## 2 System Model and Problem Formulation  
### System Model  
As illustrated in Figure 1, we investigate a multicast downlink MISO communication system equipped with a four-node STAR-RIS. An \(N\)-antenna BS exploits an \(M\)-element STAR-RIS to transmit the confidential signals to two single-antenna receivers, which are called \(\text{Bob}_{T}\) and \(\text{Bob}_{R}\), in presence of two single-antenna eavesdroppers, namely, \(\text{Eve}_{T}\) and \(\text{Eve}_{R}\). We consider that the STAR-RIS concurrently serves both the transmission and reflection regions. Specifically, the receiver situated in the \(l\)-th region is designated as the \(l\)-th Bob, where \(l\in\{T,R\}\). The baseband equivalent channels from STAR-RIS to BS, Bob, and Eve are represented as \(\mathbf{h}_{BS}\in\mathbb{C}^{M\times N}\), \(\mathbf{h}_{Sl}\in\mathbb{C}^{M\times 1}\), \(\mathbf{g}_{l}\in\mathbb{C}^{M\times 1}\). It is assumed that the direct channel between the BS and the user, denoted by \(\mathbf{h}_{Bl}\in\mathbb{C}^{N\times 1}\), \(\mathbf{h}_{l}\in\mathbb{C}^{N\times 1}\). Assuming the availability of perfect channel state information (CSI) of these links is obtained by BS [24], the STAR-RIS is configured via a controller, typically realized through a field-programmable gate array. For CP scheme, where each transmitting and reflecting coefficients element \(m\in\mathfrak{M}\triangleq\{1,2,...,M\}\), its reflection and transmission coefficients are given by \(\beta_{m}^{T}e^{\phi_{m}^{T}}\) and \(\beta_{m}^{R}e^{\phi_{m}^{R}}\), respectively, where \(\beta_{m}^{T}\), \(\beta_{m}^{R}\in[0,1]\) and \(\phi_{m}^{T}\), \(\phi_{m}^{R}\in[0,2\pi)\) denote the amplitude and phase configurations, respectively. Due to energy conservation, passive lossless STAR-RISs have to meet the following constraints \((\theta_{m}^{T})^{2}+(\theta_{m}^{R})^{2}=1\). In the case of a passive and lossless STAR-RIS, the electrical and magnetic impedances manifest as purely imaginary values, i.e., \(\beta_{m}^{T}\theta_{m}^{R}\cos(\phi_{m}^{T}-\phi_{m}^{R})=0\) which \(\phi_{m}^{T}\) and \(\phi_{m}^{R}\) are coupled by the condition \(\left|\phi_{m}^{T}-\phi_{m}^{R}\right|=\frac{\pi}{2},\frac{3\pi}{2}\). For the IP scheme, the transmission and reflection phases are independent of each other. \(\mathbf{\Theta}_{I}=\mathit{Diag}\left(\beta_{1}^{I}e^{\phi_{1}^{I}},...,\beta _{M}^{I}e^{\phi_{M}^{I}}\right)\in\mathbb{C}^{M\times M}\) denotes the coefficient for the \(l\)-th Bob. BS exploits the multiple beamforming vectors \(\mathbf{w}_{T},\mathbf{w}_{R}\in\mathbb{C}^{N\times 1}\) to send the confidential signals \(x_{T}\) and \(x_{R}\) with \(\mathbb{E}\left\{\left|s_{T}\right|^{2}\right\}=\mathbb{E}\left\{\left|s_{R} \right|^{2}\right\}=1\) to \(\mathit{Bob}_{T}\) and \(\mathit{Bob}_{R}\), respectively. Neglecting the signals reflected by the STAR-RIS twice or more due to severe path loss, the received signals at the user and the eavesdropper can be, respectively, expressed as  
\[y_{BJ} =(\mathbf{h}_{Bl}^{H}+\mathbf{h}_{Sl}^{H}\mathbf{\Theta}_{I} \mathbf{h}_{BS})_{\sum i}\mathbf{w}_{i}x_{i}+n_{BJ} \tag{1}\] \[y_{EJ} =(\mathbf{h}_{I}^{H}+\mathbf{g}_{I}^{H}\mathbf{\Theta}_{I} \mathbf{h}_{BS})_{\sum i}\mathbf{w}_{i}x_{i}+n_{EJ} \tag{2}\]  
where \(n_{BJ},n_{EJ}\sim\mathcal{CN}(0,\sigma^{2})\) represent the additive white Gaussian noise (AWGN) at both reflection and transmission, \(i\in\{T,R\}\) denote reflection and transmission areas. Using \(\mathbf{\theta}^{H}=\mathit{diag}(\mathbf{\Theta})^{T}\), we have \(\mathbf{h}_{Bl}^{H}+\mathbf{h}_{Sl}^{H}\mathbf{\Theta}_{I}\mathbf{h}_{BS}= \mathbf{h}_{Bl}^{H}+\mathbf{\theta}^{H}\mathbf{H}_{I}\) and \(\mathbf{h}_{I}^{H}+\mathbf{g}_{I}^{H}\mathbf{\Theta}_{I}\mathbf{h}_{BS}= \mathbf{h}_{I}^{H}+\mathbf{\theta}^{H}\mathbf{G}_{I}\), where \(\mathbf{H}_{I}=\mathrm{Diag}(\mathbf{h}_{Sl}^{H})\mathbf{h}_{BS}\), and \(\mathbf{G}_{I}=\mathrm{Diag}(\mathbf{g}_{I}^{H})\mathbf{h}_{BS}\), respectively.