Feedback Capacity of Gaussian Multiple-Access Wiretap Channel with Degraded Message Sets

The Schalkwijk-Kailath (SK) feedback scheme is a capacity-achieving coding scheme for the point-to-point white Gaussian channel with feedback. Recently, it has been shown that the SK scheme, which is not designed with consideration of secrecy, already achieves perfect weak secrecy by itself, i.e., the secrecy capacity of the Gaussian wiretap channel with feedback equals the capacity of the same model without secrecy constraint. In this paper, we propose a capacity-achieving SK type feedback scheme for the two-user Gaussian multiple-access channel with degraded message sets (GMAC-DMS). Similarly to the inherent secrecy nature of the classical SK scheme, we show that the proposed scheme is also secure by itself, which indicates that the feedback secrecy capacity of the two-user Gaussian multiple-access wiretap channel with degraded message sets (GMAC-WT-DMS) equals the capacity of the same model without secrecy constraint.

I. INTRODUCTION 1 Schalkwijk and Kailath [1] showed that although feedback does not increase the capacity of the white Gaussian channel, it helps to improve the channel encoding-decoding performance. Recently, [2] found the inherent secrecy nature of the Schalkwijk-Kailath (SK) feedback scheme [1], i.e., the SK scheme, which is not designed with consideration of secrecy, achieves perfect weak secrecy by itself. The inherent secrecy nature found in [2] indicates that the secrecy capacity of the white Gaussian wiretap channel with feedback equals the capacity of the same model without secrecy constraint, and it can be achieved by the SK feedback scheme. The follow-up work of [2] includes: • For the dirty paper channel with feedback, [4] found the inherent secrecy nature of a capacity-achieving SK type feedback scheme proposed in [3].
• [5] proposed a capacity-achieving SK type feedback scheme for the colored Gaussian channel, and showed that the inherent secrecy nature of this scheme holds. Although the inherent secrecy nature of SK type feedback schemes has been found for the point-to-point Gaussian channels, such a topic is mostly open for multi-user Gaussian channel models. In this paper, we focus on the two-user Gaussian multiple-access channel with degraded message sets (GMAC-DMS), and would like to answer the following two open questions: • Does there exist a capacity-achieving SK type feedback scheme for the two-user GMAC-DMS? • Does the inherent secrecy nature still hold for this capacity-achieving scheme, namely, the capacityachieving scheme of the two-user GMAC-DMS also achieves the feedback secrecy capacity region of the two-user Gaussian multiple-access wiretap channel with degraded message sets 2 (GMAC-WT-DMS)? We summarize our contributions as follows.
• A capacity-achieving SK type feedback scheme is proposed for the two-user GMAC-DMS. • This capacity-achieving scheme is shown to be secure by itself, i.e., the proposed capacity-achieving feedback scheme for the two-user GMAC-DMS also achieves the feedback secrecy capacity region of the two-user GMAC-WT-DMS, which indicates that the feedback secrecy capacity region equals the capacity region of the same model without secrecy constraint.
II. PRELIMINARIES: THE SK SCHEME FOR THE WHITE GAUSSIAN CHANNEL WITH FEEDBACK For the white Gaussian channel with feedback (see Figure  1), at each time i (i ∈ {1, 2, ..., N }), the channel input-output relationship is given by where X i is the channel input subject to average power constraint P , Y i is the channel output, and η 1,i ∼ N (0, σ 2 1 ) is the white Gaussian noise and it is independent identically distributed (i.i.d.) across the time index i. The message W is uniformly distributed in W = {1, 2, ..., |W|}. The channel input X i is a function of the message W and the feedback . The receiver generates an estimation W = ψ(Y N ), where ψ is the receiver's decoding function, and the average decoding error probability equals P e = 1 |W| w∈W P r{ψ(y N ) = w|w sent}. (2. 2) The capacity of the white Gaussian channel with feedback is denoted by C f g , and it equals the capacity C g of the white Gaussian channel, which is given by In [1], it has been shown that SK scheme achieves C f g , and this classical scheme is briefly described below.
Since W takes values in W = {1, 2, ..., 2 N R }, divide the interval [−0.5, 0.5] into 2 N R equally spaced sub-intervals, and the center of each sub-interval is mapped to a message value in W. Let θ be the center of the sub-interval with respect to (w.r.t.) the message W (the variance of θ approximately equals 1 12 ). At time 1, the transmitter sends The receiver obtains Y 1 = X 1 + η 1,1 , and gets an estimation of θ by computinĝ 12P . At time 2 ≤ k ≤ N , the receiver obtains Y k = X k + η 1,k , and gets an estimation of θ k by computinĝ where k =θ k − θ. (2.6) yields that Meanwhile, for time 2 ≤ k ≤ N , the transmitter sends where α k−1 V ar( k−1 ).
In [1], it has been shown that if R < 1 2 log(1 + P σ 2 1 ), the decoding error P e of the above coding scheme doubly exponentially decays to zero as N → ∞. For the GMAC-DMS with feedback (see Figure 2), at each time i (i ∈ {1, 2, ..., N }), the channel input-output relationship is given by

