## 3 Discussion and Example
### A Binary Example  
We give an example of a binary symmetric channel with causal channel states. The channel model is shown in Figure 4. Let the main channel be a binary symmetric channel (BSC). Its crossover probability is affected by the channel states. The wiretap channel is also assumed to be a BSC with crossover probability \(q\). More precisely, define:\[p(y|x,s=i)=\left\{\begin{array}{ll}(1-p)(1-i)+pi,&\mbox{if $y=x$}\\ (1-p)i+p(1-i),&\mbox{otherwise}\end{array}\right. \tag{47}\]  
and:  
\[p(z|y)=\left\{\begin{array}{ll}1-q,&\mbox{if $z=y$}\\ q,&\mbox{otherwise}\end{array}\right. \tag{48}\]  
where \(i\in\{0,1\}\), \(0\leq p\leq 1\) and \(0\leq q\leq 1\).  
It is assumed that the channel from the action to the channel states is a BSC with crossover probability equal to \(\alpha\), where \(0\leq\alpha\leq 1\). In this example, the parameter, \(\alpha\), is fixed as 0.2 (the other value of \(\alpha\) could also be assumed). Similar to the arguments in [1, 18, 19], the maximum values of \(H(A|Z)\), \(I(U;Y)\) and \(I(U;Y)-I(U;Z)\) are achieved when \(g_{1}:\mathcal{U}\rightarrow\mathcal{A}\) and \(g_{2}:\mathcal{U}\times\mathcal{S}\rightarrow\mathcal{X}\) are deterministic mappings. We choose \(g_{1}\) and \(g_{2}\) as:  
\[g_{1}(u=i)=i\] \[g_{2}(u=i,s=j)=i+j\ (\mbox{mod }2)\]  
where \(i,j\in\{0,1\}\). Let \(B\thicksim\mbox{Bernoulli}(\beta)\), where \(0\leq\beta\leq 1\). Let the function, \(f\), be a one-to-one mapping for simplicity. Here, we set:  
\[f:\left\{\begin{array}{ll}0\to 0\\ 1\to 1\end{array}\right.\]  
From the above conditions, we see that the random variables, \(B\), \(A\) and \(U\), share the same distribution. Then, the joint distribution \(p(a,b,s,u,x,y,z)=p(z|y)p(y|x,s)p(x|u,s)p(s|a)p(a|u)p(u)1_{\{b=f(a)\}}\) can be calculated. The joint distributions \(p(u,y)=\sum\limits_{a,b,s,x,z}p(a,b,s,u,x,y,z)\) and \(p(u,z)=\sum\limits_{a,b,s,x,y}p(a,b,s,u,x,y,z)\) can also be obtained. By some mathematical calculation and Theorem 3, we can get the maximum sum secrecy rate of the example with causal channel states for given \(p,q\) as:  
\[R_{1}+R_{2}=\max\limits_{0\leq\beta\leq 1}\{\frac{5}{2}[h(q*(\beta*p))-h(\beta*p)] -\frac{3}{2}[h(p*q)-h(p)]\} \tag{49}\]  
under the constraint \(H(B)=h(\beta)\geq R_{1}\) and the secrecy constraint:  
\[R_{e}\leq\min\{\max\limits_{0\leq\beta\leq 1}\{h(p*q)-h(p), \frac{5}{2}[h(q*(\beta*p))-h(\beta*p)]-\frac{3}{2}[h(p*(0.2*q))-h(0.2*q)]\}\] \[\max\limits_{0\leq\beta\leq 1}\{h(\beta)-1+h(q*(\beta*p)),1- \frac{5}{2}[h(q*(\beta*p))-h(\beta*p)]\}\} \tag{50}\]  
Figure 4: The binary symmetric channel with information embedding on actions.  
where \(p*q=p+q-2 pq\) and \(h(p)\) is the binary entropy function, _i.e._, \(h(p)=-p\log p-(1-p)\log(1-p)\).  
The tradeoff between the sum secrecy rate and the information embedding rate under the secrecy constraint is shown in Figure 5. It can be seen that the sum secrecy rate is reduced when the equivocation rate, \(R_{e}\), increases. In practical communication systems involving security, we always desire a bigger secure transmission rate when the extent of information leakage is at a reasonable level. Moreover, it can be seen that when the information embedding rate, \(R_{1}\), goes up, the sum secrecy rate also decreases. This tells us that the communication requirements of Decoder 1 have a negative impact on improving the secrecy transmission rate of the given communication link.