# Covert Communications in Active-RIS-Aided NOMA Systems: Element Grouping or Not?
## 2 System Model  
The considered uplink NOMA system consists of a multi-antenna BS ([PERSON]), a pair of NOMA users ([PERSON] and [PERSON]), and a warden ([PERSON]), as shown in Figure 1. We assume that [PERSON] is a covert user who has high-level of privacy and security requirements and [PERSON] is a public user who only has transmission reliability requirements. For instance, [PERSON] is a law enforcement node needing to hide its transmission pattern, while [PERSON] transmits live streams possessing low or no confidential requirements. The ARIS is deployed between the NOMA users and [PERSON] to not only aid the uplink transmissions from [PERSON] and [PERSON], but also reflect jamming signals towards [PERSON] to enhance the covert communication performance. In the considered system, the ARIS consists of \(N\) reflecting elements, [PERSON] is equipped with \(M\) receive antennas, and the remaining nodes ([PERSON], [PERSON], and [PERSON]) are each equipped with a single antenna. All the links associated with the ARIS are modeled to experience Rician fading, while all the direct links from the NOMA users ([PERSON] and [PERSON]) to [PERSON] and [PERSON] are modeled to experience Rayleigh fading. We assume that all the links follow quasi-statistic block fading. By exploiting the existing channel estimation methods for RIS-aided systems [37; 38; 39], [PERSON] and [PERSON] acquire the channel state information (CSI) associated with the ARIS and that of the direct links. To characterize the performance limit of the considered system, we further ignore the channel estimation error and assume that [PERSON] has perfect CSI of [PERSON] and [PERSON] [37; 38; 39; 29; 31].  
Let \(\mathbf{h}_{ib}\in\mathbb{C}^{M\times 1}\) and \(h_{iv}\in\mathbb{C}\) denote the direct links from node \(i\) to [PERSON] and [PERSON], respectively, with \(i\in\{a,c\}\) representing [PERSON] and [PERSON], respectively. Similarly, the channels from node \(i\) to the ARIS are denoted by \(\mathbf{h}_{ir}\in\mathbb{C}^{N\times 1}\), and the channels from the ARIS to [PERSON] and [PERSON] are denoted by \(\mathbf{H}_{rb}\in\mathbb{C}^{M\times N}\) and \(\mathbf{h}_{rv}^{H}\in\mathbb{C}^{1\times N}\), respectively. If [PERSON] and [PERSON] transmit the information signals simultaneously in a transmission block, the arrived signal at the ARIS can be written as:  
\[\mathbf{y}_{r}=\sqrt{P_{a}}\mathbf{h}_{ar}x_{a}+\sqrt{P_{c}}\mathbf{h}_{cr}x_ {c}+\mathbf{n}_{r}, \tag{1}\]  
where \(x_{i}\sim\mathcal{CN}(0,1)\) is the signal transmitted by node \(i\) with \(i\in\{a,c\}\), \(P_{i}\) is the transmit power of node \(i\) satisfying \(0\leq P_{i}\leq P_{i}^{\text{max}}\) with \(P_{i}^{\text{max}}\) denoting the maximum transmit power, and \(\mathbf{n}_{r}\sim\mathcal{CN}(\mathbf{0}_{N},\sigma_{r}^{2}\mathbf{I}_{N})\) is additive noise at the ARIS with zero mean and variance \(\sigma_{r}^{2}\). After the amplitude and phase-shift adjustments, the departure signal from the ARIS can be expressed as:  
\[\mathbf{\hat{y}}_{r}=\sqrt{P_{a}}\mathbf{\Theta}\mathbf{h}_{ar}x_{a}+\sqrt{P_ {c}}\mathbf{\Theta}\mathbf{h}_{cr}x_{c}+\mathbf{\[PERSON]}\mathbf{n}_{r}, \tag{2}\]  
where \(\mathbf{\[PERSON]}\) is the amplitude-phase-shift coefficient matrix of the ARIS.  
The power of the departure signal at the ARIS is constrained by  
\[P_{a}\|\mathbf{\Theta}\mathbf{h}_{ar}\|^{2}+P_{c}\|\mathbf{\Theta}\mathbf{h}_ {cr}\|^{2}+\|\mathbf{\Theta}\|_{F}^{2}\sigma_{r}^{2}\leq P_{r}^{\text{max}}, \tag{3}\]  
where \(P_{r}^{\text{max}}=P_{\text{ris}}-NP_{\text{circuit}}\) is the maximum power budget at the ARIS with \(P_{\text{ris}}\) and \(P_{\text{circuit}}\) denoting the total power of the ARIS and the power consumption per circuit element, respectively [34]. In the rest of this section, we present two ARIS-aided transmission schemes in detail, namely, the w-EG scheme and the w/o-EG scheme. Each scheme is discussed separately in the following two subsections.