New Result on the Feedback Capacity of the Action-Dependent Dirty Paper Wiretap Channel

The Gaussian wiretap channel with noncausal state interference available at the transmitter, which is also called the dirty paper wiretap channel (DP-WTC), has been extensively studied in the literature. Recently, it has been shown that taking actions on the corrupted state interference of the DP-WTC (also called the action-dependent DP-WTC) helps to increase the secrecy capacity of the DP-WTC. Subsequently, it has been shown that channel feedback further increases the secrecy capacity of the action-dependent DP-WTC (AD-DP-WTC), and a sub-optimal feedback scheme is proposed for this feedback model. In this paper, a two-step hybrid scheme and a corresponding new lower bound on the secrecy capacity of the AD-DP-WTC with noiseless feedback are proposed. The proposed new lower bound is shown to be optimal (achieving the secrecy capacity) and tighter than the existing one in the literature for some cases, and the results of this paper are further explained via numerical examples.


Introduction
Dirty paper coding is one of the most important pre-coding schemes in wireless communications and has a wide range of applications in information hiding [1]. Dirty paper coding was first investigated by Costa in his well-known paper on the dirty paper channel (DPC) [2], where a corrupted Gaussian state interference of a white Gaussian channel is noncausally known at the transmitter and not available at the receiver. Costa showed that the capacity of the DPC equals the capacity of the same model without state interference, which indicates that though the receiver does not know the state interference, it can still be perfectly removed by using dirty paper coding.
Note that in [2], the channel state is assumed to be generated by nature. However, in some practical scenarios, the state is affected or controlled by the communication systems, e.g., in intelligent reflecting surface-aided communication systems, the state is formed in part by the reflecting phase shift which is actually controlled by the transceiver [3]. Such a case was first investigated by Weissman in his paper on the action-dependent dirty paper channel (AD-DPC) [4], where the corrupted state interference of the DPC is affected and noncausally known by the transmitter. In [4], a lower bound on the capacity of AD-DPC was proposed, and the capacity was fully determined in [5].
In recent years, the study of the above dirty paper channels under additional secrecy constraints has received a lot attention. Specifically, [6] studied the discrete memoryless wiretap channel with state interference noncausally known by the transmitter, and obtained upper and lower bounds on the secrecy capacity. The authors of [7] extended the model studied in [6] to the broadcast situation, and also provided bounds on the secrecy capacity region of this extended model. The authors of [8,9] studied the Gaussian case of [6], namely, the dirty paper wiretap channel, and proposed bounds on the secrecy capacity. Furthermore, [8,9] pointed out that the proposed secret dirty paper coding increases the secrecy capacity of the Gaussian wiretap channel [10]. The above works mainly adopted the tools in [11,12] for establishing the secrecy rate/capacity. Very recently, the authors of [13] studied the AD-DPC with an additional eavesdropper, which is also called the actiondependent dirty paper wiretap channel (AD-DP-WTC), and proposed lower and upper bounds on the secrecy capacity. Note that the capacity results given in [9,13] indicate that: • There is a penalty term between secrecy capacity and capacity of the same model without secrecy constraints. • If the eavesdropper's channel is less noisy than the legitimate receiver's, the secrecy capacity may equal zero, i.e., no positive rate can be guaranteed for secure communications.
Very recently, it has been shown that channel feedback is an effective way to enhance the secrecy capacities of the DP-WTC [14,15] and the multi-input multi-output (MIMO) X-channels [16]. In [16], it has been shown that feedback increases the secure degrees of freedom (SDoF) region of the MIMO X-channel with secrecy constraints, which indicates that feedback also enlarges the secrecy capacity of the same model, even if the eavesdropper's channel is less noisy than the legitimate receiver's. The authors of [14][15][16] mainly adopted the idea of a generating secret key from the channel feedback and using this key to encrypt the transmitted message. Subsequently, [13] showed that a variation of the classical Schalkwijk-Kailath (SK) feedback scheme [17] for the point-to-point white Gaussian channel achieves the secrecy capacity of the DP-WTC with noiseless feedback, and the secrecy capacity equals the capacity of the same model without secrecy constraints, i.e., to achieve perfect secrecy, no rate needs to be sacrificed even if the eavesdropper gains advantage over the legitimate receiver. In addition, the authors of [13] also proposed a sub-optimal SK type feedback scheme for the AD-DP-WTC with noiseless feedback (see Figure 1), i.e., this scheme achieves a lower bound on the secrecy capacity of the AD-DP-WTC with noiseless feedback, and the secrecy capacity remains open.
In this paper, we derive a new lower bound on the secrecy capacity of the AD-DP-WTC with noiseless feedback, which is based on a hybrid two-step feedback scheme. The proposed new lower bound is shown to be optimal and tighter than the existing one in the literature for some cases, and the results of this paper are further explained via numerical examples. The remainder of this paper is organized as follows. A formal definition of the model studied in this paper and previous results are introduced in Section 2. The main result and the corresponding proof are given in Sections 3 and 4, respectively. Section 5 includes the summary of all results in this paper and discusses future work.

