Securing Reconfigurable Intelligent Surface-Aided Cell-Free Networks

Recently, there have been several innovative works considering the security performance of the cell-free networks [17-18]. Specifically, [17] studied the achievable secrecy rates and compared that with the co-located massive MIMO networks, where the active eavesdropper (Eve) was assumed. The authors in [18] considered the spatially correlated Rayleigh fading channels with active Eve, and proposed a distributed conjugate beamforming (BF) technique to obtain the minimum mean square error (MMSE). Furthermore, their research verified the correlation of ergodic secrecy rate with active Eve and channel fading conditions.

Meanwhile, RIS-aided secure issues have also received great attention recently [19-23]. Specifically, in [19], the authors considered a RIS-aided multiple-input single-output (MISO) system, in which an Eve eavesdrops the confidential message from the access point (AP) to a specific user. The authors then proposed a joint AP active BF and RIS passive BF scheme to maximize the secrecy rate of all users. Under the same system model, the authors in [20] considered AP-RIS as a rank one channel, and proposed a projection gradient descent algorithm to maximize the secrecy rate. To further enhance the PLS performance, artificial noise was introduced in [21] to deliberately destroy Eve’s channel. In [22], RIS-aided multi-user two-way communication secure system was considered, where one user signal was used to interfere with Eve wiretapping other users’ information. Considering the practical dynamic environment, a novel deep reinforcement learning (DRL)-based secure BF design method was proposed to guarantee the security by dynamically adjusting to the beam according the environment [23].

However, most of the recent related works only consider the single BS in RIS communications, without touching the cooperative secure BF problem. It is of significance to investigate RIS-aided cell-free networks, in order to further boost the network capacity or lower energy consumption. Nonetheless, the joint BF design at the BSs and RISs is much more challenging for the multi-BS and multi-RIS systems. In this paper, we aim to address this by studying the secrecy transmission problem in a RIS-based cell-free networks. The main contributions are summarized as follows:

• $\bullet$

  We construct a RIS-based cell-free network under the presence of an Eve, where all RISs and BSs rely on the central processing unit (CPU) to cooperatively serve all users at the same time. Based on this, we formulate a maximum weight sum secrecy rate (WSSR) problem by jointly optimizing the active cooperating BF at BSs and passive cooperating BF at RISs.

• $\bullet$

  To address the formulated non-trivial WSSR maximization problem, we propose a joint optimization framework based on all known channel state information (CSI). Specifically, we decompose the original problem into two sub-problems, and then each sub-problem is transformed into a solvable convex problem through semi-definite relaxation (SDR) and successive convex approximation (SCA) techniques. Finally, the solution of each sub-problem is updated though the proposed alternating optimization (AO) method until convergence.

• $\bullet$

  Due to the high-dimensional channel maxtrix of RISs, the channel estimation is a great challenge. In addition, it is also unrealistic to continuously acquire all RIS CSI at each small coherence time. To overcome this issue, we propose an extension of the joint optimization framework based on assigning RIS. Specifically, we first obtain all CSI once at the beginning of each large coherence time, and then introduce the matching factor of user and RIS to transform the original optimization problem into a mixed-integer non-linear programming (MINP) one. Next, the upper bound of the objective function is obtained by applying the SCA method, while the linear conic relaxation (LCR) method is applied to relax zero-one constraint for obtaining a suitable matching factor. Based on the matching factor, the maximum WSSR is obtained by jointly optimizing the BF of the BSs and the RISs in each small coherence time.

The rest of this paper is organized as follows: Section II proposes a downlink RIS-aided cell-free secure system model, and formulates a joint optimization of the maximization WSSR problem. A joint optimization framework is proposed in Section III, and the extension framework of the assigning RIS scheme is proposed in Section IV. Section V gives a supplement to the framework. Section VII provides simulation results to verify the security performance of the above frameworks. Section VIII concludes the paper.

