# Some New Results on the Multiple-AccessWiretap Channel

^{*}

## Abstract

**:**

## 1. Introduction

- The first is that two users wish to transmit their corresponding messages to a destination, and meanwhile, they also receive the channel output. Each user treats the other user as a wiretapper and wishes to keep its confidential message as secret as possible from the wiretapper. This model is usually called the MAC with confidential messages, and it was studied by Liang and Poor [18]. An inner bound on the capacity-equivocation region is provided for the model with two confidential messages, and the capacity-equivocation region is still not known. Furthermore, for the model of MAC with one confidential message [18], both inner and outer bounds on the capacity-equivocation region are derived. Moreover, for the degraded MAC with one confidential message, the capacity-equivocation region is totally determined.
- The second is that an additional wiretapper has access to the MAC output via a wiretap channel, and therefore, how to keep the confidential messages of the two users as secret as possible from the additional wiretapper is the main concern of the system designer. This model is usually called the multiple-access wiretap channel (MAC-WT). An inner bound on the secrecy capacity region of the degraded Gaussian MAC-WT was provided in [19], and a n-letter form of the secrecy capacity region of the degraded Gaussian MAC-WT was shown in (Theorem 6 in [20]). Moreover, an inner bound on the secrecy capacity region of the general Gaussian MAC-WT was provided in [21]. In [22,23], the MAC-WT with partially cooperating encoders (one encoder is allowed to conference and the other does not transmit any message) was studied, and inner and outer bounds on the capacity-equivocation region of this model were provided. The MAC-WT with two conference links between the encoders was investigated in [24], and inner and outer bounds on the secrecy capacity region were established for this model. Besides these works on the discrete memoryless and Gaussian cases of MAC-WT, He et al. [25] studied the MIMO MAC-WT, where the channel matrices of the legitimate users are fixed and revealed to all of the terminals, whereas the channel matrices of the eavesdropper are arbitrarily varying and only known to the eavesdropper. Recently, Zaidi et al. ([26,27]) investigated the secrecy problem of MIMO x-channels with output feedback and delayed CSI (an extension of the model of MAC-WT). The optimal sum secure degrees of freedom (SDoF) region was characterized in [26,27], and the artificial noise technique was used to construct the corresponding encoding-decoding scheme.

^{N}denotes a random N-vector (U

_{1}, ...,U

_{N}), and u

^{N}= (u

_{1}, ..., u

_{N}) is a specific vector value in that is theN-th Cartesian power of . ${U}_{i}^{N}$ denotes a randomN−i+1-vector (U

_{i}, ...,U

_{N}), and ${u}_{i}^{N}=({u}_{i},\dots ,{u}_{N})$ is a specific vector value in ${\mathcal{U}}_{i}^{N}$. Let P

_{V}(v) denote the probability mass function Pr{V = v}. Throughout the paper, the logarithmic function is to base two.

## 2. Degraded Discrete Memoryless Multiple-Access Wiretap Channel

^{D}composed of all achievable secrecy pairs (R

_{1},R

_{2}) in the model of Figure 1 is characterized in Theorem 1, where the achievable secrecy pair (R

_{1},R

_{2}) is defined in Definition 4.

#### Definition 1. (Channel encoder)

_{1}and W

_{2}take values in , , respectively. W

_{1}and W

_{2}are independent and uniformly distributed over their ranges. The input of Encoder 1 (Encoder 2) is W

_{1}(W

_{2}), while the output of Encoder 1 (Encoder 2) is ${X}_{1}^{N}\hspace{0.17em}({X}_{2}^{N})$. We assume that the encoders are stochastic encoders, i.e., the encoder ${g}_{i}^{N}\hspace{0.17em}(i=1,2)$ is a matrix of conditional probabilities ${g}_{i}^{N}\hspace{0.17em}({x}_{i}^{N}\mid {w}_{i})$, where ${x}_{i}^{N}\in {\mathcal{X}}_{i}^{N}$, w

_{i}∈ , and ${g}_{i}^{N}\hspace{0.17em}({x}_{i}^{N}\mid {w}_{i})$ is the probability that the message w

_{i}is encoded as the channel input ${x}_{i}^{N}$. Note that ${X}_{1}^{N}$ is independent of ${X}_{2}^{N}$. The transmission rates of the confidential messages are ${\scriptstyle \frac{\text{log}\Vert {\mathcal{W}}_{1}\Vert}{N}}$ and ${\scriptstyle \frac{\text{log}\Vert {\mathcal{W}}_{2}\Vert}{N}}$.

