Feedback Schemes for the Action-Dependent Wiretap Channel with Noncausal State at the Transmitter

In this paper, we propose two feedback coding schemes for the action-dependent wiretap channel with noncausal state at the transmitter. The first scheme follows from the already existing secret key based feedback coding scheme for the wiretap channel. The second one follows from our recently proposed hybrid feedback scheme for the wiretap channel. We show that, for the action-dependent wiretap channel with noncausal state at the transmitter, the second feedback scheme performs better than the first one, and the capacity results of this paper are further explained via a Gaussian example, which we call the action-dependent dirty paper wiretap channel with noiseless feedback.


Introduction
Using channel feedback to enhance the physical layer security (PLS) of a communication system was first proposed by Ahlswede and Cai [1], who re-visited the foundation of the PLS-the wiretap channel model [2]-by considering a noiseless feedback channel from the legitimate receiver to the transmitter. Ahlswede et al. [1] showed that, since the eavesdropper does not know the feedback, the legitimate receiver's feedbackcan be used to generate secret keys shared between the transmitter and the legitimate receiver, and these keys can be used to encrypt the transmitted message. Using the feedback scheme in [1], it has been shown that the secrecy capacity (channel capacity with perfect secrecy constraint) of the wiretap channel can be enhanced. Furthermore, Ahlswede et al. [1] showed that this usage of feedback is optimal (achieving the secrecy capacity of the wiretap channel with noiseless feedback) if the channel is physically degraded (the eavesdropper's received signal is a degraded version of the legitimate receiver's). In recognition of this, Ardestanizadeh et al. [3] further pointed out that, if the noiseless feedback channel can be used to transmit anything the legitimate parties wish, the best choice of the legitimate parties is to send pure random bits (secret key) over the feedback channel. Subsequently, Schaefer et al. [4] extended the work of [3] to a broadcast situation, where two legitimate receivers of the broadcast channel independently send their secret keys to the transmitter via two noiseless feedback channels, and these keys help to enhance the achievable secrecy rate region of the broadcast wiretap channel [5]. Other related works in the PLS of the feedback channels include those by [6][7][8], who introduced channel state information (CSI) into various feedback channel models. Recently, Dai et al. [9] showed that, for the general wiretap channel (without physically degraded assumption), a better usage of the feedback is to generate not only key but also (1) How should the feedback scheme in [9] be extended to the action-dependent wiretap channel with noncausal state? (2) For the action-dependent wiretap channel with noncausal state, does the hybrid feedback scheme in [9] still gain advantages over the traditional one used in [1][2][3][4][5][6][7][8]?
The main contribution of this paper includes: (1) We propose a new lower bound on the secrecy capacity of the action-dependent wiretap channel with noncausal state and noiseless feedback, which is constructed according to a hybrid feedback scheme similar to that in [9]. (2) From a Gaussian example, which is also called the action-dependent dirty paper wiretap channel with noiseless feedback, we show that our new lower bound on the secrecy capacity is larger than the secret key based lower bound. Moreover, we find that our new lower bound achieves the secrecy capacity for some special cases. The remainder of this paper is organized as follows. Section 2 is about the problem formulation and the main result of this paper. The achievability proof of our new lower bound on the secrecy capacity of the action-dependent wiretap channel with noncausal state and noiseless feedback is provided in Section 3. A Gaussian example and numerical results are provided in Section 4. Final conclusions are presented in Section 5.