Notation: In this paper, boldface letters indicate matrices or vectors. $\mathbb{C}^{M\times N}$ denotes a complex-valued matrix of size $M\times N$. $\mathbf{A}^{T}$, $\mathbf{A}^{H}$, Tr$(\mathbf{A})$ respectively denote transpose, conjugate transpose, and trace of matrix $\mathbf{A}$. For a complex value $x$, $\angle\left(x\right)$ represents its phase, $\left|x\right|$ represents its modulus, and $\Re(x)$ is its real part. For a vector $\mathbf{a}$, ${\left[{\bf{a}}\right]_{i}}$ represents its $i$-th element, $\left\|{\bf{a}}\right\|$ represents its Euclidean norm, and ${\rm{diag}}\left(x\right)$ represents the diagonal operation. For a matrix ${\bf{A}}$, ${{\bf{A}}_{i,j}}$ is the element in its $i$-th row and $j$-th column, and ${\bf{A}}\succeq 0$ represents the semi-definite matrix. ${{\bf{1}}_{L}}$ is a vector of length $L$ with all elements 1, ${{\bf{0}}_{L\times M}}$ represents an all-zero element matrix of size $L\times M$, ln$(\cdot)$ denotes the natural logarithm, and ${\nabla_{x}}$ is the gradient of $x$.

As shown in Fig. 1, in the PLS-based RIS-aided cell-free network we assume that there are $B$ BSs, $R$ RISs, and $K$ users, which are randomly deployed in a given area. The BSs are connected to the CPU through backhaul links [24] and RISs are wired or wirelessly connected by the CPU or BSs [25]. For simplification, we assume that there exits a single Eve for wiretapping all users’ information. In addition, the BSs and RIS are equipped with $M$ and $N$ antennas, respectively, while the users and Eve are equipped with single antenna, respectively. Let ${\cal B}=\left\{{1,\ldots,B}\right\}$, ${\cal R}=\left\{{1,\ldots,R}\right\}$, ${\cal N}=\left\{{1,\ldots,N}\right\}$ and ${\cal K}=\left\{{1,\ldots,K}\right\}$ denote the index of BSs, RISs, RIS elements and users, respectively.

Without loss of generality, we assume that there exist both direct links (from BSs to users/Eve) and reflect links (from BSs to RISs and from RISs to users/Eve). The direct channels from the $b$-th BS to the $k$-user and Eve are denoted as ${\mathbf{H}}_{b,k}\in{{\mathbb{C}}^{M\times{1}}}$ and ${\mathbf{H}}_{b,e}\in{{\mathbb{C}}^{M\times{1}}}$, respectively. ${\mathbf{G}}_{b,r}\in{{\mathbb{C}}^{N\times{M}}}$, ${\mathbf{F}}_{r,k}\in{{\mathbb{C}}^{N\times{1}}}$ and ${\mathbf{F}}_{r,e}\in{{\mathbb{C}}^{N\times{1}}}$ respectively represent link channels from the $b$-th BS to the $r$-th RIS, from the $r$-th RIS to the $k$-th user, and from the $r$-th RIS to the Eve. To characterize the performance of the considered RIS-aided cell-free secrecy network, all CSI is assumed available via the advanced channel estimation techniques. Here, we assume that all legitimate users and Eve do not collude [26].

The reflection phase matrix of the $r$-th IRS is represented by ${{\bf{\Theta}}_{r}}={\rm{diag(}}{\theta_{r{\rm{,}}1}},\ldots,{\theta_{r{\rm{,}}N}})\in{{\mathbb{C}}^{N\times{N}}}$, where ${\theta_{r{\rm{,}}n}}$ is the reflection phase of the $n$-th reflection element of the $r$-th IRS. Each reflecting unit of RISs adjusts the phase of the incident signal to suppress the Eve’s reception and enhance the users’ reception. Consider that the signal power reflected by RISs twice or more is much lower than the channel reflected by RISs once [27], [28], they are ignored here. We further assume ideal RIS here, i.e., each element ${\theta_{r{\rm{,}}n}}$ can be independently and continuously regulated [29], i.e.,

where $\forall r\in{\cal R}$, $\forall n\in{\cal N}$. To maximize the utility of the RISs, suppose ${\kappa_{r,n}}=1$. Note that it is unrealistic to achieve continuous adjustment of the reflection element phase in practice [30], [31], so we will refer to the low-resolution discrete phase shift in Section V.