III. CAPACITY-ACHIEVING FEEDBACK SCHEME FOR THE TWO-USER GMAC-DMS A. Problem formulation
where X 1,i and X 2,i are the channel inputs subject to average power constraints P 1 and P 2 , respectively, Y i is the channel output, and The channel input X 1,i is a function of the message W 1 and the feedback Y i−1 , and the channel input X 2,i is a function of the messages W 1 , W 2 and the feedback Y i−1 . The receiver generates an estimation where ψ is the receiver's decoding function, and the average decoding error probability equals (3.2) A rate pair (R 1 , R 2 ) is said to be achievable if for any and sufficiently large N , there exist channel encoders and decoder such that The capacity region C f gmac−dms of the GMAC-DMS with feedback is composed of all such achievable rate pairs. In addition, note that the model of GMAC-DMS is defined almost in the same fashion as GMAC-DMS with feedback, except that the channel input X 1,i is a function of the message W 1 and X 2,i is a function of the messages W 1 and W 2 . The capacity region of GMAC-DMS is denoted with C gmac−dms .

B. A SK-type feedback scheme for the two-user GMAC-DMS with noiseless feedback
In this subsection, we propose a two-step SK type feedback scheme for the two-user GMAC-DMS with noiseless feedback, and in the next subsection, we will show that this two-step SK type scheme achieves the feedback capacity C f gmac−dms . The two-step SK type feedback scheme is described below.
The main idea of the two-step SK type feedback scheme is briefly illustrated by Figure 3. In Figure 3, the common message W 1 is encoded by both transmitters, and the private message W 2 is only available at Transmitter 2. Specifically, Transmitter 1 uses power P 1 to encode W 1 and the feedback Y N as X N 1 . Transmitter 2 uses power (1 − ρ 2 )P 2 to encode W 2 and Y N as V N , and power ρ 2 P 2 to encode W 1 and Y N as U N , where 0 ≤ ρ ≤ 1 and Here note that the power of X N 2 equals P 2 , and it can be checked by [10, eq. (5.38), p.33] and the properties in [10, Appendix B]. Since W 1 is known by Transmitter 2, the codeword X N 1 and U N can be subtracted when applying SK scheme to W 2 , i.e., for the SK scheme of W 2 , the equivalent channel model has input V N , output Y N = Y N −X N 1 −U N , and channel noise η N 1 . In addition, since W 1 is known by both transmitters and W 2 is only available at Transmitter 2, for the SK scheme of W 1 , the equivalent channel model has inputs X N 1 and U N , output Y N , and channel noise η N 1 + V N , which is non-white Gaussian noise since V N is not i.i.d. generated. Furthermore, observing that Hence for the SK scheme of W 1 , the input of the equivalent channel model can be viewed as X * i . Since X 1,i is known by Transmitter 2, let Then we have ρ i = 1, which leads to P * i = P * and X * i ∼ N (0, P * ). The encoding and decoding procedure of Figure 3 is described below.
Since W j (j = 1, 2) takes values in W j = {1, 2, ..., 2 N Rj }, divide the interval [−0.5, 0.5] into 2 N Rj equally spaced subintervals, and the center of each sub-interval is mapped to a message value in W j . Let θ j be the center of the sub-interval w.r.t. the message W j (the variance of θ j approximately equals 1 12 ).

Decoding:
The receiver uses a two-step decoding scheme. First, from (2.6), we observe that at time k (3 ≤ k ≤ N ), the receiver's estimationθ 1,k of θ 1 is given bŷ where k−1 =θ 1,k−1 − θ 1 and it is computed by (3.22), and The following lemma 1 shows that the decoding error probability of θ 1 can be arbitrarily small if R 1 < 1 2 log(1 + ) is satisfied. Lemma 1: For the two-step SK type feedback scheme described above, let P e1 be the decoding error probability of Proof: See our full paper [10, Appendix B]. Second, after decoding W 1 and the corresponding codewords X 1,k and U k for all 1 ≤ k ≤ N , the receiver subtracts X 1,k and U k from Y k , and obtains Y k = V k + η 1,k . At time k (1 ≤ k ≤ N ), the receiver's estimationθ 2,k of θ 2 is given bŷ

25)
where k−1 =θ 2,k−1 − θ 2 and it is computed by (3.18), and The decoding error probability P e of the receiver is upper bounded by P e ≤ P e1 + P e2 , where P ej (j = 1, 2) is the receiver's decoding error probability of W j . From the classical SK scheme [1] (also introduced in Section II), we know that the decoding error probability P e2 of W 2 tends to 0 as N → ∞ if R 2 < 1 2 log(1+ (1−ρ 2 )P2 σ 2 1 ), and hence we omit the derivation here.
Now we have shown that if R 1 < 1 2 log(1 + ), the decoding error probability P e of the receiver tends to 0 as N → ∞, i.e., the following region is achievable.