Model Formulation
For the AD-DP-WTC with noiseless feedback (see Figure 1), the i-th (i ∈ {1, 2, . . . , N}) channel inputs and outputs are given by where X i is the output of the channel encoder subject to an average power constraint P , A i is the output of the action encoder subject to an average power constraint , Y i and Z i are channel outputs, respectively, at the legitimate receiver and the eavesdropper, and W i , η 1,i , η 2,i are independent Gaussian noises and are independently identically distributed (i.i.d.) across the time index i. Note that At time instant i (i ∈ {1, 2, . . . , N}), a corrupted state interference S i is generated through a white Gaussian channel with i.i.d. noise W i ∼ N (0, σ 2 w ) and channel input A i , where A i is a (stochastic) function of the message M. Since the corrupted state interference S N = (S 1 , . . . , S N ) is noncausally known by the channel encoder, the i-th channel input X i is a (stochastic) function of the message M, S N , and the feedback The legitimate receiver generates an estimationM = ψ(Y N ), where ψ is the legitimate receiver's decoding function, and the average decoding error probability equals The eavesdropper's equivocation rate of the message M is denoted by A rate R is said to be achievable with perfect weak secrecy if for any and sufficiently large N, there exists a channel encoder-decoder such that The secrecy capacity of the AD-DP-WTC with feedback, which is the maximum achievable secrecy rate defined in (4), is denoted by C f sag . A new lower bound, R f sag on C f sag , will be given in the next section.

SK Feedback Scheme for the Point-to-Point White Gaussian Channel
For the point-to-point white Gaussian channel with feedback, at time i (i ∈ {1, 2, . . . , N}), channel input and output are given by where X i is the channel input subject to an average power constraint P ( 1 The channel input X i is a function of the message M and the feedback Y i−1 . It is well known that the capacity C f g of the white Gaussian channel with feedback equals the capacity C g of the same model without feedback, i.e., The authors of [17] showed that the classical SK scheme achieves C f g , and this scheme is described below. Since M takes the values in M = {1, 2, . . . , 2 NR }, we divide the interval [−0.5, 0.5] into 2 NR equally spaced sub-intervals, and the center of each sub-interval is mapped to a message value in M. Let θ be the center of the sub-interval with respect to (w.r.t.) the message M.
In [17], it has been shown that the decoding error P e of the above coding scheme is upper bounded by where Q(x) is the tail of the unit Gaussian distribution evaluated at x, and From (13) and (14), we conclude that if P e → 0 as N → ∞. Recently, [18] showed that the above classical SK scheme, which is not designed with the consideration of secrecy, already achieves the secrecy capacity of the Gaussian wiretap channel with noiseless feedback, i.e., the corresponding secrecy capacity equals the capacity C f g of the same model without secrecy constraints.