Let $s_{k}$ denote the transmit signal for the $k$-user, where ${\mathbb{E}}\left\{{{{\left|{{s_{k}}}\right|}^{2}}}\right\}=1$. The transmit signals by the $b$-th BS can be expressed as

where ${{\bf{w}}_{b,k}}\in{{\mathbb{C}}^{M\times{1}}}$ represents the precoding vector for the $k$-user. Thus, received signal of the $k$-th user can be expressed as

Thus, the achievable secrecy rate from BSs to the $k$-th user can be expressed as

where ${\left[z\right]^{+}}=\max\left({z,0}\right)$.

In this paper, we aim to maximize the WSSR by jointly optimizing the BF vector ${\bf{w}}_{k}$ and the phase shift $\boldsymbol{\mu}$. Mathematically, the optimization problem can be formulated as follows

where ${\eta_{k}}\left(0\leq{\eta_{k}}\leq 1\right)$ represents the weight of the $k$-th user, ${P_{b}}$ represents the maximum available transmit power of the $b$-th BS, and ${\bf{W}}$ is defined as ${\bf{W}}{\rm{=}}{\left[{{\bf{w}}_{1}^{T},\ldots,{\bf{w}}_{K}^{T}}\right]^{T}}$. Note that in (9a) we omit ${\left[\cdot\right]^{+}}$, because the optimal value should be non-negative.

Due to the coupling of ${\bf{W}}$ and $\boldsymbol{\mu}$ in the objective function (9a) and the unit modulus constraint of $C2$, it is difficult to solve problem ${\cal P}^{o}$. Here we take the AO method to decouple variables ${\bf{W}}$ and $\boldsymbol{\mu}$, and decompose the original problem ${\cal P}^{o}$ into two sub-problems. We then fix other variables in each iteration of the sub-problems and solve each convex problem using SDR and SCA methods. The detail will be given in the following section.

Firstly, we fix the phase shift variable $\boldsymbol{\mu}$ and solve the BF matrix ${\bf{W}}$ in the original problem ${\cal P}^{o}$. Here, we give the following two inequalities near the fixed point $\left\{{\hat{a},\hat{b}}\right\}$:

Based on the above, we have the following inferences. Firstly, the information rate of the $k$-th user with fixed point $\left\{{{\bf{w}}_{k}^{t},{\boldsymbol{\mu}^{t}}}\right\}$ can be approximated as

where

Then, the rate of negative weight of the Eve eavesdropping the $k$-th user’s information can be written as

The first term of the (13) can be approximated as

where

In (15), ${\boldsymbol{\xi}_{e,k}}$ is calculated as ${\boldsymbol{\xi}_{e,k}}={\tilde{\mathbf{h}}}_{e,k}^{H}{\boldsymbol{\mu}^{t}}{\left({{\boldsymbol{\mu}^{t}}}\right)^{H}}{{\tilde{\mathbf{h}}}_{e,k}}$. The second term on the right side of (13) can be approximately as

where

Finally, based on the above inferences, the WSSR maximization problem ${{\cal P}^{o}}$ can be reformulated as the following sub-problem ${{\cal P}^{1}}$:

Here, ${{\cal P}^{1}}$ is a convex problem, which can be solved by standard convex optimization techniques or CVX toolbox.

