows. Section II introduces the channel model and the system model. Outage constrained robust design problem is formulated for the active eavesdropper model in Section III. Section IV further investigates the average eavesdropping rate maximization problem under the passive eavesdropping. Finally, Section V and Section VI show the numerical results and conclusions, respectively.
Notations: The following mathematical notations and symbols are used throughout this paper. Vectors and matrices are denoted by boldface lowercase letters and boldface uppercase letters, respectively. The symbols X * , X T , X H , and ||X|| F denote the conjugate, transpose, Hermitian (conjugate transpose), Frobenius norm of matrix X, respectively. The symbol ||x|| 2 denotes 2-norm of vector x. The symbols Tr{·}, Re{·}, |·|, λ(·), and (·) denote the trace, real part, modulus, eigenvalue, and angle of a complex number, respectively. diag(x) is a diagonal matrix with the entries of x on its main diagonal. [x] m means the m th element of the vector x. The Kronecker product and the Hadamard product between two matrices X and Y is denoted by X ⊗ Y and X ⊙ Y, respectively. X Ymeans that X − Y is positive semidefinite. Additionally, the symbol C denotes complex field, R represents real field, and

A. Channel Model
Define the set of all LUs as K = {1, 2, ..., K}, and denote set K −K = K/{K} and set K +E = K ∪ {E}. By denoting {D i , θ i } ∀i∈K +E as the distances and the azimuth angles respectively from the BS to the LUs and the ED, as shown in Fig. 1, then the corresponding channels {g i ∈ C N ×1 } ∀i∈K +E follow the Rician fading distribution [15]: where ̺ 0 is the pathloss at the reference distance of d 0 , α BS and K BS are the pathloss exponent and the Rician factor of the BS-related links, respectively. It is assumed that the BS is equipped 6 with a uniform linear array (ULA). Then, the line-of-sight (LoS) component is given by g LOS i = 1, e −jπ sin θ i , · · · , e −j(N −1)π sin θ i , and the non-LoS component is drawn from a Rayleigh fading, i.e., Furthermore, by denoting {D IRS , θ IRS } as the distance and the azimuth angle from the BS to the IRS, it is straightforward to obtain the distances {d IRS,i } ∀i∈K +E and the azimuth angles {ϕ i } ∀i∈K +E from the IRS to the LUs and the ED as shown in Fig. 2, i.e., The corresponding channels {h i ∈ C N ×1 } ∀i∈K +E are given by where α IRS and K IRS are the pathloss exponent and the Rician factor of the IRS-related links, respec- where 1 ≤ x ≤ M x , 1 ≤ y ≤ M y and φ is the elevation angle observed at the IRS side.

B. Signal Transmission
To achieve high success rate of attack, the ED can locate the line between the BS and legitimate user. In this situation, the signal received by the ED is highly correlated with that of this user [11], [12], thus posing great threats to the system. As shown in Fig. 1, we assume that the ED hides at the line connecting the BS and one of the users, denoted as user K, which leads to θ E ≈ θ K , g LOS E ≈ g LOS K and D E ∈ (0, D K ). When the Rician factor K BS is sufficiently large, the channel g E is approximately equal to the channel g K .
In order to achieve high-quality secure communication, the angle aware user cooperation (AAUC) scheme [15] is adopted here. In particular, in the first phase, the BS multicasts the common signal to all users except user K. In the second phase, the helping users (∀k ∈ K −K ) forward the decoded common signal to user K via the IRS. In this work, in order to implement the AAUC scheme without consuming extra energy, the LUs adopt the hybrid information and energy harvesting receiving mode which splits the received signal into two power streams with power splitting ratios t k and 1 − t k . The former is used for decoding the signal and the latter is for energy harvesting.

