Feedback Coding Schemes for the Broadcast Channel with Mutual Secrecy Requirement

Recently, the physical layer security (PLS) of the communication systems has been shown to be enhanced by using legal receiver's feedback. The present secret key based feedback scheme mainly focuses on producing key from the feedback and using this key to protect part of the transmitted message. However, this feedback scheme has been proved only optimal for several degraded cases. The broadcast channel with mutual secrecy requirement (BC-MSR) is important as it constitutes the essence of physical layer security (PLS) in the down-link of the wireless communication systems. In this paper, we investigate the feedback effects on the BC-MSR, and propose two inner bounds and one outer bound on the secrecy capacity region of the BC-MSR with noiseless feedback. One inner bound is constructed according to the already existing secret key based feedback coding scheme for the wiretap channel, and the other is constructed by a hybrid coding scheme using feedback to generate not only keys protecting the transmitted messages but also cooperative messages helping the receivers to improve their decoding performance. The performance of the proposed feedback schemes and the gap between the inner and outer bounds are further explained via two examples.


I. INTRODUCTION
Besides reliability, introducing an additional secrecy criteria into a physically degraded 1broadcast channel, Wyner [1] first studied the secure transmission over the wiretap channel (WTC).Later, on the basis of [1], Csiszár and Körner [2] studied the WTC without the "physically degraded" assumption and with an additional common message available at both the legal receiver and the wiretapper.The outstanding work [1]- [2] reveals the reliability-security trade-off of the communication channels in the presence of a wiretapper.The follow-up study of the WTC mainly focuses on the multi-user channel in the presence of a wiretapper (e.g.multiple-access wiretap channel [3], [4], relay-eavesdropper channel [5], [6], broadcast wiretap channel [7], [8], two way wiretap channel [9], [10], etc.), and the multi-terminal security, see [11]- [15].
In recent years, the effect of legal receiver's feedback on the PLS of communication channels attracts a lot of attention.For the WTC with noiseless feedback (WTC-NF), Ahlswede and Cai [16] pointed out that to enhance the secrecy capacity of the WTC, the best use of the legal receiver's feedback channel output is to generate random bits from it and use these bits as a key by the transmitter protecting part of the transmitted message.Using this secret key based feedback scheme, Ahlswede and Cai [16] determined the secrecy capacity of the physically degraded WTC-NF, and the secrecy capacity of the general WTC-NF has not been determined yet.On the basis of [16], Ardestanizadeh et al. investigated the WTC with rate limited feedback [17] where the legal receiver is free to use the noiseless feedback channel to send anything as he wishes (up to a rate R f ).For the degraded case, they showed that the best choice of the legal receiver is sending a key through the feedback channel, and if the legal receiver's channel output Y 1 is sent, the best use of it is to extract a key.Later, Schaefer et al. [18] extended the work of [17] to a broadcast situation, where two legitimate receivers of the broadcast channel independently sent their secret keys to the transmitter via two noiseless feedback channels, and these keys help to increase the achievable secrecy rate region of the broadcast wiretap channel [7].Cohen er al. [19] generalized Ardestanizadeh et al.'s work [17] by considering the WTC with noiseless feedback, and with causal channel state information (CSI) at both the transmitter and the legitimate receiver.Cohen er al. [19] showed that the transmitted message can be protected by two keys, where one is generated from the noiseless feedback, and the other is generated by the causal CSI.They further showed that these two keys increases the achievable secrecy rate of the WTC with rate limited feedback [17].Other related works in the WTC with noiseless feedback and CSI are investigated in [20]- [21].Here note that for the WTC-NF, the present literature ( [16]- [21]) shows that the secrecy capacity is achieved only for the degraded case, i.e., Ahlswede and Cai's secret key based feedback coding scheme [16] is only optimal for the degraded channel models.Finding the optimal feedback coding scheme for the general channel models needs us to exploit other uses of the feedback.
The broadcast channel with mutual secrecy requirement (BC-MSR) is an important model for the PLS in the down-link of the wireless communication systems.The already existing literature [12], [13] provides inner and outer bounds on the secrecy capacity region of BC-MSR.To investigate the feedback effects on the BC-MSR (see Figure 1), in this paper, two feedback strategies for the BC-MSR are proposed.One is an extension of the already existing secret key based feedback scheme for the WTC [16], and the other is a hybrid coding scheme using the feedback to generate not only keys but also cooperative messages helping the receivers to improve their decoding performance.Two inner bounds on the secrecy capacity region of the feedback model shown in Figure 1 are constructed with respect to the proposed two feedback coding schemes.Moreover, for comparison, we also provide a corresponding outer bound.These  Now the remiander of this paper is organized as follows.Necessary mathematical background, the previous non-feedback coding scheme for the BC-MSR and a generalized Wyner-Ziv coding scheme are provided in Section II.Section III is for the model formulation and the main results.
Section IV shows the details about the proof of the main results.Two examples and numerical results are shown in Section V, and a summary of this work is given in Section VI.