Similarly, by fixing the BF matrix ${\bf{W}}$, we can rewrite problem ${{\cal P}^{o}}$ as a sub-problem w.r.t. ${\boldsymbol{\mu}}$ around the fixed point $\left\{{{\bf{w}}_{k}^{t},{\boldsymbol{\mu}^{t}}}\right\}$ as follows:

where the calculation of ${\alpha_{k}^{t}}$, ${\beta_{k}^{t}}$ and $\chi_{k}^{t}$ is the same as (12b), (12d) and (17b) in the previous sub-section. Besides,

For $\ln\left({1+\sum\limits_{j=1,j\neq k}^{K}{{{\left|{{{\left({{\boldsymbol{\mu}^{t}}}\right)}^{H}}{{\tilde{\mathbf{h}}}_{e,k}}{{{\bf{w}}_{j}^{t}}}}\right|}^{2}}}}\right)$ in (19a), it can also be approximated as

where ${{\bf{\Psi}}_{k}}{\bf{\Psi}}_{k}^{H}=\sum\limits_{j=1,j\neq k}^{K}{{{\tilde{\mathbf{h}}}_{e,k}}{\bf{w}}_{j}^{t}{{({\bf{w}}_{j}^{t})}^{H}}{\tilde{\mathbf{h}}}_{e,k}^{H}}$. Omitting the constant term, we can rewrite the problem ${{\cal P}^{2}}$ as

For brevity, we denote $\boldsymbol{\varpi}=\sum\limits_{j=1}^{K}{{\bf{w}}_{j}^{t}{{({\bf{w}}_{j}^{t})}^{H}}}$, and then ${{\beta_{k}}+{{\left|{{\alpha_{k}}}\right|}^{2}}}={\boldsymbol{\mu}^{H}}{{\tilde{\mathbf{h}}}_{k}}\boldsymbol{\varpi}{\tilde{\mathbf{h}}}_{k}^{H}\boldsymbol{\mu}+1$, ${\chi_{k}}={\boldsymbol{\mu}^{H}}{{\tilde{\mathbf{h}}}_{e,k}}\boldsymbol{\varpi}{\tilde{\mathbf{h}}}_{e,k}^{H}\boldsymbol{\mu}$. Next, we denote

Hence, we can simplify problem ${{\cal P}^{3}}$ as follows:

where ${\bf{A}}=\sum\limits_{k=1}^{K}{{\eta_{k}}{{\bf{A}}_{k}}}$, and ${\bf{v}}=\sum\limits_{k=1}^{K}{{\eta_{k}}}{{\bf{v}}_{k}}$.

The simplified sub-problem ${{\cal P}^{4}}$ is a quadratic programming (QP) problem with unit modulus constraint. Next, we will employ the alternating direction method of multipliers (ADMM) technology to solve it. Firstly, we introduce a slack variable $\mathbf{p}\in{{\mathbb{C}}^{NR+1}}$, and the problem ${{\cal P}^{4}}$ can be rewritten as

The augmented Lagrangian equation of problem ${{\cal P}^{5}}$ can be expressed as

where $\boldsymbol{\lambda}\in{{\mathbb{C}}^{NR+1}}$ and ${\delta}\geq 0$ represent Lagrangian multipliers. We put the superscript $t$ of each variable to represent the index of the iteration, and then the process of each iteration is as follows:

1) Optimizing ${\mathbf{p}}$: Here, the first-order optimization solves the gradient $\partial{\mathbf{{\cal G}}}/\partial{\boldsymbol{p}}$ as follows:

Let (28) be equal to zero, we can obtain

2) Optimizing ${\boldsymbol{\mu}}$: The solution of (27b) is similar to the above, but with the constraint of (25c). Its solution is equivalent to

i.e.,

where $\forall n\in\left\{{1,\ldots,NR}\right\}$, and ${\left|{{\boldsymbol{\mu}^{t+1}}}\right|_{NR+1}}=1$.

3) Optimizing ${\boldsymbol{\lambda}}$: From (28), we can solve (27c) as

The above three sub-problems can be solved in closed form. Finally, to make the ADMM algorithm converge, we have the following lemma [30].