1) Multicasting Phase:
In this phase, the BS multicasts the signal s to the helping LUs through beamforming vector f ∈ C N ×1 which is limited to the maximum transmit power P max , i.e., ||f|| 2 2 ≤ P max . Since g E ≈ g K , the beamforming f needs to satisfy |g H K f|= 0 to ensure that |g H E f|≈ 0. Let Q ∈ C N ×(N −1) be the orthogonal matrix which spans the null space of g K by using the QR decomposition, i.e., Q H Q = I. Then, we can design f = Qz, where z ∈ C (N −1)×1 is a newly introduced variable.
Therefore, the signal received by LU k is given by g H k Qz + n k , where n k is the received noise with the noise power of σ 2 k . By adopting the hybrid receiving mode and let t = [t 1 , ..., t K−1 ] T where t k is the power splitting ratio of LU k, the achievable rate at LU k = K is where the factor 1/2 is due to the two transmission phases. The harvested power at LU k = K is 2) User Cooperation Phase: In this phase, the helping LUs (∀k ∈ K −K ) forward the signal s to LU K through a beamforming vector w ∈ C (K−1)×1 = [w 1 , ..., w K−1 ] T by using the power harvested in the multicasting phase. Since LU K is randomly selected by the ED and assume that there are many obstacles in the communication environment, such as indoor applications, the direct links between the helping LUs and the LU K may be blocked. To address this issue, an IRS can be installed on the building with a certain height, and thus the IRS is capable of reflecting the signals forwarded by the helping LUs to LU K. Denote by e the reflection coefficient vector of the IRS, where |e m | 2 = 1, ∀m = 1, · · · , M. Then, the signal received by LU K is given by is the cascaded LU-IRS-LU (LIL) channel, and n K ∼ CN(0, σ 2 K ) is the noise. The corresponding achievable rate is On the other hand, the signal recieved by the ED is is the cascaded LU-IRS-ED (LIE) channel, and n E ∼ CN(0, σ 2 E ) is the received noise at the ED.
The corresponding eavesdropping rate is Finally, the secrecy rate of this system under the AAUC scheme can be expressed as [12]: In the following two sections, we consider the system design for two ED models: the active eavesdropper model and the passive eavesdropper model.

III. ED MODEL I: ACTIVE EAVESDROPPER MODEL
In this section, we consider the active attack case, in which the ED pretends to be an LU sending pilot signals to the transmitters (including the BS and the helping LUs) during the channel estimation procedure [11], [12]. It is reasonable to assume that the BS is capable of addressing this attack by using the multi-antenna technique, so as to obtain perfect CSI of the system. Nevertheless, the signle-antenna helping LUs only have the imperfect CSI of LU K and the ED due to their limited anti-interference ability.

A. Channel Uncertainties
Based on the above assumption, the cascaded channels can be modeled as where H K and H E are the estimated cascaded channels, are the unknown cascaded channel errors. △ K k and △ E k are the unknown cascaded LIL and LIE channel error vectors at LU k, respectively.
where Σ K k ∈ C M ×M and Σ E k ∈ C M ×M are positive semidefinite error covariance matrices. Note that the CSI error vectors of different LUs are independent with each other. Therefore, we have where Σ K and Σ E are block diagonal matrices, i.e.,

B. Outage constrained beamforming design
Under the statistical CSI error model, we develop a probabilistically robust algorithm for the secrecy rate maximization problem, which is formulated as where ρ ∈ (0, 1] is the secrecy rate outage probability. Problem (11) is difficult to solve due to the computationally intractable rate outage probability constraint (11b), the non-convex unit-modulus constraint (11d), and the non-convex power constraint (11f).
Firstly, we replace constraint (11b) with the development of a safe approximation consisting of three steps in the following.
Step 1: Decouple the Probabilistic Constraint: First of all, based on the independence between Step 2: Convenient Approximations: To address the non-concavity of R k − R E , ∀k ∈ K K , we need to construct a sequence of surrogate functions of {R i } ∀i∈K +E . More specifically, we need the following lemmas.
Lemma 1 [16] The quadratic-over-linear function x 2 y is jointly convex in (x, y), and lower bounded by its linear first-order Tayler approximation 2x (n) y (n) x − ( x (n) y (n) ) 2 y at fixed point (x (n) , y (n) ).
By substituting x with g H k Qz and y with 1/t k , we utilize Lemma 1 to obtain a concave lower bound of rate R k (z, t k ) for ∀k ∈ K −K . The lower bound is given by for any feasible solution {z (n) , t

Lemma 2
The upper bound of rate R E (w, e) is given by where a E is the auxiliary variable.
Proof: See Appendix A.

