## 3 Discussion and Example
### Discussion
Note that it is always desired to find a smaller upper bound to approach the secrecy capacity._  
Similarly, substituting \(R_{1}=0\) into \(\mathcal{R}_{\text{o}c}\), we can get an upper bound on the secrecy capacity of the model in Figure 3 with causal states, which is:  
\[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]5 \[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]6  
where \(\mathcal{R}_{cd}\) is the capacity-equivocation region of the model with causal states. \(C^{\prime}_{uc}\) coincides with the upper bound on the secrecy capacity of the model in [18] with causal states. Then, we study a special channel model where the \"action\" is removed, _i.e._, the wiretap channel with noncausal channel state information. In this model, the channel state is generated by nature. This model is a special case of the wiretap channel with information embedding on actions by eliminating the action encoder and the mapping, \(f\). It is also a special case of the model in Figure 3 without action. Setting the random variable, \(A\), in Equation (28) to be a constant, we get a new outer bound of the wiretap channel with noncausal channel state information as:  
\[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]7  
The outer bound in [20] was derived as \(C^{\prime}_{\text{dain}}=\max_{p(u,v,q,x,s)}\min\{I(U;Y)-I(U;S),I(U;Y)-I(V;Z|Q)\}\). Comparing the two bounds, we see that \(C^{\prime\prime}_{un}\leq C^{\prime}_{\text{dain}}\). This is stated in the following corollary. \[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]8 The wiretap channel with action-dependent states [18]. **Corollary 5**: _For the wiretap channel with noncausal channel state information [20], the new upper bound on the secrecy capacity \(C^{\prime\prime}_{un}\leq C^{\prime}_{dain}\)._  
In addition, in the case of no actions, our model turns into a special case that was also studied in [21]. The comparison between our results and those in [21] for this case is stated as follows. * The achievable rate-equivocation region of the special case obtained from [21] is contained in that obtained from our results. * We also provide an outer bound for this special case. [21] obtained an achievable rate region for the broadcast wiretap channel with the asymmetric side information, which was:  
\[C_{un}=\max_{p(x|u,s)p(q|v)p(v|u)p(u|s,a)p(a)}\min\{I(U;Y)-I(U;S|A),I(U;Y)-I(V; Z|Q)-I(U;S|V)\} \tag{21}\]9 \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]0 \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]1 (32) \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]2 \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]3  
We first give an equivalent expression of the achievable rate region \(\mathcal{R}_{I}\) as follows. \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]4 \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]5 \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]6 (35) \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]7 \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]8 (36) \[C_{lc}=\max_{p(u,a)p(x|u,s)}\min\{I(U;Y)-I(U;Z),H(A|Z)\} \tag{22}\]9 (37) \[C_{uc}=\max_{p(x|s,u)p(q|v)p(v|u)p(u|a)p(a)}I(U;Y)-I(V;Z|Q) \tag{23}\]0 (38) \[C_{uc}=\max_{p(x|s,u)p(q|v)p(v|u)p(u|a)p(a)}I(U;Y)-I(V;Z|Q) \tag{23}\]1  
where \(R_{e1}\) and \(R_{e2}\) are the equivocation rates of the messages, \(m_{1}\) and \(m_{2}\), respectively.