# Covert Communications in Active-RIS-Aided NOMA Systems: Element Grouping or Not?
## 3 Problem Formulation and Proposed Solutions  
### Problem Formulation  
The aim of the covert communication design is to maximize the covert communication rate \(R_{a}\) and \(\hat{R}_{a}\) for the w-EG and w/o-EG schemes, respectively, by jointly optimizing the NOMA transmit power, ARIS beamforming, and receive beamforming subject to the constraint on the covertness level, quality-of-service (QoS) requirements of Carol, power budget at the ARIS, and SIC processing constraint at Bob. For the w-EG scheme, we formulate the covert communication rate \(R_{a}\)'s maximization problem as follows:  
\[(\mathrm{P1}) :\max_{\hat{p}_{a},P_{c},\mathbf{w},\mathbf{\Theta}}R_{a}\] (15a) s.t. \[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\] (15b) \[\|\mathbf{g}_{cb}\|^{2}>\|\mathbf{g}_{ab}\|^{2},\] (15c) \[R_{c}\geq R_{c}^{\min},\] (15d) \[\tilde{\alpha}_{n}+\tilde{\alpha}_{n}=1,\ \forall n\in\mathcal{N},\] (15e) \[\tilde{\alpha}_{n},\tilde{\alpha}_{n}\in\{0,1\},\ \forall n\in\mathcal{N},\] (15f) \[\tilde{\beta}_{n},\tilde{\beta}_{n}\in[0,\eta_{n}],\ \forall n\in \mathcal{N},\] (15g) \[\|\mathbf{w}\|^{2}=1,\] (15h) (3) and (10). (15i)  
In problem (P1), constraint (15b) limits the maximum transmit power of node \(i\), constraint (15c) guarantees a successful SIC at Bob, constraint (15d) ensures the minimum rate \(R_{c}^{\min}\) of Carol, constraint (15e) and (15f) make each element work in at least one mode, i.e., information signal reflection or jamming signal reflection, constraint (15g) sets the maximum signal amplification for each element, (15h) sets the unit power constraint on the receive beamformer, and constraint (15i) ensures the maximum reflection power of the ARIS and the required covertness level, respectively. For the w/o-EG scheme, the covert communication rate \(\hat{R}_{a}\)'s maximization problem is formulated as follows:  
more general case, i.e., \(\gamma_{n}^{r}\gamma_{n}^{j}\geq 0\). To make the equation hold if and only if \(\gamma_{n}^{r}\) and \(\gamma_{n}^{j}\) equal to 0, we add the following constraint to ensure that each element works in at least one mode:  
\[\gamma_{n}^{r}+\gamma_{n}^{j}\geq 1,\ \forall n\in\mathcal{N}. \tag{26}\]  
At the same time, it is necessary to penalize the part of \(\gamma_{n}^{r}\gamma_{n}^{j}>0\), and we obtain its upper bound by the AGM inequality, which is given by  
\[\gamma_{n}^{r}\gamma_{n}^{j}\leq\frac{1}{2}(\varphi_{n}\gamma_{n}^{r})^{2}+ \frac{1}{2}\bigg{(}\frac{\gamma_{n}^{j}}{\varphi_{n}}\bigg{)}^{2}=\mathcal{A} \Big{(}\gamma_{n}^{r},\gamma_{n}^{j}\Big{)},\forall n\in\mathcal{N}, \tag{27}\]  
where the equality holds if and only if \(\varphi_{n}=\sqrt{\gamma_{n}^{j}\Big{/}\gamma_{n}^{r}}\). Then, a penalty term \(\sum_{n=1}^{N}\mathcal{A}\Big{(}\gamma_{n}^{r},\gamma_{n}^{j}\Big{)}\) is introduced into the objective function to deal with the binary integer constraint. Let \(\|\mathbf{A}_{m}\|_{*}\) and \(\|\mathbf{A}_{m}\|_{2}\) denote the nuclear norm and spectral norm of \(\mathbf{A}_{m}\), respectively. To deal with the rank-one constraint, we add a penalty term \(\sum_{m\in\{r,j\}}\mathcal{R}\big{(}\mathbf{A}_{m},\mathbf{A}_{m}^{(k)}\big{)}\) into the objective function, where \(\mathbf{A}_{m}^{(k)}\) is the solution obtained in the \(k\)th iteration and \(\mathcal{R}\big{(}\mathbf{A}_{m},\mathbf{A}_{m}^{(k)}\big{)}=\|\mathbf{A}_{m} \|_{*}+\widehat{\mathbf{A}}_{m}^{(k)}\) with \(\widehat{\mathbf{A}}_{m}^{(k)}=-\|\mathbf{A}_{m}^{(k)}\|_{2}-\text{Tr}\big{(} \mathbf{v}_{\max}^{(k)}\mathbf  
\[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\]0 (31a) s.t.