Lemma 3
The lower bound of rate R K (w, e) is given by where a K and v are the auxiliary variables.
Proof: See Appendix B.
For the convenience of derivations, we assume that . Furthermore, the error vectors in (10) can be rewritten as vec( Now, substituting (15) and (17) into (13), then (13) is replaced by Step 3: A Bernstein-Type Inequality-Based Safe Approximation: The outage probabilities in (19) are characterized by quadratic inequalities, which can be safely approximated by using the following lemma.
Before using Lemma 4, we need the following simplified derivations for LU k, ∀k ∈ K −K , i.e., By substituting (21) into (20) and introducing slack variables {x E , y E }, the constraints for ∀k ∈ K −K in (19a) are transformed into the following deterministic form: On the other hand, the simplified derivations for LU K are given by By substituting the above equations into (20) and introducing slack variables {x K , y K }, the constraint for LU K in (19b) is transformed into the following deterministic form: Then, to handle the non-convex power constraint (11f), we replace the right hand side of (11f) with its linear lower bound When rank(W) ≈ 1, the relationship between w and W is given by the following constraint: As for constraint (27a), since λ max (W) is a convex function of W [16], the left hand side of (27a) is concave, which is the difference between a linear function and a convex function. Hence, we need to construct a convex approximation of constraint (27a). To address this issue, we introduce the following lemma.
Lemma 5 Denote by v max the eigenvector corresponding to the maximum eigenvalue of a matrix V, we have for any Hermitian matrix Z.

Let d (n)
W be the eigenvector corresponding to the maximum eigenvalue of the feasible point W (n) . Then, by using Lemma 5, the surrogate convex constraint of (27a) is given by Now, we consider constraint (28a). By appying the first-order Tayler approximation to ||w|| 2 , we obtain the following convex approximation of the constraint in (28a) as  [13], [19], [20] to relax the unit-modulus constraint (11d). Comparing with the SDR technique, the penalty CCP method is capable of finding a feasible solution to meet constraint (11d).
In particular, the constraint of (11d) can be relaxed by where e [j] m is any feasible solution and b = [b 1 , ..., b 2M ] T is slack vector variable. The proof of (32) can be found in [13], [19]. Following the penalty CCP framework, the subproblem for optimizing e is formulated as

IV. ED MODEL II: PASSIVE EAVESDROPPER MODEL
In this section, we focus on the transmission design for the passive attack, which is more practical and more challenging to address, since the passive ED can hide itself and its CSI is not known [11], [12]. The authors of [15] proposed to exploit the angular information of the ED to combat its passive attack, which is also applicable here. In this section, the cascaded LIL channel H K and the channel H IRS are assumed to be perfect, which is reasonable due to the fact that the pilot information for channel estimation for LUs is known at the BS.

A. Average eavesdropping rate Maximization
The signal received by the ED is formulated as Since the ED is passive, we can only detect the activity of the ED on the line segment between the BS and LU K without knowing its exact location. This detection of a passive attack is based on spectrum sensing [21]. Hence, the average eavesdropping rate is considered which can be computed as follows [15], [22], [23]: With (34), we formulate the following optimization problem: The main challenge to solve Problem (35) is from the average eavesdropping rate containing the integration over D E and the expectation over h E . To address this issue, we use Jensen's inequality to construct an upper bound of R av E (w, e) given by where can be computed via one-dimension integration. According to (2), we define where h E describes the LoS component and is the mean of channel h E . Moreover, R E is a positive semi-definite covariance matrix representing the spatial correlation characteristics of the NLoS component. Therefore, we have h E ∼ CN(h E , R E ) [24], and further get The surrogate functions of R k (z, t k ) for ∀k ∈ K −K are given by R k (z, t k |z (n) , t

B. Proposed Algorithm
k ) given in (14), and those of R K (w, e) and R E (w, e) are given in the following lemma by using the first-order Taylor approximation.
Lemma 6 Let {w (n) , e (n) } be any feasible solution, then R K (w, e) is lower bounded by a concave surrogate function R K w, e|w (n) , e (n) defined by where q (n) ) is upper bounded by a convex surrogate function R E w, e|w (n) , e (n) given by R up E w, e|w (n) , e (n) = where q Furthermore, we have the following proposition.

Proposition 1
The functions { R k , R K , R up E } preserve the first-order property of functions {R k , R K , R E }, respectively. Let's take R K and R K as an example Proof: See Appendix B in [15].
which is convex and can be solved by using CVX.
Giving {z, w, t} and combining (14) Denote the solutions to P1 and P2 as e # 1 and e # 2 , respectively. In addition, let us denote the objective function value of Problem (44) as obj(e), which is a function of e. If obj(e # 1 ) ≥ obj(e # 2 ), then the optimal solution of Problem (44) is given by e # 1 . Otherwise, the optimal solution is e # 2 . The solutions to Problem P1 and Problem P2 are given in the following lemma.