A. Notations and Basic Lemmas
Notations: For the rest of this paper, 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 1 represents a RV, and y 1 represents a value in Y 1 .Similarly, Y N 1 represents a random N -vector (Y 1,1 , ..., Y 1,N ), and y N 1 = (y 1,1 , ..., y 1,N ) represents a vector value in Y N 1 (the N -th Cartesian power of Y 1 ).In addition, for an event X = x, its probability is denoted by P (x).In the remainder of this paper, the base of the log function is 2.
An independent identically distributed (i.i.d.) generated vector x N according to the probability where π x N (x) is the number of x showing up in x N .The set composed of all typical vectors x N is called the strong typical set, and it is denoted by T N (P (x)).The following lemmas related with T N (P (x)) will be used in the rest of this paper.
generated random vectors with respect to (w.r.t.) the probabilities P (x) and P (y), respectively.
Here notice that X N is independent of Y N (l).Then there exists ν > 0 satisfying the condition that lim Lemma 2: (Packing Lemma [23]): Let X N and Y N (l) (l ∈ L and |L| ≤ 2 N R ) be i.i.d.generated random vectors w.r.t. the probabilities P (x) and P (y), respectively.Here notice that X N is independent of Y N (l).Then there exists ν > 0 satisfying the condition that lim , where ϕ(ν) → 0 as ν → 0.
Remark 1: From Lemma 3, it is easy to see that there are at least colors and at most γ colors mapped by | and applying the properties of the conditional strong typical set [23], we see that where 1 and 2 tend to zero while N → ∞.

B. Non-feedback coding scheme for the BC-MSR
For the model of Figure 1 without feedback, a hybrid coding scheme combining Marton's binning technique for the general broadcast channel [22] with the random binning technique for the wiretap channel [1] is proposed in [12], [13].In this subsection, we review this hybrid coding scheme.

Definitions:
The message W j (j = 1, 2) is conveyed to Receiver j, and it is uniformly drawn from the set {1, ..., 2 N R j }.The randomly generated W j , which is used for confusing the illegal receiver 2 , is 2 The idea of using random messages to confuse the wiretapper is exactly the same as the random binning technique used in Wyner's wiretap channel [1], where this randomly produced message is analogous to the randomly chosen bin index used in the random binning scheme.
Moreover, similar to the coding scheme in Marton's achievable region for the broadcast channel [22], the message W j , which enables its codeword U N j to be jointly typical with other codewords, chooses values from the set {1, ..., 2 N R j }.