Lemma 1: The above ADMM algorithm converges, no matter whether $\boldsymbol{\mu}$ belongs to a continuous set, as long as the penalty factor $\delta$ satisfies:

Proof: The certification process can refer to [32].

Here, we briefly summarize the proposed scheme for solving the original problem ${\cal P}^{o}$. First, we solve sub-problem ${\cal P}^{1}$ by fixing $\boldsymbol{\mu}$, and then solve sub-problem ${\cal P}^{5}$ based on obtaining $\mathbf{W}$ from the ${\cal P}^{1}$. For Solving ${\cal P}^{5}$, the ADMM method is applied to alternately iterating (27a), (27b), and (27c) until the result converges. Next, ${\cal P}^{1}$ is re-solved based on obtaining $\boldsymbol{\mu}$ from the ${\cal P}^{5}$, and the above process is repeated until convergence.

In this section, we propose an assigning RIS optimization scheme that balances the security performance and cost as shown in Fig. 2. Specifically, we select RISs with better secure service quality for joint optimization based on the feature that the RIS with poor service quality has less impact on user secrecy rate. At the start of a large coherence time, a full channel estimation is performed and the RIS with better service quality is selected for each user. After that, in each repeated small coherence time, we only obtain the CSI related to the allocated RIS and perform joint optimization problem. Here we omit the channel estimation of RISs with poor service quality in each small coherence time, which greatly decreases the overhead with a small performance loss. Meanwhile, those RISs that are not screened for user $k$ will not provide services to it. Because the phase shift of the unscreened RIS without joint optimization is actually equivalent to a random phase shift, the impact on the user’s secrecy rate is very limited. The specific performance loss and impact will be shown in detail later in Section VI.

To determine the matching problem among RISs and users, we introduce a matching factor variable $l_{r,k}$ $\left(l\in\{0,1\}\right)$ to indicate whether the $k$-th user and the $r$-th RIS are grouped. When $l_{r,k}$ is $0$, it means that the $r$-th RIS does not serve the $k$-th user and vice versa. For convenience, we define $\textbf{L}_{k}=[{l_{1,k}},\ldots,{l_{R,k}}]^{T}$. To express the matching problem, we define the virtual received signal of the $k$-th user and the Eve eavesdropping on the $k$-th user as follows

The virtual achievable secret rate of user $k$ can be written as

where the virtual SINRs of ${{\hat{\gamma}}_{k}}$ and ${{\hat{\gamma}}_{e,k}}$ are the same as (7a) and (7b), respectively, which will be given in the following sub-section. We introduce a parameter ${R_{{\rm{assign}}}}$ to indicate the maximum number of RISs served by each user, where ${R_{{\rm{assign}}}}\leq R$. Therefore, the maximum WSSR problem with assigning RIS is formulated as

Due to the nonconvex objective function (36a), it is obvious that ${{\cal P}_{assign}}$ is a nonconvex MINP, which is difficult to solve directively. Next, an effective algorithm is proposed to deal with it.

When the variable ${l_{r,k}}$ is fixed, we only need to multiply both ${{\bf{F}}_{r,k}}$ and ${{\bf{F}}_{r,e,k}}$ by a coefficient ${l_{r,k}}$ to replace the original ${{\bf{F}}_{r,k}}$ and ${{\bf{F}}_{r,e,k}}$, and then the problem ${{\cal P}_{assign}}$ can be transformed into the same form as the problem ${{\cal P}^{o}}$ that can be solve by the proposed Algorithm 1 proposed in Section III.

In this section, we mainly focus on the problem of optimizing ${\textbf{{L}}_{k}}$ when ${\bf{W}}$ and $\boldsymbol{\mu}$ are fixed. To transform ${{\cal P}_{assign}}$ into a solvable convex problem, we rewrite the mathematical form of the channel. The reflection channels of the $k$-th user and the Eve eavesdropping on the $k$-th user are denoted as