Lemma 7
The optimal solution of P1 is given by where is the price introduced by the price mechanism [25].
The optimal solution of P2 is given by where ̺ opt 2 is the price and Proof: See Appendix C.
Since e # 1 and e # 2 are the globally optimal solutions of P1 and P2, respectively, hence e # is the globally optimal solution of Problem (44). The optimal price parameter can be obtained by using the bisection search method detailed in [25].

V. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we provide numerical results to evaluate the performance of the proposed schemes.
The results are obtained by using a computer with a 1.99 GHz i7-8550U CPU and 16 GB RAM.
The simulated system setup is measured with polar coordinates: The BS is located at (0 m, 0 m) and the IRS is placed at (50 m, 0 m) with elevation angle φ = 2π 3 ; K LUs are randomly and uniformly distributed in an area with D k ∼ U(20 m, 40 m) and θ k ∼ U(− π 2 , π 2 ) for ∀k ∈ K, where U is the uniform distribution. The ED is located at (D E , θ K ) with D E ∈ (0, D K ). The pathloss at the distance of 1 m is −30 dB, the pathloss exponents are set to α BS = α IRS = 2.2 and the Rician 20 factor is 5. The transmit power budget at the BS is P max = 30 dBm and the noise powers are For the statistical CSI error model, the variances of 1) and δ E ∈ [0, 1) measure the relative amount of CSI uncertainties. In addition, the outage probability of secrecy rate is ρ = 0.05.

A. Robust secrecy rate in ED Model I
In order to verify the performance of the proposed outage constrained beamforming in the AAUC, the case of N = 8 and K = 5 is simulated. For comparison, we also consider the "Non-robust" as the benchmark scheme, in which the estimated cascaded LIL and LIE channels are naively regarded as perfect channels, resulting in the following problem where R K (e) = 1 2 log 2 (1 + |e H H K w| 2 /σ 2 K ) and R E (e) = 1 2 log 2 (1 + |e H H E w| 2 /σ 2 E ). Problem (50) can be solved by using the proposed low complexity algorithm used to solve Problem (35). Fig. 3 investigates the feasibility rate and the maximum secrecy rate versus the distance of the ED, in which the coordinate of X-axis is set to the ratio of D E /D K . The feasibility rate is defined as the ratio of the number of channel realizations that have a feasible solution to the outage constrained problem of (11) to the total number of channel realizations. It is observed from Fig. 3(a) that the closer the ED is to LU K, the lower the feasibility rate will be, which means that the location of the eavesdropper imposes a great threat to the security system. From Fig. 3(b), we can see that the secrecy rate drops fast when the ED approaches LU K, and this secrecy rate reduces to zero when the channel error is large. At this situation, the whole system is no longer suitable for secure communication.
Next, the performance versus the size of the IRS, i.e., M, is verified in Fig. 4. Assume that the ED is located at D E /D K = 0.5. In Fig. 4(a), the feasibility rate decreases rapidly with M when the level of channel uncertainty is high. In Fig. 4(b), the case of δ K = δ E = 0 is regarded as the perfect cascaded CSI case, and its maximum secrecy rate increases with M, which is consistent with that of Fig. 6 in [14]. The performance of δ K = δ E = 0 can be used as the performance upper bound of the proposed outage constrained beamforming method. The maximum secrecy rate under small channel uncertainty levels, e.g., (δ K = {0.1, 0.2}, δ E = 0) or (δ K = 0, δ E = 0.1), also increases with M. In addition, it is observed that black lines of δ E = 0 are higher than blue lines of δ K = 0, which means that the negative impact of cascaded LIL channel error on secrecy rate is smaller than that of the cascaded LIE channel error. Furthermore, when δ E increases to 0.1 or larger, secrecy rate starts to decrease at large M. The reason is that increasing M can not only enhance the secrecy rate due to its increased beamforming gain, but also increase the cascaded channel estimation error.
Therefore, at small channel uncertainty level, especially that of the ED, the benefits brought by the increased M outweighs its drawbacks, and vice versa. This observation provides useful guidance to carefully chose the size of the IRS according to the level of the channel uncertainty.