C. Capacity of the two-user GMAC-DMS with noiseless feedback
The following Theorem 1 determines the capacity region C f gmac−dms of the two-user GMAC-DMS with feedback.
The achievability of C f gmac−dms directly follows from the two-step SK type feedback scheme proposed in the preceding subsection. The converse proof of C f gmac−dms consists of two parts. First, from the converse proof of the bounds on R 2 and R 1 + R 2 in GMAC with feedback [9, pp. 627-628], we conclude that C f gmac−dms is outer bounded by Next, it is easy to check that the outer bound C f −out gmac−dms in (3.28) is exactly the same as the inner bound C f −in gmac−dms in (3.27), which completes the proof.
Remark 1: • The capacity region C gmac−dms of the GMAC-DMS equals C f −out gmac−dms in (3.28), and the proof is briefly explained as follows. First, note that in [7], it has been shown that the capacity C mac−dms of the discrete memoryless multiple-access channel with degraded message sets is given by for some joint distribution P X1X2 (x 1 , x 2 ). Substituting X 1 ∼ N (0, P 1 ), X 2 ∼ N (0, P 2 ) and (3.1) into (3.29), defining ρ = E[X1X2] √ P1P2 , and applying the encodingdecoding scheme of [7], an achievable region which equals C f −out gmac−dms is obtained. Next, applying the converse proof of the bounds on R 2 and R 1 + R 2 in GMAC with feedback [9, pp. 627-628], we can conclude that C gmac−dms is outer bounded by C f −out gmac−dms , which completes the proof. • In [8], it has been shown that feedback does not increase the capacity C mac−dms of the discrete memoryless multiple-access channel with degraded message sets. Since C gmac−dms = C f −out gmac−dms and C f gmac−dms = C f −out gmac−dms , we can conclude that feedback also does not increase the capacity of the GMAC-DMS. For the Gaussian multiple-access wiretap channel with degraded message sets (GMAC-WT-DMS) and noiseless feedback (see Figure 4), at each time i (i ∈ {1, 2, ..., N }), the channel input-output relationships are given by where X 1,i and X 2,i are the channel inputs subject to average power constraints P 1 and P 2 , respectively, Y i and Z i are the channel outputs of the legitimate receiver and the wiretapper, respectively, and η 1,i ∼ N (0, σ 2 1 ), η 2,i ∼ N (0, σ 2 2 ) are i.i.d. channel noises across the time index i. The channel encoders and decoder are defined in the same fashion as those in Section III. The wiretapper's equivocation rate of the messages W 1 and W 2 is defined as A rate pair (R 1 , R 2 ) is said to be achievable with perfect weak secrecy if for any and sufficiently large N , there exist channel encoders and decoder such that 3) The secrecy capacity region C f s,gmac−dms of the GMAC-WT-DMS with feedback is composed of all achievable secrecy rate pairs (R 1 , R 2 ) defined in (4.3). The following Theorem 2 establishes that the secrecy constraint does not reduce the capacity of GMAC-DMS with feedback.
Theorem 2: C f s,gmac−dms = C f gmac−dms , where C f s,gmac−dms is the secrecy capacity region of the GMAC-WT-DMS with feedback, and C f gmac−dms is given in Theorem 1.
Proof: Since C f s,gmac−dms ⊆ C f gmac−dms , we only need to show that any achievable rate pair (R 1 , R 2 ) in C f gmac−dms satisfies the secrecy constraint in (4.3). In the preceding section, we introduce a two-step SK type scheme for the GMAC-DMS with feedback, and show that this scheme achieves C f gmac−dms . In this new scheme, the transmitted codewords X 1,i , U i and V i at time i (1 ≤ i ≤ N ) can be expressed as those in [10, eq. (5.38), p.33]. From these expressions, we can conclude that for 3 ≤ i ≤ N , θ 1 and θ 2 are not contained in the transmitted X 1,i , U i and V i . Hence following similar steps in [10, eq. (3.18), p.17] and choosing sufficiently large N , we can prove that 1 N H(W 1 , W 2 |Z N ) ≥ R 1 + R 2 − , which completes the proof.

V. CONCLUSION AND FUTURE WORK
In this paper, a capacity-achieving SK type feedback scheme is proposed for the GMAC-DMS with feedback, and it is also shown to achieve the secrecy capacity region of the GMAC-WT-DMS with feedback. One possible future work of this paper is that whether one can identify dualities of some kind between the GMAC and the Gaussian broadcast models when feedback and secrecy constraint are considered.