## 2 Convert Information Mapping Aided GSDM
### Detection Algorithm for the Proposed Scheme  
Based on (12), the receive signal at Bob aided by CIM at the \(k\)-th slot can be expressed as  
\[\begin{split}\mathbf{r}_{B}^{k}&=\mathbf{H}(\Theta_{B}) \mathbf{s}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}}}}}}}}}^{s}+ \mathbf{n}_{B}}\\ &=\mathbf{H}(\Theta_{B})\mathbf{W}\mathbf{A}\mathbf{\chi}_{\mathcal{I }_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{ \mathcal{I}}}}}}}}^{s}+\mathbf{n}_{B}.\end{split} \tag{14}\]  
Assume that the optimal detector of maximum likelihood (ML) is adopted to detect the composition of unit column vectors corresponding to the receiver subset \(l\) and the combination of APM symbols \(s\), which can be formulated as  
\[\langle\hat{l},\hat{s}\rangle=\arg\min_{l\in\mathbb{L},s\in\mathbb{S}}\left\{ \left\|\mathbf{r}_{B}^{k}-\mathbf{H}(\Theta_{B})\mathbf{W}\mathbf{A}\mathbf{\chi}_{ \mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}_{\mathcal{I}} }}}}^{s}\right\|^{2}}\right\}, \tag{15}\]  
where \(\mathbb{L}=\{1,2,\ldots,2f_{2}\}\) and \(\mathbb{S}=\left\{1,2,\ldots,2^{k_{3}}\right\}\) denotes the set of the indices of receivers subsets at Bob and the set of constellations combinations of \(M\)-ary APM symbols, respectively.  
Furthermore, the index of receiver group can be set as  
\[\hat{g}_{k}=\left\{\begin{array}{cc}0,&\text{if }\hat{l}\leq f_{2}\\ 1,&\text{if }\hat{l}>f_{2}.\end{array}\right. \tag{16}\]  
Figure 2: An example with \(N_{r}=4\) for the proposed CIM-GSDM scheme.  
In particular, the bit \(K_{1}\) for determining the receive group index is decided by \(\hat{g}_{k}\) and \(\hat{g}_{k-1}\), which can be given by  
\[K_{1}=\left\{\begin{array}{cc}1,&\hat{g}_{k}=\hat{g}_{k-1}\\ 0,&\hat{g}_{k}\
eq\hat{g}_{k-1}.\end{array}\right. \tag{17}\]  
In addition, the index of receiver subset \(\hat{p}\) in the receive group \(\mathcal{I}_{g_{k}}\) can be detailed as follows:  
\[\hat{p}=\mathrm{MOD}_{\mathcal{I}_{2}}\Big{(}\hat{l}-1\Big{)}+1, \tag{18}\]  
where \(\mathrm{MOD}_{\mathcal{I}_{2}}(b)\) returns the remainder after division of \(b\) by \(a\). As a result, \(\hat{p}\) can be exploited to demodulating the index bits \(k_{2}\) of the receive subset.  
With respect to Eve, the receive signal at Eve at the \(k\)-th slot can be expressed as  
\[\begin{split}\mathbf{r}_{E}^{k}&=\mathbf{H}(\Theta_{ E})\mathbf{s}_{\mathcal{I}_{2g_{k}}}^{s}+\mathbf{n}_{E}\\ &=\mathbf{H}(\Theta_{E})\mathbf{W}\mathbf{A}\mathbf{x}_{\mathcal{ I}_{2g_{k}}}^{s}+\mathbf{n}_{E}.\end{split} \tag{19}\]  
Eve is possible to obtain channel state information (CSI) of the system and achieves to calculate the beamforming matrix \(W\). However, the interference matrix is still hard to confirm. As a result, the ML detector employed at Eve can be represented as  
\[\langle\hat{l},\hat{s}\rangle=\arg\min_{\hat{l}\in\mathbb{L},s\in\mathbb{S}} \bigg{\{}\Big{\|}\mathbf{r}_{E}^{k}-\mathbf{H}(\Theta_{E})\mathbf{W}\mathbf{ A}\mathbf{x}_{\mathcal{I}_{g_{k}}}^{s}\Big{\|}^{2}\bigg{\}}. \tag{20}\]  
In general, the CIM-GSDM system are much more safe than the original system. On the one hand, only part of the disturbed receivers at Bob are activated and adjusted by the covert information, which is hard to be ensured at Eve. On the other hand, due to the interference matrix added based on (19) and (20), the accuracy of detection performance of APM symbols suffers reduction significantly. That is to say, except to \(k_{1}\) and \(k_{2}\) bits, [PERSON] can hardly obtain \(k_{3}\) bits neither. By contrast, [PERSON] is able to demodulate all the bits successfully since [PERSON] knows the rules. As a benefit, the proposed strategy of CIM-GSDM can let [PERSON] communicate with [PERSON] smoothly while disturbing [PERSON], which enhances the security of transmission effectively.