Previous Results on the AD-DP-WTC with Feedback
For the capacity C f sag of the AD-DP-WTC with feedback, [13] showed that it is bounded by where ). (18) Note that C ag is the capacity of the AD-DPC without feedback, and is given in [5].
Since the capacity C f sag of the AD-DP-WTC with feedback is no larger than that of the same model without secrecy constraints, and feedback does not increase the capacity of the AD-DPC [4], and C ag serves as a trivial upper bound on C f sag . The lower bound C f sag ≥ L is derived by constructing for ρ * 2 1 + ρ * 2 2 = 1, and for ρ * 2 1 + ρ * 2 2 < 1, where G is randomly generated according to G ∼ N (0, P(1 − ρ * 2 1 − ρ * 2 2 )) and it is independent of A and W. Substituting (19) and (20) into (1), the AD-DP-WTC with feedback is equivalent to the Gaussian wiretap channel with feedback with input A, channel noise W + G + η 1 + η 2 , channel output Y of the legitimate receiver, and the channel output Z of the eavesdropper. Directly applying the SK scheme introduced in the preceding subsection, we conclude that L is achievable. Moreover, from [18], we know that the SK scheme also achieves the secrecy capacity of the Gaussian wiretap channel with noiseless feedback, which indicates that L is achievable with perfect weak secrecy, and the proof is completed.
Note that the above lower and upper bounds on C f sag do not meet in general, and exploring a tighter lower bound on C f sag is the motivation of this paper.

A New Lower Bound on the Secrecy Capacity of the AD-DP-WTC with Feedback
Theorem 1. A new lower bound on the secrecy capacity C f sag of the AD-DP-WTC with feedback is given by Remark 1. Comparing the new lower bound in (21) with the upper bound C ag in (16), it is easy to see that there still exists a gap between the two bounds due to the penalty term 1 2 .
The following Corollary 1 shows that the proposed new lower bound in (21) is optimal for a special case. Moreover, the following Corollary 2 shows that the new lower bound in (21) is tighter than the existing lower bound in (16) when σ 2 w tends to infinity.
which indicates that the secrecy capacity is determined for this case.
where L is the existing lower bound defined in (18).
Proof. Define the gap R G between the new lower bound in (21) and the existing lower bound in (18) by • For the case that the maximum is achieved when ρ 2 • For the case that the maximum is achieved when Combining (25) and (26), we conclude that lim σ 2 The proof of Corollary 2 is completed. Proof sketch of Theorem 1. The main idea of the achievable scheme is briefly illustrated by Figures 2 and 3. In Figure 2, we split the message M into two parts The sub-message M 1 is available at both the action encoder and channel encoder, and the sub-message M 2 is only available at the channel encoder. Let Moreover, M 1 is also encoded as A i with power P A , and which indicates that U i is a deterministic function of A i . The i-th (i ∈ {1, 2, . . . , N}) channel inputs and outputs are rewritten as where W i = G i + W i is an i.i.d Gaussian noise process with zero mean and variance σ 2 w = (σ w + ρ 2 1 P) 2 , and A * i = A i + U i is subject to an average power constraint P * = P A + ρ 2 2 P + 2 ρ 2 2 PP A . In Figure 3, since M 1 is known by the channel encoder, the codeword A * i = A i + U i can be subtracted when applying an SK type feedback scheme to M 2 , i.e., the transmission of M 2 is through an equivalent channel with input V i , output Moreover, since M 1 and S N are known by the channel encoder, W N = S N − A N and In addition, the transmission of M 1 is through an equivalent channel with inputs Then, applying an SK type scheme to M 2 , and Wyner's random binning scheme [11] together with Feinstein's greedy coding scheme [19] to M 1 , the lower bound in Theorem 1 is obtained. The detail of the proof is given in the next section.  Figure 4, we see that our new scheme performs better than the existing one when the noise variance σ 2 w is large enough.
Decoding procedure: The legitimate receiver does a two-step decoding scheme. First, by applying Feinstein's decoding rule [19] to the decoding of A * N 2 . According to Feinstein's lemma [19], the decoding error probability of M 1 and M * 1 , denoted by P e1 , can be arbitrarily small if where Y N 2 = (Y 2 , Y 3 , . . . , Y N ). Second, after decoding M 1 , the legitimate receiver obtains A * k for all 2 ≤ k ≤ N, and subtracts A * k from Y k , then the legitimate receiver obtains Y k = Y k − A * k = V k + η 1,k + W k . At time 1, the legitimate receiver obtains Y 1 = V 1 + W 1 + η 1,1 and computeŝ At time k (2 ≤ k ≤ N), the legitimate receiver's estimationθ k of θ is given bŷ where where (c) follows from (31). Now we bound the decoding error probability P e2 of M 2 as follows. Fromθ N = θ + ε N and the definition of θ, we have where (d) following from Q(x) is the tail of the unit Gaussian distribution evaluated at x, and (e) is from Lemma 1. Since Q(x) is decreasing while x is increasing, from (42), we can conclude that if P e2 → 0 as N is large enough. Note that the total decoding error probability P e of M = (M 1 , M 2 ) is upper bounded by P e ≤ P e1 + P e2 . From the above analysis, we conclude that P e → 0 as N tends to infinity if (38) and (43) are guaranteed.
The first part

