Information embedding on actions is a new channel model in which a specific decoder is used to observe the actions taken by the encoder and retrieve part of the message intended for the receiver. We revisit this model and consider a different scenario where a secrecy constraint is imposed. By adding a wiretapper in the model, we aim to send the confidential message to the receiver and keep it secret from the wiretapper as much as possible. We characterize the inner and outer bounds on the

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... ty-equivocation region of such a channel with noncausal (and causal) channel state information. Furthermore, the lower and upper bounds on the sum secrecy capacity are also obtained. Besides, by eliminating the specific decoder, we get a new outer bound on the capacity-equivocation region of the wiretap channel with action-dependent states and prove it is tighter than the existing outer bound. A binary example is presented to illustrate the tradeoff between the sum secrecy rate and the information embedding rate under the secrecy constraint. We find that the secrecy constraint and the communication requirements of information embedding have a negative impact on improving the secrecy transmission rate of the given communication link. Keywords: information embedding; wiretap channel; action-dependent states; sum secrecy capacity Recently, [11, 12] explored information embedding on actions in the channel with noncausal action-dependent states; see Figure 1 . In this new setup, an additional decoder was introduced to observe a function of the actions taken by the encoder. It tried to get part of the transmitted message. Actually, the idea of "information embedding" on actions in such a channel is related to the classical topic of information hiding (e.g., [13] [14] [15] [16] [17] ) and could be explained by the following example. In communication networks, probing the congestion state requires sending training packets to the nearest router. Meanwhile, the router (the "recipient" of the actions) may need to obtain partial information, such as the header of the packet, to find the address of the intended receiver. Since the actions play the role of providing necessary information about the message for the router, it is natural to ask how much information could be embedded in the actions without affecting the system performance. [11] got the capacity-cost region and showed that the communication requirements of the action-cribbing decoder were generally in conflict with the goal of improving the efficiency of the communication link. However, the above action-dependent channel models [1-12] considered no secrecy constraint, which was extremely important in communications. For instance, the broadcast nature of wireless networks gives rise to the hidden danger of information leakage to malicious receiver when broadcasting the sensitive data and acquiring the state information. Recent works [18, 19] studied the secure communication problems in channels with action-dependent states. [18] added a wiretapper to the model in [1] and got the inner and outer bounds on the capacity-equivocation region. The capacity-equivocation region is the set of all the achievable rate pairs (R, R e ), where R and R e are the rates of the confidential message and wiretapper's equivocation about the message. [19] studied the effects of feedback on the secrecy capacity, which is the maximum rate of data transmission at which the message can be communicated in perfect secrecy, of the wiretap channel with action-dependent states.