Encoding procedure:
The transmitter selects q N (w 0 ), u N 1 (w 1 , w 1 , w 1 ) and u N 2 (w 2 , w 2 , w 2 ) to transmit.Here notice that w 0 , w 1 and w 2 are randomly chosen from the sets {1, 2, ..., 2 N R 0 }, {1, 2, ..., 2 N R 1 } and {1, 2, ..., 2 N R 2 }, respectively, and the indexes w 1 and w 2 are chosen by finding a pair of )) satisfying the condition that given q N (w 0 ), (u N 1 , u N 2 , q N (w 0 )) are jointly typical.If multiple pairs exist, choose the pair with the smallest indexes; if no such pair exists, proclaim an encoding error.On the basis of the covering lemma (see Lemma 1), this kind of encoding error tends to zero if (2.4) Decoding procedure: First, Receiver j (j = 1, 2) chooses a unique q N jointly typical with y N j .if more than one or no such q N exists, declare an decoding error.From packing lemma (see Lemma 2), this kind of decoding error tends to zero if (2.5) After decoding q N , Receiver j seeks a unique u N j satisfying the condition that (u N j , q N , y N j ) are jointly typical.From the packing lemma, this kind of decoding error tends to zero if (2.6) Once u N j is decoded, Receiver j extracts w j in it.

Equivocation analysis:
The Receiver 2's equivocation rate ∆ 1 , which is denoted by where (a) follows from H(W 1 |U N 1 ) = 0, (b) follows from the generation of Q n , U n 1 , U n 2 and the channel is memoryless, and (c) follows from that given w1 , q N , u N 2 and y n 2 , Receiver 2 tries to find out only one u n 1 that is jointly typical with y n 2 , q n , u n 2 , and implied by the packing lemma, we see that Receiver 2's decoding error tends to zero if then applying Fano's lemma, (2.9) and From (2.4), (2.6), (2.8), (2.9), (2.10) and (2.11), the achievable secrecy rate region C bc−msr for the BC-MSR [12] is obtained, and it is given by Combining the above coding scheme for C bc−msr with the already existing secret key based feedback scheme for the WTC [16], it is not difficult to propose a secret key based feedback coding scheme for the BC-MSR, which will be shown in the next section.
C. The Generalized Wyner-Ziv Coding Scheme for the Distributed Source Coding with Side Information Fig. 2: The distributed source coding with side information In this subsection, we review the generalized Wyner-Ziv coding scheme for the distributed source coding with side information [24].For the distributed source coding with side information shown in Figure 2, the source X N is correlated with the side information Y N 1 and Y N 2 , and they are i.i.d.generated according to the probability P (x, y 1 , y 2 ).Using an encoding function into three indexes W * 0 , W * 1 and W * 2 respectively choosing values from the sets {1, 2, ..., 2 N R 0 }, {1, 2, ..., 2 N R 1 } and {1, 2, ..., 2 N R 2 }.The indexes W * 0 , W * j (j = 1, 2) together with the side information Y N j are available at Receiver j.Receiver j generates a reconstruction sequence to the indexes W * 0 , W * j and the side information Y N j .The goal of the communication is that the reconstruction sequence V N j is jointly typical with the source X N according to the probability P (v j |x) × P (x).
A rate triplet (R 0 , R 1 , R 2 ) is said to be achievable if for any > 0, there exists a sequence of encoding and reconstruction functions (φ, ϕ 1 , ϕ 2 ) such that as N → ∞.The following generalized Wyner-Ziv Theorem [24] provides an achievable region R inner consisting of achievable rate triplets (R 0 , R 1 , R 2 ) for this distributed source coding with side information problem.
Here note that the generalized Wyner-Ziv coding scheme described above indicates that in a broadcast channel, each receiver's channel output can be viewed as side information helping the receiver to decode an estimation of the channel input, and this estimation of the channel input helps the receiver to improve his decoding performance.Motivated by this generalized Wyner-Ziv coding scheme, in the next section, a hybrid feedback strategy for the BC-MSR is proposed, which combines the already existing secret key based feedback scheme for the WTC [16] and the generalized Wyner-Ziv coding scheme with the previous non-feedback coding scheme for the BC-MSR described in Subsection II-B.

