## 3 Discussion and Example
### Discussion  
**Corollary 1**: _The lower and upper bounds on the sum secrecy capacity of the model in Figure 2 with noncausal action-dependent states are:_  
\[C_{ln}=\max_{p(x|u,s)p(u|s,a)p(a)}\min\{I(U;Y)-\max\{I(U;Z),I(U;S|A)\},H(A|Z)\} \tag{20}\]  
_and:_  
\[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]respectively._  
_Proof_: _According to the definition of Formula (5), \(\mathcal{R}_{in}\) and \(\mathcal{R}_{on}\), we can easily get Equations (20) and (21)._  
Similarly, we have the following corollary for the causal case. **Corollary 2**: _The lower and upper bounds on the sum secrecy capacity of the model in Figure 2 with causal action-dependent states are:_  
\[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]  
_and:_  
\[C_{uc}=\max_{p(x|s,u)p(q|v)p(v|u)p(u|a)p(a)}I(U;Y)-I(V;Z|Q) \tag{23}\]  
_respectively._  
_Proof_: _According to the definition of Formula (5), \(\mathcal{R}_{ic}\) and \(\mathcal{R}_{oc}\), we can easily get Equations (22) and (23)._  
According to [11], the capacity region of the model in Figure 1 (without the secrecy constraint) is:  
\[\mathcal{R}_{E}=\{(R_{1},R_{2})\] \[R_{1}\leq H(B) \tag{24}\] \[R_{1}+R_{2}\leq I(U;Y)-I(U;S|A)\} \tag{25}\]  
Then, the corresponding capacity is:  
\[C_{E}=\max_{p(u,a,x,s)}I(U;Y)-I(U;S|A) \tag{26}\]  
Compare Formula (26) with (20); we can get \(C_{ln}\leq C_{E}\). This implies that the secrecy constraint reduces the secrecy transmission rate of the communication link. Therefore, once the problem of information leakage is considered, the system designer has to trade off between the transmission rate and data security. Moreover, without the secrecy constraint, we have the following corollary. **Corollary 3**: _Without considering the secrecy constraint (i.e., ignoring \(R_{e}\) in \(\mathcal{R}_{in}\)), we arrive at the results in [11]._  
_Proof_: _This corollary is verified by setting \(R_{e}=0\) in \(\mathcal{R}_{in}\)._  
When no message needs to be embedded in the actions, the model in Figure 2 turns to the wiretap channel with action-dependent states [18]; see Figure 3. [18] gave an upper bound on the secrecy capacity of the model with noncausal states as:  
\[C_{dain}=\max_{p(u,v,q,a,x,s)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V;Z|Q)\} \tag{27}\]  
Substituting \(R_{1}=0\) into \(\mathcal{R}_{on}\), we get a new upper bound on the secrecy capacity of the model in Figure 3 with noncausal states. This new upper bound is:  
\[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]0 \[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]1where \(\mathcal{R}_{nd}\) is the capacity-equivocation region of the model with noncausal states. Note that the difference between Equations (28) and (27) is the extra term, \(I(U;S|V)\). The cause of the emergence of this term is stated in detail at the end of Appendix A. Then, we give the following corollary. **Corollary 4**: _For the wiretap channel with noncausal action-dependent states shown in Figure 3, the new upper bound on the secrecy capacity \(C^{\prime}_{un}\leq C_{\text{dain}}\)._  
_Proof_: From the two Formulas, (27) and (28), we can get the difference between \(C^{\prime}_{un}\) and \(C_{\text{dain}}\) as:_  
\[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]2 \[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]3 \[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]4  
_It is easy to see that \(\Lambda_{1}\geq 0\), i.e., our new upper bound \(C^{\prime}_{un}\leq C_{\text{dain}}\).