B. Average secrecy rate in ED Model II
This subsection evaluates the performance of the proposed scheme under the passive attack. In order to evaluate the performance of the proposed low complexity algorithm, we consider two benchmark algorithms which are given by: 1) The "SOCP" based scheme, i.e., the CVX is used to solve the SOCP version of Problem (43). 2) The "Random" phase scheme, in which the phases of the reflection elements are randomly generated.
Under the case of N = 8 and K = 5, the average secrecy rate versus M is shown in Fig. 5(a). It can be seen that the performance of the proposed algorithm is almost the same as that of the SOCP algorithm, and both of them outperform the scheme with random phases. Moreover, increasing the size of the IRS can significantly increase the average secrecy rate of the system. Accordingly, Fig.   5(b) describes the CPU time required for these three algorithms. The simulation setup is the same as that of Fig. 5(a). It is observed from Fig. 5(b) that the proposed algorthm with closed-form solution requires much less CPU running time than the SOCP scheme. In addition, the CPU running time of the SCOP-based algorithm increases significantly with the size of the IRS, but the proposed algorithm is not sensitive to the size of the IRS. This is because the computational complexity of the SOCP depends on M, while that of the closed-form solution does not.
Finally, Fig. 6 illustrates the performance and the complexity versus the number of users in the case of M = 64. Again, the proposed algorithm can achieve the same performance of the SOCP-based scheme with less CPU running time. However, the average secrecy rate is saturated when the number of users is greater than 7.

VI. CONCLUSIONS
In this work, we have proposed a two-phase IRS-aided communication system to realize the secure communication under the active attacks and passive eavesdropping. In order to address the cascaded channel error caused by the active attacks, we maximize the secrecy rate subject to secrecy rate outage probability constraints, which has been tackled by using the Bernstein-Type inequality. For the case of the partial CSI of the ED, average secrecy rate maximization problem was considered, which is addressed by the proposed low-complexity algorithm. It was shown that the negative effect of the ED's channel error is greater than that of the LU. In addition, the number of elements on the IRS has a negative impact on system performance when the channel error is large. This conclusion provides an engineering insight for the careful selection of the number of elements at the IRS.
APPENDIX A THE PROOF OF LEMMA 2 To begin, we introduce the following lemma.
Lemma 8 Let x ≥ 0 be a positive real number, and consider the function g 1 (a, x) = −ax + ln a + 1, By appying Lemma 8, we can construct an upper bound of rate R E (w, e) as where the equality (a) is from Lemma 8.
Hence, the proof is completed.

APPENDIX B THE PROOF OF LEMMA 3
To prove Lemma 3, we first introduce the following lemma.
By appying Lemma 9, we can construct a lower bound of rate R K (w, e) as = −a K min v g 2 (v, e H H K w) + ln a K + 1 2 ln 2 = a K max v −g 2 (v, e H H K w) + ln a K + 1 2 ln 2 ≥ a K −g 2 (v, e H H K w) + ln a K + 1 2 ln 2 , for any v ≥ 0 = 1 2 ln 2 (−a K |v| 2 |e H H K w| 2 −σ 2 K a K |v| 2 +2a K Re ve H H K w − a K + ln a K + 1) where Equality (a) is from Lemma 8, and Equality (b) is from Lemma 9.
Hence, the proof is completed.
APPENDIX C THE PROOF OF LEMMA 7 To begin with, we solve P1: P1 is equivalent to min e e H A E e (53a) e H A K e ≥ e 2R − 1. (53c) Step 1: Construct a surrogate problem: Under the MM algorithm framework [27], we have the following lemma.
Lemma 10 [28], [29] Given A A 0 and x, then quadratic function x H A 0 x is majorized by x H Ax− at feasible point e (n) . To deal with the non-convex constraint (53c), we replace e H A K e with its linear lower bound, resulting in the following equivalent constraint (53c) ⇒ 2Re{e (n),H A K e} ≥ e 2R − 1 + e (n),H A K e (n) .
The optimal ̺ opt 1 is determined by using the bisection search method, the detailed information about which can be found in [25].  (58) Step 2: Closed-form solution: By omitting the constant, Problem (57) is then equivalent to where ̺ 2 is a nonnegative price. The globally optimal solution is given by e # 2 (̺ opt 2 ) = exp{j arg(c + ̺ 2 (λ max (A K )I−A K )e (n) )} where the optimal ̺ opt 2 is determined by using the bisection search method. Hence, the proof is completed.