Problem Formulation and New Result
Notations: For the rest of this manuscript, the random variables (RVs), values and alphabets are written in uppercase letters, lowercase letters and calligraphic letters, respectively. The random vectors and their values are denoted by a similar convention. For example, Y represents a RV, and y represents a value in the alphabet Y. Similarly, Y N represents a random N-vector (Y 1 , ..., Y N ), and y N = (y 1 , ..., y N ) represents a vector value in Y N (the Nth Cartesian power of Y). In addition, for an event X = x, its probability is denoted by P(x). In the remainder of this manuscript, the base of the log function is 2.
Model description: In Figure 1, the channel is discrete memoryless, i.e., the overall channel transition probability is given by where s i ∈ S, x i ∈ X , y i ∈ Y and z i ∈ Z. The message W is uniformly distributed in its alphabet W = {1, 2, ..., |W |}, and a stochastic action encoder encodes W into an action sequence A N . The channel state sequence S N is generated through a discrete memoryless channel (DMC) A N → S N with transition probability P(s|a). Since S N is non-causally known by the channel encoder and the legitimate receiver's channel output is sent back to the transmitter, the ith (i ∈ {1, 2, ..., N}) channel input , where f i is a stochastic encoding function. The legitimate receiver produces an estimationŴ = ψ(Y N ) (ψ is the legitimate receiver's decoding function), and the average decoding error probability equals The eavesdropper's equivocation rate of the message W is formulated as Given a positive number R, if for arbitrarily small and sufficiently large N, there exist a pair of channel encoder and decoder described above such that we say R is achievable with weak perfect secrecy. The secrecy capacity C f sa consists of all achievable weak secrecy rates, and bounds on C f sa are given in the following theorems and corollary.
Proof. The lower bound R f * sa is achieved by combining the binning scheme in [13] with the hybrid coding scheme in [9], and the details about the proof are in Section 3.
The following lower bound R f * * sa in Corollary 1 can be directly obtained from Theorem 1 by letting V be constant, and this lower bound can be viewed as a secret key based lower bound (application of the secret key based feedback strategy [1] to the model of Figure 1) on C f sa .
and the joint distribution is denoted by P(u, a, s, x, y, z) = P(y, z|x, s)P(x|u, s)P(u|a, s)P(a, s).
Remark 1. Note that [21] also proposed a secret key based lower bound on the secrecy capacity of the physically degraded action-dependent wiretap channel with noncausal state and noiseless feedback. However, we should point out that the model studied in [21] assumes the action encoder is a deterministic encoder, i.e., if the eavesdropper knows A N , he also knows the message W. Hence, our lower bound R f * * sa generalizes that in [21] as the deterministic action encoder is a special case of the stochastic one studied in this paper and there is no physically degraded assumption in this paper.
Besides the above lower bounds on C f sa , the following theorem shows a simple upper bound on C f sa .
and the joint distribution is denoted by Equation (8).
Proof. Since C f sa cannot exceed the capacity of the model in Figure 1 without eavesdropper, we know that C f sa is upper bounded by the capacity of the action-dependent channel with feedback. In [13], it has been shown that feedback does not increase the capacity of the action-dependent channel (max(I(U; Y) − I(U; S|A))), hence Theorem 2 is proved. The proof of Theorem 2 is completed.
In Section 4, the above proposed hybrid lower bound R f * sa ise compared with the secret key based lower bound R f * * sa via a Gaussian example, and we show which feedback strategy performs better.

Proof of Theorem 1
In this section, the hybrid feedback strategy for the wiretap channel [9] and the binning scheme for the action-dependent channel with noncausal state at the transmitter [13] are combined to show the achievability of Theorem 1. The rest of this section is organized as follows. The code-book construction and the transmission scheme are described in Section 3.1, and the equivocation analysis of the proposed scheme is shown in Section 3.2.

Code-Book Construction and Transmission Scheme
Definitions and notations: • Similar to the coding scheme in [9], suppose that the overall transmission consists of B blocks, and the codeword length in each block is N.
respectively. In addition, let X B = (X 1 , ...,X B ) be a collection of the random vectors X N for all blocks. Analogously, we have The vector value is written in lower case letter.

