# Covert Communications in Active-RIS-Aided NOMA Systems: Element Grouping or Not?
## 2 System Model
### w-EG Scheme
The achievable rates of decoding of \(x_{a}\) and \(x_{c}\) are given by \(R_{a}=\log_{2}(1+\gamma_{a})\) and \(R_{c}=\log_{2}(1+\gamma_{c})\), respectively, where \(\gamma_{a}\) and \(\gamma_{c}\) are the received SINRs given by \(\gamma_{a}=\frac{P_{a}|\mathbf{w}^{H}\mathbf{g}_{ab}|^{2}}{P_{a}|\mathbf{w}^{ H}\mathbf{z}_{ab}|^{2}+\|\mathbf{w}^{H}\mathbf{H}_{rb}\mathbf{\Theta}\|^{2} \sigma_{r}^{2}+\sigma_{b}^{2}}\) and \(\gamma_{c}=\frac{P_{c}|\mathbf{w}^{H}\mathbf{g}_{cb}|^{2}}{P_{a}|\mathbf{w}^{ H}(\mathbf{g}_{ab}+\mathbf{z}_{ab})|^{2}+P_{c}|\mathbf{w}^{H}\mathbf{z}_{ab}|^{2}+ \|\mathbf{w}^{H}\mathbf{H}_{rb}\mathbf{\[PERSON]}\|^{2}\sigma_{r}^{2}+\sigma_{b}^{2}}\), respectively.  
In order to judge whether [PERSON] transmits an information signal to [PERSON] or not, [PERSON] performs a binary decision according to the Neyman-Pearson criterion [41], which results in two hypotheses as follows:  
\[\mathcal{H}_{0}:\ y_{w} =\sqrt{P_{c}}x_{c}(g_{cw}+z_{cw})+\mathbf{h}_{rw}^{H}\mathbf{\Theta} \mathbf{n}_{r}+n_{w}, \tag{5}\]  
\[\mathcal{H}_{1}:\ y_{w} =\sqrt{P_{a}}(g_{aw}+z_{aw})+\sqrt{P_{c}}(g_{cw}+z_{cw})\] \[+\mathbf{h}_{rw}^{H}\mathbf{\Theta}\mathbf{n}_{r}+n_{w}, \tag{6}\]  
where \(g_{iw}=h_{iw}+\mathbf{h}_{rw}^{H}\mathbf{\Theta}_{i}\mathbf{h}_{ir}\) with \(i\in\{a,c\}\), \(z_{iw}=\mathbf{h}_{rw}^{H}\mathbf{\Theta}_{j}\mathbf{h}_{ir}\) with \(i\in\{a,c\}\), and \(n_{w}\sim\mathcal{CN}(0,\sigma_{w}^{2})\) denotes additive noise at Willie with zero mean and variance \(\sigma_{w}^{2}\). The likelihood functions of the received signals under \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\) can be written as \(p_{0}(y_{w})=e^{-|y_{w}|^{2}/\sigma_{0}^{2}}/\pi\sigma_{0}^{2}\) and \(p_{1}(y_{w})=e^{-|y_{w}|^{2}/\sigma_{1}^{2}}/\pi\sigma_{1}^{2}\), respectively, with \(\sigma_{0}^{2}=P_{c}|g_{cw}+z_{cw}|^{2}+\|\mathbf{h}_{rw}^{H}\mathbf{\[PERSON]}\|^{2} \sigma_{r}^{2}+\sigma_{w}^{2}\) and \(\sigma_{1}^{2}=P_{a}|g_{aw}+z_{aw}|^{2}+P_{c}|g_{cw}+z_{aw}|^{2}+\|\mathbf{h}_{ rw}^{H}\mathbf{\[PERSON]}\|^{2}\sigma_{r}^{2}+\sigma_{w}^{2}\). Under the Neyman-Pearson criterion, [PERSON] makes the likelihood ratio test \(\frac{p_{1}(y_{w})}{p_{0}(y_{w})}\underset{\mathcal{D}_{1}}{\geq 1}\) to minimize the detection error probability (DEP), where \(\mathcal{D}_{0}\) and \(\mathcal{D}_{1}\) denote the binary decisions endorsing \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\), respectively. In practice, [PERSON] utilizes its average received power \(P_{w}\overset{\bar{M}_{0}\rightarrow\infty}{=}\frac{1}{\bar{M}}\sum_{m=1}^{M}|y _{w}(m)|^{2}\) as the decision statistic with \(\bar{m}\) denoting the transmission block index. Then, the optimal decision rule for [PERSON] can be rewritten as \(P_{w}\gtrless\phi^{*}\), where \(\phi^{*}=\frac{\sigma_{0}^{2}\sigma_{1}^{2}}{\sigma_{1}^{2}-\sigma_{0}^{2}}\text {In}\frac{\sigma_{1}^{2}}{\sigma_{0}^{2}}\) is the optimal detection threshold for [PERSON]'s decision, which results in the following minimum DEP [41]:  
\[\xi^{*} = \text{Pr}(\mathcal{D}_{1}|\mathcal{H}_{0})+\text{Pr}(\mathcal{D} _{0}|\mathcal{H}_{1}) \tag{7}\] \[= \text{Pr}(P_{w}\geq\phi^{*}|\mathcal{H}_{0})+\text{Pr}(P_{w}\leq \phi^{*}|\mathcal{H}_{1})\] \[= 1+\left(\frac{\sigma_{1}^{2}}{\sigma_{0}^{2}}\right)^{-\frac{ \sigma_{1}^{2}}{\sigma_{1}^{2}-\sigma_{0}^{2}}}-\left(\frac{\sigma_{1}^{2}}{ \sigma_{0}^{2}}\right)^{-\frac{\sigma_{0}^{2}}{\sigma_{1}^{2}-\sigma_{0}^{2}}},\]  
where \(\text{Pr}(\mathcal{D}_{1}|\mathcal{H}_{0})\) and \(\text{Pr}(\mathcal{D}_{0}|\mathcal{H}_{1})\) denote the false alarm and missed detection probabilities, respectively.