#### Definition 2. (Channels)

_{Y}

_{|}

_{X}

_{1}

_{,X}

_{2}(y|x

_{1}, x

_{2}). Note that ${P}_{{Y}^{N}\mid {X}_{1}^{N},{X}_{2}^{N}}({y}^{N}\mid {x}_{1}^{N},{x}_{2}^{N})={\prod}_{n=1}^{N}{P}_{{Y}_{n}\mid {X}_{1,n,}{X}_{2,n}}({y}_{n}\mid {x}_{1,n},{x}_{2,n})$. The inputs of the MAC are ${X}_{1}^{N}$ and ${X}_{2}^{N}$, while the output is Y

^{N}.

_{Z}

_{|}

_{Y}(z|y). The wiretapper’s equivocation to the confidential messages W

_{1}and W

_{2}is defined as:

#### Definition 3. (Decoder)

_{D}: → × , with input Y

^{N}and outputs W̆

_{1}, W̆

_{2}. Let P

_{e}be the error probability of the receiver, and it is defined as Pr{(W

_{1},W

_{2}) ≠ (W̆

_{1}, W̆

_{2})}.

#### Definition 4. (Achievable secrecy pair (R_{1},R_{2}) in the model of Figure 1)

_{1},R

_{2}) (where R

_{1},R

_{2}> 0) is called achievable if, for any ε > 0 (where ε is an arbitrary small positive real number and ε → 0), there exists a channel encoder-decoder (N,Δ, P

_{e}), such that:

^{D}, which is composed of all achievable secrecy pairs (R

_{1},R

_{2}) in the model of Figure 1.

#### Theorem 1

^{D}is as follows,

#### Proof

- Case 1: the pair (R
_{1}= I(X_{1}; Y |U) − I(X_{1};Z|U,X_{2}),R_{2}= I(X_{2}; Y |X_{1}, U) − I(X_{2};Z|U)) is achievable. - Case 2: the pair (R
_{1}= I(X_{1}; Y |X_{2}, U) − I(X_{1};Z|U),R_{2}= I(X_{2}; Y |U) − I(X_{2};Z|U,X_{1})) is achievable.

^{*}is encoded as u

^{N}, and the channel input ${x}_{1}^{N}$ represents the superposition code in which the confidential message w

_{1}is superimposed on w

^{*}. In addition, the channel input ${x}_{2}^{N}$ represents the random binning codeword encoded by the confidential message w

_{2}.

^{*}is encoded as u

^{N}, and the channel input ${x}_{2}^{N}$ represents the superposition code in which the confidential message w

_{2}is superimposed on w

^{*}. In addition, the channel input ${x}_{1}^{N}$ represents the random binning codeword encoded by the confidential message w

_{1}.