of (44) is bounded by
where ( f ) follows from the Markov chain M 1 → Z N 2 → Z 1 and Z N 2 = (Z 2 , Z 3 , . . . , Z N ), (g) is due to the fact that A * N 2 is a deterministic function of M 1 , M * 1 , and (h) follows from Fano's inequality when the codeword A * N 2 is generated by Feinstein's greedy construction [19], i.e., if given M 1 , the eavesdropper's decoding error probability P ew of M * 1 is arbitrarily small (P ew ≤ ) as N tends to infinity, then using Fano's inequality, we have Formula (45) indicates that N is sufficiently large, if we have where → 0 as N → ∞. From (46) and (48), we conclude that Substituting (50) into (38) and noting that the maximum of R 1 is achieved when N → ∞, then we have The fundamental limit of R 1 in (51) is upper bounded by Proof. The Proof of Lemma 2 is in Appendix B.

Conclusions and Future Work
This paper shows a new lower bound on the secrecy capacity of the AD-DP-WTC with noiseless feedback by proposing a novel two-step hybrid feedback scheme. The numerical result shows that when the noise variance σ 2 w is large enough, our new feedback scheme performs better than the existing one in the literature. Moreover, when the eavesdropper's channel noise variance is fairly large, our new feedback scheme is almost optimal (the new lower bound almost equals the upper bound). Possible future work could be to extend the proposed feedback scheme to the multiple-access situation.
Author Contributions: G.X. did the theoretical work, performed the experiments, analyzed the data, and drafting the work; B.D. designed the work, did the theoretical work, analyzed the data, interpreted the data for the work, and revised the work. Both of the authors approve the version to be published and agree 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. Both authors have read and agreed to the published version of the manuscript. and α 1 = Var(ε 1 ), and hence α 1 = E( In the same way, since where where (a) follows from (34), and Y k−1 = V k−1 + η 1,k−1 = P α k−2 ε k−2 + η 1,k−1 . Now, it remains to further compute the second part of Lemma 1. According to (35), we have V k = P α k−1 ε k−1 , and it can be further expressed as where (b) follows from (34), and Therefore, we conclude that where (c) follows from V 2 = P α 1 . From (A4) and (33), we see that V k is a function of (η 1,1 , η 1,2 , . . . , η 1,k−1 ), i.e., Finally, according to (A2) and (A4), we complete the proof of Lemma 1.
where (b) follows from the entropy power inequality, (c) follows from N−1 is increasing and where 2 , and the last inequality of (A8) is from Jensen's inequality.