## 2 Channel Model and Main Results
### Symbol Notations and Channel Model  
Throughout this paper, we use calligraphic letters, e.g., \(\mathcal{X}\), \(\mathcal{Y}\), to denote the finite sets and \(\|\mathcal{X}\|\) to denote the cardinality of the set, \(\mathcal{X}\). Uppercase letters, e.g., \(X\), \(Y\), are used to denote random variables taking values from finite sets, e.g., \(\mathcal{X}\), \(\mathcal{Y}\). The value of a random variable, \(X\), is denoted by the lowercase letter, \(x\). We use \(Z_{i}^{j}\) to denote the \((j-i+1)\)-vectors \((Z_{i},Z_{i+1},...,Z_{j})\) of random variables for \(1\leq i\leq j\) and will always drop the subscript when \(i=1\). Moreover, we use \(X\sim p(x)\) to denote the probability mass function of the random variable, \(X\). For \(X\sim p(x)\) and \(0\leq\epsilon\leq 1\), the set of the typical \(N\)-sequences \(x^{N}\) is defined as \(T_{X}^{N}(\epsilon)=\{x^{N}:|\pi(x|x^{N})-p(x)|\leq\epsilon p(x)\text{ for all }x\in\mathcal{X}\}\), where \(\pi(x|x^{N})\) denotes the frequency of occurrences of letter \(x\) in the sequence, \(x^{N}\) (for more details about typical sequences, please refer to [23,24]). The set of the conditional typical sequences, e.g., \(T_{Y|X}^{N}(\epsilon)\), follows similarly. In this paper, it is assumed that the base of the \(log\) function is two. The wiretap channel with information embedding on actions is depicted in Figure 2. We aim to send the confidential message \((M_{1},M_{2})\) to the legitimate receiver through such a channel and keep it secret from the wiretapper as much as possible. Part of the message embedded in the actions needs to be retrieved by the specific decoder. We use equivocation at the wiretapper to measure the secrecy of the confidential message. Concretely, the model of wiretap channel with information embedding on actions is specified by \(\{\mathcal{A},\mathcal{B},\mathcal{S},f,p(s|a),\mathcal{X},p(y,z|s,x), \mathcal{Y},\mathcal{Z}\}\). To send the message \((M_{1},M_{2})\), an action sequence, \(A^{N}(M_{1},M_{2})\), is first selected by the encoder. Then, the generation of the channel states, \(S^{N}\), is affected by the actions, instead of by nature. The channel states, \(S^{N}\), are generated through a discrete memoryless channel (DMC) \(p(s^{N}|a^{N}(m_{1},m_{2}))=\prod\limits_{i=1}^{N}p(s_{i}|a_{i})\). The stochastic channel encoder, \(\varphi\), is specified by a matrix of conditional probability distributions, \(\varphi(x^{N}|m_{1},m_{2},s^{N})\). Note that \(\sum\limits_{x^{N}}\varphi(x^{N}|m_{1},m_{2},s^{N})=1\), and \(\varphi(x^{N}|m_{1},m_{2},s^{N})\) is the probability that the message \((m_{The main channel is a DMC with discrete input alphabet \(\mathcal{X}\times\mathcal{S}\) and output alphabet \(\mathcal{Y}\). The channel is memoryless in the sense that \(p(y^{N}|x^{N},s^{N})=\prod\limits_{i=1}^{N}p(y_{i}|x_{i},s_{i})\), where \(y^{N}\in\mathcal{Y}^{N}\), \(x^{N}\in\mathcal{X}^{N}\) and \(s^{N}\in\mathcal{S}^{N}\). Decoder 1 observes signal \(B^{N}\) as a deterministic function of the actions, \(A^{N}\), _i.e._, \(B^{N}=f(A^{N})\). It estimates part of the transmitted message. Decoder 1 is specified by \(\psi_{1}:\mathcal{B}^{N}\rightarrow\mathcal{M}_{1}\). The output of Decoder 1 is \(\hat{M}_{1}\). The probability of the error of Decoder 1 is defined as \(P_{e1}=Pr\{\hat{M}_{1}\
eq M_{1}\}\). The legitimate receiver decodes the message \((\hat{\hat{M}}_{1},\hat{\hat{M}}_{2})\) by Decoder 2 (see Figure 2). Decoder 2 is specified by \(\psi_{2}:\mathcal{Y}^{N}\rightarrow\mathcal{M}_{1}\times\mathcal{M}_{2}\). The probability of the error of Decoder 2 is defined as \(P_{e2}=Pr\{(\hat{M}_{1},\hat{\hat{M}}_{2})\
eq(M_{1},M_{2})\}\). The wiretap channel is also a DMC with transition probability \(p(z^{N}|y^{N})=\prod\limits_{i=1}^{N}p(z_{i}|y_{i})\), where \(z^{N}\in\mathcal{Z}^{N}\) is the observation of the wiretapper. The uncertainty of the message for the wiretapper is measured by \(\lim\limits_{N\rightarrow\infty}\Delta=\lim\limits_{N\rightarrow\infty}\frac {H(M_{1},M_{2}|Z^{N})}{N}\).