Decoding scheme:
The decoding procedure starts from block B. At block B, the legitimate receiver chooses ā u B (1, 1, 1, w * B , w * * B−1 ) which is jointly typical withȳ B andā B (1, 1, 1). For the case that more than one or no suchū B exists, declare a decoding error. Based on the Packing Lemma [22] and a similar argument in [13], this kind of decoding error approaches to zero when After decodingū B , the legitimate receiver extracts w * * B−1 from it. Then, he tries to select only onē v B−1 (w * * B−1 , w * * * B−1 ) such that given w * * B−1 ,v B−1 is jointly typical withȳ B−1 . For the case that more than one or no suchv B−1 exist, declare a decoding error. Based on the Packing Lemma [22], this kind of decoding error approaches to zero when R * * * ≤ I(V; Y).
After obtaining such uniquev B−1 , the legitimate receiver tries to find only one pair of (ū B−1 ,ā B−1 ) such that (ȳ B−1 ,ā B−1 ,v B−1 ,ū B−1 ) are jointly typical. Based on the Packing Lemma [22] and a similar argument in [13], this kind of decoding error approaches to zero when After decodingū B−1 , the legitimate receiver picks out w B−1,1 , w B−1,2 ⊕ k B−1 , w * * B−2 from it. Note that the legitimate receiver has full knowledge of k B−1 = g B−1 (ȳ B−2 ), and hence he obtains the message w B−1 = (w B−1,1 , w B−1,2 ). Analogously, the legitimate receiver decodes the messages w B−2 , w B−3 , ..., w 1 , and the decoding procedure is completed. For convenience, the encoding and decoding schemes are explained by the following Figures 2 and 3, respectively.

Equivocation Analysis
The overall equivocation ∆, which is denoted by ∆ = 1 BN H(W|Z B ), is given by where (a) is due to the definitionsW 1 = (W 1,1 , ..., W B,1 ) andW 2 = (W 1,2 , ..., W B,2 ). The term H(W 1 |Z B ) in Equation (15) can be bounded by where (b) is implied by H(W 1 |U B ) = 0, (c) is due to the construction of U B and the channel is memoryless, and (d) is due to that givenw 1 and z B , the eavesdropper attempts to find a unique u B that is jointly typical with his own received signals z B , and according to the Packing Lemma [22], we can conclude that the eavesdropper's decoding error tends to zero if where (e) is due to the Markov chain is from the balanced coloring Lemma [9] (p. 264), i.e., givenz i−1 andū i−1 , there are at least γ 1+δ colors, which implies that where 1 , 2 and δ approach to 0 as N goes to infinity. Substituting Equations (16) and (18) into Equation (15), we have The bound in Equation (20) indicates that if Next, from Equations (22), (10) and (14), we can conclude that Then, implied by Equations (21) and (14), we have Finally, applying Fourier-Motzkin elimination to remove R 1 , R 2 (R = R 1 + R 2 ), R , R , R * and R * * from Equations (22), (23), (24), (12), (14), (17) and (21), Theorem 1 is proved.

