## 2 Channel Model and Main Results
### Main Results  
In this subsection, four theorems are presented. Theorems 1 and 2 give the inner and outer bounds on the capacity-equivocation region for the channel model in Figure 2 with noncausal action-dependent states. For the causal case, the inner and outer bounds are characterized in Theorems 3 and 4.  
**Theorem 1**: _An achievable rate-equivocation region of the wiretap channel with information embedding on actions when the states are noncausally known to the channel encoder is the set:_  
\[\mathcal{R}_{in}=\{(R_{1},R_{2},R_{e})\] \[R_{1}\leq H(B) \tag{6}\] \[R_{1}+R_{2}\leq I(U;Y)-I(U;S|A)\] (7) \[R_{e}\leq R_{1}+R_{2}\] (8) \[R_{e}\leq I(U;Y)-\max\{I(U;Z),I(U;S|A)\}\] (9) \[R_{e}\leq H(A|Z)\} \tag{10}\]  
_where the joint distributions \(p(a,b,u,x,s,y,z)=p(z|y)p(y|x,s)p(x|u,s)p(s|u,a)p(u|a)p(a)1_{\{b=f(a)\}}\), which indicates \((A,B,U)\rightarrow(X,S)\to Y\to Z\) form a Markov chain._  
**Theorem 2**: _An outer bound on the capacity-equivocation region of the wiretap channel with information embedding on actions when the states are noncausally known to the channel encoder is the set:_  
\[\mathcal{R}_{on}=\{(R_{1},R_{2},R_{e})\] \[R_{1}\leq H(B) \tag{11}\] \[R_{1}+R_{2}\leq I(U;Y)-I(U;S|A)\] (12) \[R_{e}\leq R_{1}+R_{2}\] (13) \[R_{e}\leq I(U;Y)-I(V;Z|Q)-I(U;S|V)\} \tag{14}\]  
_where the joint distributions \(p(a,b,u,x,s,y,z)=p(z|y)p(y|x,s)p(x|u,s)p(q|v)p(v|u)p(a,u,s)1_{\{b=f(a)\}}\), which indicates \((B,A,U,V,Q)\rightarrow(X,S)\to Y\to Z\) and \(Q\to V\to U\to Y\to Z\) form Markov chains._  
_Comments:_  
\(\bullet\) Theorems 1 and 2 are proven in Appendix A.  
\(\bullet\) To exhaust \(\mathcal{R}_{in}\) and \(\mathcal{R}_{on}\), it is enough to restrict \(\mathcal{U}\), \(\mathcal{V}\) and \(\mathcal{Q}\) to satisfy:  
\[\|\mathcal{U}\|\leq\|\mathcal{A}\|\|\mathcal{X}\|\|\mathcal{S}\|+2\] \[\|\mathcal{Q}\|\leq\|\mathcal{A}\|\|\mathcal{X}\|\|\mathcal{S}\|\] \[\|\mathcal{V}\|\leq\|\mathcal{A}\|\|\mathcal{X}\|\|\mathcal{S}\| (\|\mathcal{A}\|\|\mathcal{X}\|\|\mathcal{S}\|+1)\]  
This can be easily proven by using the support lemma (see p. 310 in [25]).  
**Theorem 3**: _An achievable rate-equivocation of the wiretap channel with information embedding on actions when the states are causally known to the channel encoder is the set:_  
\[\mathcal{R}_{ic}=\{(R_{1},R_{2},R_{e})\] \[R_{1}\leq H(B)\] \[R_{1}+R_{2}\leq I(U;Y)\] \[R_{e}\leq R_{1}+R_{2}\] \[R_{e}\leq I(U;Y)-I(U;Z)\] \[R_{e}\leq H(A|Z)\} \tag{15}\]_where the joint distributions \(p(a,b,u,x,s,y,z)=p(z|y)p(y|x,s)p(x|u,s)p(s|a)p(u|a)p(a)1_{\{b=f(a)\}}\), which indicates \((A,B,U)\rightarrow(X,S)\to Y\to Z\) and \(U\to A\to S\) form Markov chains._  
**Theorem 4**: _An outer bound on the capacity-equivocation region of the wiretap channel with information embedding on actions when the states are causally known to the channel encoder is the set:_  
\[\mathcal{R}_{oc}=\{(R_{1},R_{2},R_{e})\] \[R_{1}\leq H(B) \tag{16}\] \[R_{1}+R_{2}\leq I(U;Y)\] (17) \[R_{e}\leq R_{1}+R_{2}\] (18) \[R_{e}\leq I(U;Y)-I(V;Z|Q)\} \tag{19}\]  
_where the joint distributions \(p(a,b,u,x,s,y,z)=p(z|y)p(y|x,s)p(x|u,s)p(q|v)p(v|u)p(a,u,s)1_{\{b=f(a)\}}\), which indicates \((B,A,U,V,Q)\rightarrow(X,S)\to Y\to Z\) and \(Q\to V\to U\to Y\to Z\) form Markov chains._  
_Comments:_  
\(\bullet\) Theorems 3 and 4 are proven in Appendix B.  
\(\bullet\) To exhaust \(\mathcal{R}_{ic}\) and \(\mathcal{R}_{oc}\), it is enough to restrict \(\mathcal{U}\), \(\mathcal{V}\) and \(\mathcal{Q}\) to satisfy:  
\[\|\mathcal{U}\|\leq\|\mathcal{A}\|\|\mathcal{X}\|\|\mathcal{S}\|+1\] \[\|\mathcal{Q}\|\leq\|\mathcal{A}\|\|\mathcal{X}\|\|\mathcal{S}\|\] \[\|\mathcal{V}\|\leq(\|\mathcal{A}\|\|\mathcal{X}\|\|\mathcal{S} \|)^{2}\]  
This can be easily proven by using the support lemma (see p. 310 in [25]).  
Further discussion about the theorems and the comparison with other existing results are given in Section 3.