## 3 Performance Analysis
### Eve's Average Bit Error Probability  
Analogously, the upper bound of ABEP at Eve in the CIM-GSDM system at \(k\)-slot based on (20) can be formulated as  
\[P_{E_{k}}\leqslant\frac{1}{R_{p}2^{R_{p}}}\sum_{\mathbf{x}_{T_{g_{k}}}^{i} \subseteq\mathcal{X}_{T_{m_{k}}}^{i}}\sum_{\mathbb{C}\mathcal{N}\
eq\mathbf{x }_{T_{g_{k}}}^{i}}d\left(\mathbf{x}_{T_{g_{k}}}^{i}\rightarrow\mathbf{x}_{T_{g _{k}}}^{i}\right)P_{\mathbf{H}(\Theta_{E})}\left(\mathbf{x}_{T_{g_{k}}}^{i} \rightarrow\mathbf{x}_{T_{m_{k}}}^{i}\right), \tag{27}\]  
where \(P_{\mathbf{H}(\Theta_{E})}\left(\mathbf{x}_{T_{g_{k}}}^{i}\rightarrow\mathbf{x }_{T_{m_{k}}}^{i}\right)\) denotes the PEP for [PERSON] under the channel matrix \(\mathbf{H}(\Theta_{E})\). In this scenario, the PEP in (27) is given by  
\[P_{\mathbf{H}(\Theta_{E})}\left(\mathbf{x}_{T_{g_{k}}}^{i} \rightarrow\mathbf{x}_{T_{m_{k}}}^{i}\right) \tag{28}\] \[=P\Bigg{(}\left\|\mathbf{r}_{E}^{k}-\mathbf{H}_{E}\mathbf{x}_{T_ {g_{k}}}^{i}\right\|^{2}>\left\|\mathbf{r}_{E}^{k}-\mathbf{H}_{E}\mathbf{x}_{ T_{g_{k}}}^{i}\right\|^{2}\Bigg{)}\] \[=P\Bigg{(}\left\|\mathbf{H}_{E}\mathbf{\Lambda}\mathbf{x}_{T_{g_{ k}}}^{i}+\mathbf{n}_{E}-\mathbf{H}_{E}\mathbf{x}_{T_{g_{k}}}^{i}\right\|^{2}> \left\|\mathbf{H}_{E}\mathbf{\Lambda}\mathbf{x}_{T_{g_{k}}}^{i}+\mathbf{n}_{ E}-\mathbf{H}_{E}\mathbf{x}_{T_{g_{k}}}^{i}\right\|^{2}\Bigg{)}\] \[=P\Bigg{(}\mathcal{R}\Big{[}\mathbf{n}_{E}^{H}\Big{(}\mathbf{H}_{ E}\mathbf{x}_{T_{g_{k}}}^{i}-\mathbf{H}_{E}\mathbf{x}_{T_{g_{k}}}^{i}\Big{)} \Big{]}>\frac{\left\|\mathbf{H}_{E}\mathbf{\Lambda}\mathbf{x}_{T_{g_{k}}}^{i}- \mathbf{H}_{E}\mathbf{x}_{T_{g_{k}}}^{i}\right\|^{2}-\left\|\mathbf{H}_{E} \mathbf{\Lambda}\mathbf{x}_{T_{g_{k}}}^{i}-\mathbf{H}_{E}\mathbf{x}_{T_{g_{k} }}^{i}\right\|^{2}}{2}\Bigg{)},\]  
where \(\mathbf{H}_{E}=\mathbf{H}(\Theta_{E})\mathbf{W}\) and \(\mathbf{n}_{E}\sim\mathcal{CN}\big{(}\mathbf{0},\sigma_{B}^{2}\mathbf{I}_{N \times N}\big{)}\). Moreover, \(\mathcal{R}\Big{[}\mathbf{n}_{E}^{H}\Big{(}\mathbf{H}_{E}\mathbf{x}_{T_{g_{k}} }^{i}-\mathbf{H}_{E}\mathbf{x}_{T_{g_{k}}}^{i}\Big{)}\Big{]}\) is a Gaussian random variable obeying \(\mathcal{CN}\sim\left(0,\frac{1}{2}\sigma_{E}^{2}\Big{\|}\mathbf{H}_{E}\left( \mathbf{x}_{T_{g_{k}}}^{i}\rightarrow\mathbf{x}_{T_{g_{k}}}^{i}\right)\Big{\|} ^{2}\right)\). Therefore, the PEP for Eve can be formulated as  
\[P_{\mathbf{H}(\Theta_{E})}\Big{(}\mathbf{x}_{T_{g_{k}}}^{i} \rightarrow\mathbf{x}_{T_{g_{k}}}^{i}\Big{)}=Q\Bigg{(}\frac{\left\|\mathbf{H} _{E}\mathbf{\Lambda}\mathbf{x}_{T_{g_{k}}}^{i}-\mathbf{H}_{E}\mathbf{x}_{T_{g _{k}}}^{i}\right\|^{2}-\left\|\mathbf{H}_{E}\mathbf{\Lambda}\mathbf{x}_{T_{g_ {k}}}^{i}-\mathbf{H}_{E}\mathbf{x}_{T_{g_{k}}}^{i}\right\|^{2}}{\sqrt{2}\sigma _{E}\Big{\|}\mathbf{H}_{E}\mathbf{x}_{T_{g_{k}}}^{i}-\mathbf{H}_{E}\mathbf{x}_{ T_{g_{k}}}^{i}\Big{\|}}\Bigg{)}. \tag{29}\]  
By substituting (29) to (27), the theoretical upper bound of ABEP for Eve can be calculated. By contrast, Eve can not recognize the correct covert information and its corresponding bit since [PERSON] has no idea about the rule of CIM. In this case, the average error probability of the bit for receiver group selection at Eve can be approximated by \(P_{r}=0.5\). Specifically, the ABEP which is considered the \((k-1)\)-th slot and the \(k\)-slot jointly can be detailed as  
\[\bar{P}_{r} =\frac{1}{R_{p}}\times P_{r}+\frac{\left(R_{p}-1\right)}{R_{p}} \times P_{E_{k}} \tag{30}\] \[=\frac{0.5+\left(R_{p}-1\right)P_{E_{k}}}{R_{p}}.\]  
Finally, the upper bound of ABEP at Eve is obtained.  
It should be noted that the theoretical bound of ABEP of Eve only can be attained in high SNR region due to its robust.