# Covert Communications in Active-RIS-Aided NOMA Systems: Element Grouping or Not?
## 2 System Model
### w-EG Scheme  
In this section, we investigate the impacts of the w-EG scheme on the covert communication performance of an ARIS-aided NOMA system. For the w-EG scheme,  
Figure 1: The ARIS-aided uplink NOMA system model. is the amplitude-phase-shift coefficient matrix of the ARIS with the reflecting matrix \(\mathbf{\Theta}_{r}=\mathbf{M}_{r}\mathbf{\Theta}_{r}\) and jamming matrix \(\mathbf{\Theta}_{j}=\mathbf{M}_{j}\mathbf{\hat{\Theta}}_{j}\). Specifically, \(\mathbf{M}_{r}=\text{diag}([\hat{\alpha}_{1},\hat{\alpha}_{2},\cdots,\hat{\alpha }_{N}]^{T})\) and \(\mathbf{M}_{j}=\text{diag}([\hat{\alpha}_{1},\hat{\alpha}_{2},\cdots,\hat{ \alpha}_{N}]^{T})\) are the mode selection matrices with \(\hat{\alpha}_{n},\hat{\alpha}_{n}\in\{0,1\}\), \(\hat{\alpha}_{n}+\hat{\alpha}_{n}=1\), and \(n\in\mathcal{N}\triangleq\{1,2,\cdots,N\}\). With the aid of the mode selection matrices, each element of the ARIS works in either the reflection mode or the jamming mode. The matrices \(\mathbf{\hat{\Theta}}_{r}=\text{diag}([\hat{\beta}_{1}e^{i\hat{\mathbf{\beta}}_{1}}, \hat{\beta}_{2}e^{i\hat{\mathbf{\beta}}_{2}},\cdots,\hat{\beta}_{N}e^{i\hat{\mathbf{ \beta}}_{N}}]^{T})\) and \(\mathbf{\hat{\Theta}}_{j}=\text{diag}([\hat{\beta}_{1}e^{i\hat{\mathbf{\beta}}_{1}}, \hat{\beta}_{2}e^{i\hat{\mathbf{\beta}}_{2}},\cdots,\hat{\beta}_{N}e^{i\hat{\mathbf{ \beta}}_{N}}]^{T})\) contain the reflection coefficients for enhancing the NOMA transmission quality and deteriorating Willie's detection performance, respectively, where \(\tilde{\beta}_{n},\tilde{\beta}_{n}\in[0,\eta_{n}]\) and \(\tilde{\phi}_{n},\tilde{\phi}_{n}\in[0,2\pi)\) denote the \(n\)th element's reflecting amplitude and phase shift, respectively, with \(\eta_{n}>1\) denoting the allowed maximum reflecting amplitude. At Bob, the received signal can be expressed as:  
\[\mathbf{y}_{b} =\sqrt{P_{a}}x_{a}(\mathbf{g}_{ab}+\mathbf{z}_{ab})+\sqrt{P_{c}} x_{c}(\mathbf{g}_{cb}+\mathbf{z}_{cb})\] \[+\mathbf{H}_{rb}\mathbf{\Theta}_{r}+\mathbf{n}_{b}, \tag{4}\]  
where \(\mathbf{g}_{ib}=\mathbf{h}_{ib}+\mathbf{H}_{rb}\mathbf{\Theta}_{r}\mathbf{h}_{ir}\) is the composite transmission channel from node \(i\) (\(i\in\{a,c\}\)) to Bob, \(\mathbf{z}_{ib}=\mathbf{H}_{rb}\mathbf{\Theta}_{j}\mathbf{h}_{ir}\) is the composite jamming channel from node \(i\) (\(i\in\{a,c\}\)) to Bob, and \(\mathbf{n}_{b}\sim\mathcal{CN}(\mathbf{0}_{M},\sigma_{b}^{2}\mathbf{I}_{M})\) is additive noise at Bob with zero mean and variance \(\sigma_{b}^{2}\). Then, a beamformer \(\mathbf{w}\in\mathbb{C}^{M\times 1}\) is applied at Bob to obtain \(y_{b}=\mathbf{w}^{H}\mathbf{y}_{b}\) for further processing. In the considered ARIS-aided uplink NOMA system, the user ordering is conducted based on the channel gains [29; 40]. Specifically, the two users, [PERSON] and [PERSON], are paired based on their channel gains. At Bob, to guarantee the successful successive interference cancellation (SIC), the signal of the user with the stronger channel gain is decoded first. Then, the signal of the user with the weaker channel gain is decoded at the last stage of the SIC processing. To facilitate SIC at Bob, we assume that \(\|\mathbf{g}_{cb}\|^{2}>\|\mathbf{g}_{ab}\|^{2}\) is guaranteed by not only \(\|\mathbf{h}_{cb}\|^{2}\geq\|\mathbf{h}_{ab}\|^{2}\), but also the amplitude-phase-shift coefficient matrix \(\mathbf{\Theta}\) as optimized in the next section. Consequently, \(x_{a}\) is decoded at the last stage of SIC without suffering from inter-user interference.