III. PROBLEM FORMULATION AND MAIN RESULTS
The model of BC-MSR consists of one input x N , two outputs y N 1 , y N 2 , and satisfies where Let W 1 and W 2 be the transmission messages, and their values respectively belong to the Using feedback, the transmitter produces the time-t channel input X t as a function of the messages W 1 , W 2 and of the previously received for some stochastic encoding function After N channel uses, Receiver j (j = 1, 2) decodes W j .Namely, Receiver j generates the where ψ j is Receiver j's decoding function.Receiver j's average decoding error probability is denoted by Receiver 2's equivocation rate of the message W 1 is formulated as Analogously, Receiver 1's equivocation rate oft the message W 2 is formulated as Define an achievable secrecy rate pair (R 1 , R 2 ) as below.Given two positive numbers R 1 and R 2 , if for arbitrarily small , there exist one channel encoder and two channel decoders with parameters M 1 , M 2 , N , ∆ 1 , ∆ 2 , P e,1 and P e,2 satisfying ) ) the pair (R 1 , R 2 ) is called an achievable secrecy rate pair.The secrecy capacity region C f s consists of all achievable secrecy rate pairs.We first propose a hybrid inner bound are not only used to generate secret keys protecting part of the messages, but also used to produce cooperative messages represented by V 0 , V 1 and V 2 helping the receivers to improve their decoding performances.The inner bound and the joint distribution is denoted by Proof: The coding scheme achieving the inner bound C f −in−2 s combines the already existing secret key based feedback scheme for the WTC [16] and the previous non-feedback coding scheme for the BC-MSR with the generalized Wyner-Ziv scheme described in Section II, and it can be briefly illustrated as follows.
• Encoding: The transmission is through n blocks.First, similar to the secret key based feedback scheme for the WTC [16], in each block, split the transmitted message w i (i ∈ {1, 2}) into two parts, i.e., w i = (w i,1 , w i,2 ) for each block is chosen according to the current block's w i,1 , similar auxiliary messages w i , w i shown in the non-feedback coding scheme for BC-MSR (see Section II), the encrypted w i,2 and the previous block's compressed index w * i .Moreover, the sequence q N is chosen according to the current block's randomly chosen "common message" w 0 (see the non-feedback coding scheme for BC-MSR in Section II) and the previous block's compressed index w * 0 .Here note that for the last block, we do not transmit the real message w i (i ∈ {1, 2}) to Receiver i, i.e., we transmit a constant in block n.
• Decoding: The decoding for Receiver i (i ∈ {1, 2}) begins from the last block.In block n, using a similar decoding scheme of the non-feedback coding scheme for BC-MSR, Receiver i decodes u N i and q N for block n.Then he extracts the block n − 1's compressed indexes w * i and w * 0 from the decoded u N i and q N of block n, respectively.Next, similar to the generalized Wyner-Ziv coding scheme, Receiver i views the received signal y N i of block n − 1 as side information.Given block n − 1's w * i , w * 0 and y N i , Receiver i seeks a unique pair of (v N i , v N 0 ) in block n − 1 satisfying the condition that (v N i , v N 0 , y N i ) are jointly typical.Once v N i of block n − 1 is decoded, Receiver i decodes q N for block n − 1 by finding a unique q N satisfying the condition that (q N , y N i , v N i ) are jointly typical.After q N for block n − 1 is decoded, Receiver i decodes u N i for block n − 1 by finding a unique u N i satisfying the condition that (u N i , q N , y N i , v N i ) are jointly typical.Once Receiver i decodes u N i and q N for block n − 1, he obtains the transmitted message w i for block n − 1 and extracts the block n − 2's compressed indexes w * i and w * 0 .Repeating the above decoding procedure, Receiver i obtains all the messages.Details about the proof are in Section IV.
Then, we propose a secret key based inner bound C f −in−1 s on C f s , where the feedback is used to produce keys, and these keys together with the random binning technique prevent each receiver's intended message from being eavesdropped by the other receiver.The inner bound C f −in−1 s is shown in the following Theorem 3. where and the joint distribution is denoted by P (q, u 1 , u 2 , x, y 1 , y 2 ) = P (y 1 , y 2 |x)P (x|u 1 , u 2 )P (u 1 , u 2 |q)P (q), (3.12) Proof: The coding scheme achieving the inner bound C f −in−1 s combines the already existing secret key based feedback scheme for the WTC [16] with the previous non-feedback coding scheme for the BC-MSR (see Section II).Letting , where the joint distribution is denoted by and Q may be assumed to be a (deterministic) function of U 1 and U 2 .
Proof: See Appendix A.
IV. PROOF OF THEOREM 2 The messages are conveyed to the receivers via n blocks.In block i (1 2 are denoted by Xi , Ȳ1,i , Ȳ2,i , Qi , Ū1,i , Ū2,i , V0,i , V1,i and V2,i , respectively.In addition, let X n = ( X1 , ..., Xn ) be a collection of the random sequences X N for all blocks.Similarly, define The value of the random vector is written in lower case letter.