#### Remark 1

- The MAC-WT was first investigated by Tekin and Yener [19,21]. In [21], an achievable secrecy rate region (inner bound on the secrecy capacity region) is given by:$$\begin{array}{l}{\mathcal{R}}^{Di}=\{({R}_{1},{R}_{2}):\\ {R}_{1}\le I({X}_{1};Y\mid {X}_{2})-I({X}_{1};Z)\\ {R}_{2}\le I({X}_{2};Y\mid {X}_{1})-I({X}_{2};Z)\\ {R}_{1}+{R}_{2}\le I({X}_{1},{X}_{2};Y)-I({X}_{1},{X}_{2};Z)\}\end{array}$$Letting U be a constant, it is easy to see that the region $\mathcal{R}$
^{D}of Theorem 1 reduces to $\mathcal{R}$^{Di}, i.e., $\mathcal{R}$^{Di}⊆ $\mathcal{R}$^{D}. - Note that the above $\mathcal{R}$
^{Di}is constructed according to the random binning technique. In this paper, we combine the artificial noise technique (the dummy message w^{*}can be also viewed as an artificial noise) with the classical random binning technique to construct the encoding scheme of the model of Figure 1. To be more specific, first, we randomly choose a dummy message (artificial noise) w^{*}. Second, the transmitted codeword is constructed by using the double binning technique, where the index of the bin is related to w^{*}and the index of the sub-bin is related to the transmitted message w_{1}or w_{2}. Finally, we randomly choose a codeword in sub-bin w_{1}or w_{2}to transmit. By using this double binning technique, we prove that $\mathcal{R}$^{D}is achievable. Here, note that the double binning technique (combination of artificial noise and binning) is also used in [22,23]. By using the Markov chain (X_{1},X_{2}) → Y → Z and letting R_{e}= R_{1}, V = const, V_{1}= X_{1}, V_{2}= X_{2}and C_{12}= 0, it is easy to see that the third inequality of (Theorem 2 in [22]) reduces to R_{1}≤ I(X_{1}; Y |X_{2}, U) − I(X_{1};Z|U), and it is coincident with the first inequality of $\mathcal{R}$^{D}. - The region $\mathcal{R}$
^{D}is convex. The proof is directly obtained by introducing a time sharing random variable into Theorem 1, and thus, it is omitted here.

## 3. Degraded Gaussian Multiple-Access Wiretap Channel

#### 3.1. Secrecy Capacity Region of the Degraded Gaussian Multiple-Access Wiretap Channel

_{1}

_{,i}~ (0,N

_{1}) and η

_{2}

_{,i}~ (0,N

_{2}). The random vectors ${\eta}_{1}^{N}$ and ${\eta}_{2}^{N}$ are independent with i.i.d. components. The channel inputs ${X}_{1}^{N}$ and ${X}_{2}^{N}$ are subject to the average power constraints P

_{1}and P

_{2}, respectively, i.e.,

#### Theorem 2

#### Proof

- (Proof of ): The direct proof follows by computing the mutual information terms in Theorem 1 with the following distributions: X
_{1}= U + V, U ~ (0, αp_{1}), V ~ (0, (1 − α)p_{1}) and X_{2}~ (0, p_{2}). U, V and X_{2}are independent. The details are omitted here. The converse proof follows from Section 7, and it is omitted here, too. Thus, the proof of is completed. - (Proof of $\mathcal{B}$): The direct proof follows by computing the mutual information terms in Theorem 1 with the following distributions: X
_{2}= U + V, U ~ (0, αp_{2}), V ~ (0, (1 − α)p_{2}) and X_{1}~ (0, p_{1}). U, V and X_{1}are independent. The details are omitted here. The converse proof follows from Section 7, and it is omitted here, too. Thus, the proof of $\mathcal{B}$ is completed.

#### 3.2. Discussions

^{G}is achieved when α = 0, and it coincides with Tekin–Yener’s inner bound $\mathcal{R}$

^{Gi}, i.e., Tekin–Yener’s inner bound $\mathcal{R}$

^{Gi}is, in fact, the secrecy capacity region of the degraded Gaussian MAC-WT. The rigorous proof is as follows.

#### Proof

^{Gi}if α = 0. Thus, the proof is completed.

## 4. Power Control for Two Kinds of Optimal Points on the Secrecy Rate Region of a Special Gaussian Multiple-Access Wiretap Channel

_{M},N

_{W}~ (0, 1) and $0<\alpha \le {\scriptstyle \frac{1}{2}}$.

_{1}and p

_{2}are transmission powers for the codewords ${x}_{1}^{N}$ and ${x}_{2}^{N}$, respectively, and 0 ≤ p

_{1}, p

_{2}≤ P. Note that the region $\mathcal{R}$ is directly from [21].

#### 4.1. Max-Min Point

#### Theorem 3

#### Proof

_{1}≤ R

_{2}and R

_{1}≤ R

^{2}, respectively. In region , ${R}_{min}^{*}=\text{max\hspace{0.17em}min}\{{R}_{1},{R}_{2}\}=\text{max\hspace{0.17em}}{R}_{1}$, and in region $\mathcal{B}$, ${R}_{min}^{*}=\text{max\hspace{0.17em}min}\{{R}_{1},{R}_{2}\}=\text{max\hspace{0.17em}}{R}_{2}$.