The Action-Dependent Dirty Paper Wiretap Channel with Noiseless Feedback
The Gaussian case of the action-dependent wiretap channel with noncausal state at the transmitter and feedback, which we also call the action-dependent dirty paper wiretap channel with noiseless feedback, is depicted in Figure 4. At time i (i ∈ {1, 2, ..., N}), the inputs and outputs of this Gaussian model satisfy where X i is the channel input subject to an average power constraint P, A i is the output of the action encoder subject to an average power constraint P A , t is a constant, and W i , η 1,i , η 2,i are independent Gaussian noises and are i.i.d. across the time index i. Here, note that W i ∼ N (0, σ 2 w ), η 1,i ∼ N (0, σ 2 1 ) and η 2,i ∼ N (0, σ 2 2 ). The secrecy capacity of the action-dependent dirty paper wiretap channel with feedback is denoted by C f sag , and the lower and upper bounds on C f sag will be given in the remainder of this section. Before we show the bounds on C f sag , define where α 2 P A + γ 2 σ 2 w ≤ P, G ∼ N (0, P − α 2 P A − γ 2 σ 2 w ) and G, A, W, η 1 , η 2 are independent of each other. Note that the definitions in Equation (26) are exactly the same as those in the action-dependent dirty paper channel [13]. Further, define First, substituting Equations (26) and (25) into Equation (5), our new lower bound R f * sag on C f sag is given by the following Theorem 3.
Second, substituting Equations (26) and (25) into Equation (7), the secret key based lower bound R f * * sag on C f sag is given by the following Theorem 4.
Third, substituting Equations (26) and (25) into Equation (9), the upper bound C f −out sag on C f sag is given by Theorem 5.
Proof. Here note that Equation (9) is also the capacity of the action-dependent channel with noncausal state at the transmitter, and the capacity formula of its Gaussian case is be given in [13] by substituting Equations (26) and (25) into Equation (9) and maximizing the parameters δ and β. Now, directly using the Gaussian capacity formula in [13], we have Equation (37). The proof is completed.
Finally, to show the feedback gain, we also provide a lower bound C in sag on the secrecy capacity C sag of the action-dependent dirty paper wiretap channel (see Theorem 6).
Proof. In [20], a lower bound C in sa on the secrecy capacity C sa of the discrete memoryless actiondependent wiretap channel with noncausal state at the transmitter is provided, and it is given by Here, note that the term H(A|Z) in Equation (39) holds due to the assumption that the action encoder is a deterministic function of the transmitted message. Specifically, once the eavesdropper obtains the action sequence A N , he knows the transmitted message, hence the achievable secrecy rate cannot exceed the eavesdropper's uncertainty about A N , i.e., H(A|Z).
In this paper, we use a stochastic action encoder instead of the deterministic one in [20], which indicates that, even if the eavesdropper obtains A N , he does not know the transmitted message due to the randomness assumption of the action encoder. Hence. the term H(A|Z) no longer holds in this paper, i.e., for the action-dependent wiretap channel with noncausal state at the transmitter and stochastic action encoder, a lower bound C in * sa is given by Finally, substituting Equations (26) and (25) into Equations (40), Equation (38) is obtained. The proof is completed. Figure 5 depicts the bounds on C f sag and the lower bound C in sag on the secrecy capacity of the actiondependent dirty paper wiretap channel for σ 2 w = P A = 1, σ 2 1 = 1, σ 2 2 = 0.1, t = 0.9 and several values of P. For this case, we see that there is no positive achievable secrecy rate C in sag of the action-dependent dirty paper wiretap channel, and feedback enhances C in sag . Moreover, we see that the hybrid feedback scheme performs better than the secret key based feedback scheme, and there exists a gap between the lower and upper bounds on C f sag when P is sufficiently large. Figure 6 depicts the bounds on C f sag and the lower bound C in sag on the secrecy capacity of the actiondependent dirty paper wiretap channel for σ 2 w = P A = 1, σ 2 1 = 0.1, σ 2 2 = 0.1, t = 0.6 and P taking values in [0, 0.5]. For this case, we see that feedback enhances C in sag , and the hybrid feedback scheme performs better than the secret key based feedback scheme. Moreover, we see that, when P is small, the hybrid feedback scheme is optimal, i.e., its corresponding lower bound meets the upper bound, which implies that the secrecy capacity C f sag is determined for this case. Figure 7 is an extension of Figure 6 with P taking values in [0, 50]. We see that, when P is sufficiently large, there exists a gap between the lower and upper bounds on C f sag , and eliminating this gap still has a long way to go.

Conclusions
In this paper, we propose two achievable secrecy rates for the action-dependent wiretap channel with noncausal state at the transmitter and feedback, where one rate is achieved by using the already existing secret key based feedback strategy, and the other is achieved by using a hybrid feedback strategy. From a Gaussian example (also called the action-dependent dirty paper wiretap channel with feedback), we show that both feedback strategies proposed in this paper enhance the achievable secrecy rate of the action-dependent dirty paper wiretap channel, and the hybrid feedback strategy performs better than the secret key based feedback strategy. Moreover, we show that the hybrid feedback strategy is optimal for some special cases.
Author Contributions: H.Z. did the theoretical work, performed the experiments, analyzed the data and drafted the work; B.D. designed the work, performed the theoretical work, analyzed the data, interpreted the data for the work and revised the work; and L.Y. performed the theoretical work, interpreted the data for the work and revised the work. All authors approved the version to be published and agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.