Code-books generation:
• The message W j (j = 1, 2) is sent to Receiver j via n blocks, i.e., the message W j is composed of n components (W j = (W j,1 , ..., W j,n )), and each component W j,i (i ∈ {1, 2, ..., n}) is the message transmitted in block i.Here W j,i takes values in the set {1, ..., 2 N R j }.

A. Dueck-type Example
In this subsection, we further explain the inner and outer bounds on the secrecy capacity region of the BC-MSR with noiseless feedback via a Dueck-type example.In this example (see Figure 6), the channel input and outputs satisfy where the channel inputs X 0 , X 1 , X 2 and the channel noises Z 0 , Z 1 , Z 2 are binary random variables (taking values in {0, 1}), and the channel noises are independent of the channel inputs.
and substituting , and it is given by Here note that (5.3) is obtained when α 1 = α 2 = α 3 = 1 2 .Second, we show the hybrid inner bound C f −in−2 s on the secrecy capacity region of this Duecktype example.Substituting (5.2) , and it is given by (5.4) Third, we show a simple cut-set outer bound C out sf on the secrecy capacity region of this Dueck-type example.Since only X 0 and X 1 are transmitted to receiver 1, the transmission rate R 1 of the message W 1 is upper bounded by I(X 0 , X 1 ; Y 1 ).Analogously, the transmission rate R 2 is upper bounded by I(X 0 , X 2 ; Y 2 ).For all receivers, the sum rate R 1 + R 2 is upper bounded by I(X 0 , X 1 , X 2 ; Y 1 , Y 2 ).Now it remains to calculate these upper bounds.Since X 0 , X 1 , X 2 , Z 0 , Z 1 and Z 2 take values in {0, 1}, from (5.1), we have where (1) follows from the fact that the channel noises are independent of the channel inputs.
Analogously, we have For I(X 0 , X 1 , X 2 ; Y 1 , Y 2 ), we have where (2) also follows from the fact that the channel noises are independent of the channel inputs.Combining (5.5), (5.6) with (5.7), an outer bound C out sf on the secrecy capacity region of this Dueck-type example is given by Finally, in order to show the advantage of feedback, we also give an inner bound C in s on the secrecy capacity region of the Dueck-type example without the feedback.Here notice that in [12], an achievable secrecy rate region C in * s for the BC-MSR is proposed, and it is given by (5.9) 2) and (5.1) into (5.9),C in * s reduces to C in s , and it is given by Here note that (5.10) is obtained when α 1 = α 2 = α 3 = 1 2 .In order to compare the above bounds, we further define the channel noises Z 0 , Z 1 and Z 2 in the following two cases: • Case 1: The channel noise Z 2 only depends on Z 1 , i.e., there exists a Markov chain Z 0 → • Case 2: The channel noises Z 1 and Z 2 only depend on the noise Z 0 , i.e., there exists a In the remainder of this subsection, we show the numerical results on the above two cases of this Dueck-type example, see the followings.