#### 4.2. Single User Point

_{1}(or R

_{2}) with the help of the senders, i.e., ${R}_{su,i}^{*}=\text{max\hspace{0.17em}}{R}_{i}\hspace{0.17em}(i=1,2)$.

#### Theorem 4

- If $0\le \alpha \le {\scriptstyle \frac{3-\sqrt{5}}{2}}$,$${R}_{su,2}^{*}=\{\begin{array}{ll}{\scriptstyle \frac{1}{2}}\hspace{0.17em}\text{log}(1+\alpha P),\hfill & if\hspace{0.17em}0\le P\le {\scriptstyle \frac{\alpha}{1-2\alpha}},\hfill \\ {\scriptstyle \frac{1}{2}}\hspace{0.17em}\text{log}(1+\alpha P),\hfill & if\hspace{0.17em}{\scriptstyle \frac{\alpha}{1-2\alpha}}\le P\le {\scriptstyle \frac{1-2\alpha}{{\alpha}^{2}}},\hfill \\ {\scriptstyle \frac{1}{2}}\hspace{0.17em}\text{log\hspace{0.17em}}{\scriptstyle \frac{1+P}{1+\alpha P}},\hfill & P>{\scriptstyle \frac{1-2\alpha}{{\alpha}^{2}}}.\hfill \end{array}$$

- If ${\scriptstyle \frac{3-\sqrt{5}}{2}}\le \alpha \le {\scriptstyle \frac{1}{2}}$,$${R}_{su,2}^{*}=\{\begin{array}{ll}{\scriptstyle \frac{1}{2}}\hspace{0.17em}\text{log}(1+\alpha P),\hfill & if\hspace{0.17em}0\le P\le {\scriptstyle \frac{1-\alpha}{\alpha}},\hfill \\ {\scriptstyle \frac{1}{2}}\hspace{0.17em}\text{log\hspace{0.17em}}{\scriptstyle \frac{1+2P}{1+P}},\hfill & if\hspace{0.17em}{\scriptstyle \frac{1+\alpha}{\alpha}}\le P\le {\scriptstyle \frac{\alpha}{1-2\alpha}},\hfill \\ {\scriptstyle \frac{1}{2}}\hspace{0.17em}\text{log\hspace{0.17em}}{\scriptstyle \frac{1+P}{1+\alpha P}},\hfill & P>{\scriptstyle \frac{\alpha}{1-2\alpha}}.\hfill \end{array}$$

#### Proof

#### 4.3. Numerical Examples and Discussions

_{1}+ R

_{2}is also maximized. However, for secrecy capacity, the point max(R

_{1}+ R

_{2}) does not necessarily coincide with ${R}_{min}^{*}$ all the time.

## 5. Conclusions

## 6. Direct Proof of Theorem 1

_{1}= I(X

_{1}; Y |X

_{2}, U) − I(X

_{1};Z|U),R

_{2}= I(X

_{2}; Y |U) − I(X

_{2};Z|U,X

_{1})) is achievable, and the achievability proof for the pair (R

_{1}= I(X

_{1}; Y |U) − I(X

_{1};Z|U,X

_{2}),R

_{2}= I(X

_{2}; Y |X

_{1}, U) − I(X

_{2};Z|U)) follows by symmetry.

_{1}, W

_{2}and W

^{*}(dummy message) taking values in the alphabets , and , respectively, where:

_{Z,Y,X}

_{1}

_{,X}

_{2}

_{,U}(z, y, x

_{1}, x

_{2}, u). For arbitrary ε > 0, define:

_{1}→ 0 as N → ∞.

_{2}→ Y .

_{N}

_{→∞}Δ ≥ R

_{1}+ R

_{2}and P

_{e}≤ ε are given in Section 6.2.