1) A special case of Dueck-type example with
where 0 ≤ p, q, r ≤ 1 2 .Substituting (5.16) and Z 0 → Z 1 → Z 2 into (5.3),(5.4), (5.8) and (5.10), we have ) where , C out sf and C in s for p = q = r = 0.05.Form this figure, we conclude that the hybrid feedback strategy performs better than the secret key based feedback strategy, and both of these strategies increase the secrecy rate region of the BC-MSR.Moreover, note that there is still a gap between the inner and outer bounds on the secrecy capacity region of the BC-MSR with noiseless feedback., C out sf and C in s for p = 0.25, q = 0.2 and r = 0.3.Form Figure 8, we conclude that for this case, the secrecy capacity region of the BC-MSR with noiseless feedback is determined, and this is because the outer bound C out sf meets with the hybrid strategy inner bound C in−2 sf .Also, we see that the hybrid feedback coding strategy performs better than the secret key based feedback strategy, and both of them increase the secrecy rate region of the BC-MSR.
where 0 ≤ p, q, r ≤ 1 2 .Substituting (5.16) and Z 0 → Z 1 → Z 2 into (5.3),(5.4), (5.8) and (5.10), we have (5.17) , C out sf and C in s for p = q = r = 0.05.It is easy to see that for the case Z 1 → Z 0 → Z 2 , the hybrid feedback strategy performs better than the secret key based feedback strategy, and both of these strategies increase the secrecy rate region of the BC-MSR.Also notice that there is a gap between the inner and outer bounds.

B. Blackwell-type Example
In this subsection, we explain the inner and outer bounds on the secrecy capacity region of the BC-MSR with noiseless feedback via a Blackwell-type example.In this example (see Figure 11), the channel input X chooses values from {0, 1, 2}, the channel outputs Y 1 and Y 2 choose values from {0, 1}, and they satisfy where ⊕ is the modulo addition over {0, 1}, the noises Z 1 ∼ Bern(p), Z 2 ∼ Bern(p) (0 ≤ p ≤ 0.5), and the noises Z 1 , Z 2 are mutually independent and they are independent of the channel inputs.
In addition, L is independent of the channel input and outputs.
inner and outer bounds are further illustrated via two examples (a Dueck type example and a Blackwell type example).

Fig. 1 :
Fig. 1: The broadcast channel with noiseless feedback and mutual secrecy requirement we omit the achievability proof of the inner bound C f −in−1 s here.Finally, we propose an outer bound C f −out s on C f s , see the following Theorem 4. Theorem 4: C f s ⊆ C f −out s

Fig. 6 : 1 s
Fig. 6: A Dueck-type example of the BC-MSR with noiseless feedback

Figure 8 plots
Figure 8 plots C in−1 sf , C in−2 sf

Fig. 9 :
Fig. 9: Comparison of the bounds for the case Z 1 → Z 0 → Z 2 and p = q = r = 0.05

= H(Y 1 )R 1 ≤Figure 12 .
Figure 12.From Figure12, we see that both feedback strategies increase the secrecy sum rate of C in * * s , and the hybrid feedback strategy does not always perform better than the secret key based feedback strategy in enhancing the secrecy sum rate of this Blackwell-type BC-MSR.Moreover, as shown in Figure12, there exists a gap between the secrecy sum rates of the inner and outer bounds, and eliminating this gap is still a tough work.

Fig. 12 :PROOF OF THEOREM 4 First, define Y i−1 1 = (Y 1 , 1 , 2 = (Y 2 , 1 ,
Fig. 12: The maximum secrecy sum rates of the bounds for the Blackwell-type example of the BC-MSR with noiseless feedback ).The sub-message w i,1 is encoded exactly the same as that in the non-feedback coding scheme for BC-MSR (see Section II), and w i,2 is encrypted by a key produced by the feedback channel output y N i of the previous block.Then, compress the encoded sequences u N 1 , u N 2 , q N and the feedback channel outputs y N 1 and y N i (i ∈ {1, 2}) is Receiver i's estimation of the channel input, and v N 0 is an auxiliary sequence helping Receiver i to decode v N i .Finally, the sequence u N i y 1 , y 2 ), and label them asv1,i (w * 0,1,i , w * 1,1,i , t 1,i ), where w * 0,1,i ∈ {1, 2, ..., 2 N R01 }, w * 1,1,i ∈ {1, 2, ..., 2 N R11 } and t 1,i ∈ {1, 2, ..., 2 N R 1 }.