#### 6.1. Coding Construction

**Construction of**${X}_{1}^{N}$: Generate 2

^{N}

^{(}

^{I}

^{(}

^{X}

^{1;}

^{Y}

^{|}

^{X}

^{2}

^{,U}

^{)−}

^{ε}

^{2)}i.i.d. codewords ${x}_{1}^{N}\hspace{0.17em}({\varepsilon}_{2}\to 0\mathrm{\hspace{0.17em}\u200a\u200a}\text{as\hspace{0.28em}}N\to \infty )$ according to ${\prod}_{i=1}^{N}{P}_{{X}_{1}}({x}_{1,i})$, and divide them into 2

^{NR}

^{1}bins. Each bin contains 2

^{N}

^{(}

^{I}

^{(}

^{X}

^{1;}

^{Y}

^{|}

^{X}

^{2}

^{,U}

^{)−}

^{ε}

^{2−}

^{R}

^{1)}codewords. Here, note that:

_{1}, randomly choose a codeword in bin w

_{1}to transmit.

**Construction of**U

^{N}

**(dummy message)**: Generate 2

^{NR*}i.i.d. codewords u

^{N}according to ${\prod}_{i=1}^{N}{P}_{U}({u}_{i})$. Randomly choose a u

^{N}(w

^{*}) to transmit. Note that here, U

^{N}is independent of ${X}_{1}^{N}$.

**Construction of**${X}_{2}^{N}$: Generate 2

^{N}

^{(}

^{R}

^{2+}

^{R}

^{*}+

^{R}

^{**}) i.i.d. codewords ${x}_{2}^{N}$ according to ${\prod}_{i=1}^{N}{P}_{{X}_{2}\mid U}({x}_{2,i}\mid {u}_{i})$, and divide them into 2

^{NR}

^{*}bins. Each bin contains 2

^{N}

^{(}

^{R}

^{2+}

^{R}

^{**}) codewords. Divide the codewords in each bin into 2

^{NR}

^{2}sub-bins, and each sub-bin contains 2

^{NR}

^{**}codewords.

^{*}and a given message w

_{2}, first choose the index of the bin according to w

^{*}, and then, choose the index of the sub-bin in bin w

^{*}according to w

_{2}. Finally, randomly choose a codeword in sub-bin w

_{2}to transmit.

**Decoding scheme for the legitimate receiver**: for a given y

^{N}, try to find a sequence u

^{N}(ŵ

^{*}), such that (u

^{N}(ŵ

^{*}), y

^{N}) are jointly typical. If there exists a unique sequence with the index ŵ

^{*}, put out the corresponding ŵ

^{*}, else declare a decoding error. Based on the AEPand (3), the probability Pr{ŵ

^{*}= w

^{*}} goes to one.

^{*}, the legitimate receiver tries to find a sequence ${x}_{2}^{N}\hspace{0.17em}({\widehat{w}}_{2},{\widehat{w}}^{*})$, such that $({u}^{N}({\widehat{w}}^{*}),{x}_{2}^{N}\hspace{0.17em}({\widehat{w}}_{2},{\widehat{w}}^{*}),{y}^{N})$ are jointly typical. If there exists a unique sequence with the index ŵ

_{2}, put out the corresponding ŵ

_{2}; else declare a decoding error. Based on the AEP, (2), (3), (4), (5) and the construction of ${x}_{2}^{N}$, the probability Pr{ŵ

_{2}= w

_{2}} goes to one.

_{2}and ŵ

^{*}, the legitimate receiver tries to find a sequence ${x}_{1}^{N}\hspace{0.17em}({\widehat{w}}_{1})$, such that $({u}^{N}({\widehat{w}}^{*}),{x}_{1}^{N}\hspace{0.17em}({\widehat{w}}_{1}),{x}_{2}^{N}\hspace{0.17em}({\widehat{w}}_{2},{\widehat{w}}^{*}),{y}^{N})$ are jointly typical. There exists a unique sequence with the index ŵ

_{1}; put out the corresponding ŵ

_{1}; else declare a decoding error. Based on the AEP, (1) and the construction of ${x}_{1}^{N}$, the probability Pr{ŵ

_{1}= w

_{1}} goes to one.

#### 6.2. Proof of the Achievability

_{e}≤ ε is easy to be checked. It remains to be shown that lim

_{N}

_{→∞}Δ ≥ R

_{1}+ R

_{2}; see the following.

^{N}is independent of ${X}_{1}^{N}$.

^{N}, Z

^{N}and W

_{1}, the total number of possible codewords of ${X}_{1}^{N}$ is 2

^{N}

^{(}

^{I}

^{(}

^{X}

^{1;}

^{Z}

^{|}

^{U}

^{)−}

^{ε}

^{2)}. By using Fano’s inequality and the fact that ε

_{2}→ 0 as N → ∞, we have:

^{N}and ${X}_{2}^{N}$.

^{N}, Z

^{N}, ${X}_{1}^{N}$ and W

_{2}, the total number of possible codewords of ${X}_{1}^{N}$ is 2

^{NR}

^{**}. By using Fano’s inequality and (4), we have:

## 7. Converse Proof of Theorem 1

_{1},R

_{2}) are contained in the set $\mathcal{R}$

^{D}. We will prove the inequalities of Theorem 1 in the remainder of this section.

**(Proof of**R

_{1}≤ I(X

_{1}; Y |X

_{2}, U) − I(X

_{1};Z|U)

**)**:

_{i}→ (X

_{1}

_{,i},X

_{2}

_{,i}) → Z

^{i}

^{−1}; (7) is from J is a random variable (uniformly distributed over {1, 2, ...,N}), and it is independent of ${X}_{1}^{N},{X}_{2}^{N}$, Y

_{N}and Z

_{N}; (8) is from J is uniformly distributed over {1, 2, ...,N}; and (9) is from the definitions that X

_{1}≜ X

_{1}

_{,J}, X

_{2}≜ X

_{2}

_{,J}, Y ≜ Y

_{J}, Z ≜ Z

_{J}and U ≜ (Z

^{J}

^{−1}, J).

_{e}≤ ε, ε → 0 as N → ∞, ${\text{lim}}_{N\to \infty}{\scriptstyle \frac{H({W}_{1})}{N}}={R}_{1}$ and (1), it is easy to see that R

_{1}≤ I(X

_{1}; Y |X

_{2}, U) − I(X

_{1};Z|U).

**(Proof of**R

_{2}+ I(X

_{2}; Y |X

_{1}, U) − I(X

_{2};Z|U)

**)**: The proof is analogous to the proof of R

_{1}+ I(X

_{1}; Y |X

_{2}, U) − I(X

_{1};Z|U), and it is omitted here.

**Proof of**R

_{1}+ R

_{2}≤ I(X

_{1},X

_{2}; Y |U) − I(X

_{1},X

_{2};Z|U):

_{i}→ (X

_{1}

_{,i},X

_{2}

_{,i}) → Z

^{i}

^{−1}and Z

_{i}→ (X

_{1}

_{,i},X

_{2}

_{,i}) → Z

^{i}

^{−1}; (4) is from Y

_{i}→ Y

^{i}

^{−1}→ Z

^{i}

^{−1}; (5) is from J is a random variable (uniformly distributed over {1, 2, ...,N}), and it is independent of ${X}_{1}^{N},{X}_{2}^{N}$, Y

^{N}and Z

^{N}; (6) is from J is uniformly distributed over {1, 2, ...,N}; and (7) is from the definitions that X

_{1}≜ X

_{1}

_{,J}, X

_{2}≜ X

_{2}

_{,J}, Y ≜ Y

_{J}, Z ≜ Z

_{J}and U ≜ (Z

^{J}

^{−1}, J) and the fact that P

_{e}→ 0 as N → ∞.

_{N}

_{→∞}Δ ≥ R

_{1}+ R

_{2}and (2), it is easy to see that R

_{1}+ R

_{2}≤ I(X

_{1},X

_{2}; Y |U) − I(X

_{1},X

_{2};Z|U).

## Acknowledgment

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Dai, B.; Ma, Z.
Some New Results on the Multiple-AccessWiretap Channel. *Entropy* **2014**, *16*, 4693-4712.
https://doi.org/10.3390/e16084693

**AMA Style**

Dai B, Ma Z.
Some New Results on the Multiple-AccessWiretap Channel. *Entropy*. 2014; 16(8):4693-4712.
https://doi.org/10.3390/e16084693

**Chicago/Turabian Style**

Dai, Bin, and Zheng Ma.
2014. "Some New Results on the Multiple-AccessWiretap Channel" *Entropy* 16, no. 8: 4693-4712.
https://doi.org/10.3390/e16084693