On the Capacity of the 2-User Symmetric Interference Channel with Transmitter Cooperation and Secrecy Constraints

On the Capacity of the -User Symmetric Interference Channel with Transmitter Cooperation and Secrecy Constraints

\authorblockNParthajit Mohapatra and Chandra R. Murthy
\authorblockASingapore University of Technology and Design, Singapore 487372
Indian Institute of Science, Bangalore, India, 560012
Email: parthajit@sutd.edu.sg, cmurthy@ece.iisc.ernet.in
Major portion of this work was carried out, when the first author was at the department of ECE, Indian Institute of Science, Bangalore.
Abstract

This paper studies the value of limited rate cooperation between the transmitters for managing interference and simultaneously ensuring secrecy, in the -user Gaussian symmetric interference channel (GSIC). First, the problem is studied in the symmetric linear deterministic IC (SLDIC) setting, and achievable schemes are proposed, based on interference cancelation, relaying of the other user’s data bits, and transmission of random bits. In the proposed achievable scheme, the limited rate cooperative link is used to share a combination of data bits and random bits depending on the model parameters. Outer bounds on the secrecy rate are also derived, using a novel partitioning of the encoded messages and outputs depending on the relative strength of the signal and the interference. The inner and outer bounds are derived under all possible parameter settings. It is found that, for some parameter settings, the inner and outer bounds match, yielding the capacity of the SLDIC under transmitter cooperation and secrecy constraints. In some other scenarios, the achievable rate matches with the capacity region of the -user SLDIC without secrecy constraints derived by Wang and Tse [1]; thus, the proposed scheme offers secrecy for free, in these cases. Inspired by the achievable schemes and outer bounds in the deterministic case, achievable schemes and outer bounds are derived in the Gaussian case. The proposed achievable scheme for the Gaussian case is based on Marton’s coding scheme and stochastic encoding along with dummy message transmission. One of the key techniques used in the achievable scheme for both the models is interference cancelation, which simultaneously offers two seemingly conflicting benefits: it cancels interference and ensures secrecy. Many of the results derived in this paper extend to the asymmetric case also. The results show that limited transmitter cooperation can greatly facilitate secure communications over -user ICs.

{keywords}

Interference channel, information theoretic secrecy, deterministic approximation, cooperation.

I Introduction

Interference management and ensuring security of the messages are two important aspects in the design of multiuser wireless communication systems, owing to the broadcast nature of the physical medium. The interference channel (IC) is one of the simplest information theoretic models for analyzing the effect of interference on the throughput and secrecy of a multiuser communication system. One way to enhance the achievable rate with secrecy constraints at the receivers is through cooperation between the transmitters. In this work, the role of transmitter cooperation in managing interference and ensuring secrecy is explored by studying the -user IC with limited-rate cooperation between the transmitters and secrecy constraints at the receivers. In practice, such scenarios can arise in a cellular network, where different users have subscribed to different data contents, and are served by different base stations belonging to the same service provider. In this case, it is important for the service provider to support high throughput, as well as secure its transmissions, to maximize its own revenue. In these scenarios, the transmitters (e.g., base stations) are not completely isolated from each other, and cooperation among them is possible. As the base stations can trust each other, there is no need for secrecy constraints at the transmitters. Such cooperation can potentially provide significant gains in the achievable throughput in the presence of interference, while simultaneously guaranteeing security.

To illustrate the value of transmitter cooperation in simultaneously managing interference and ensuring secrecy, a snapshot of some of the results to come in the sequel is presented in Fig. 1. Here, the capacity of the symmetric linear deterministic IC (SLDIC) with and without cooperation is plotted against , where without any secrecy constraints at the receivers [1]. Also plotted are the outer bound and the achievable rate for the -user SLDIC with secrecy constraints at the receivers, developed in Secs. III and IV, respectively. Two cases are considered: no transmitter cooperation () and with cooperation between the transmitters ( bits per channel use). The outer bounds plotted for with secrecy constraints at receivers show that, as the value of increases, there is a dramatic loss in the achievable rate compared to the case without the secrecy constraint. The performance significantly improves with cooperation, and it can be seen that it is possible to achieve a nonzero secrecy rate for all values of except . This paper presents an in depth study of the interplay between interference, security, and transmitter cooperation in the -user IC setting. It demonstrates that having a secure cooperative link in a network can significantly improve the achievable secrecy rate.

Fig. 1: Data rate normalized by for the -user SLDIC. Here, is the capacity of the cooperative link between the transmitters,  bits, and is set based on the value of .

Past work: The interference channel has been extensively studied over the past few decades, to understand the effects of interference on the performance limits of multi-user communication systems. The capacity region of the Gaussian IC (GIC) without secrecy constraints at receiver remains an open problem, even in the user case, except for some special cases such as the strong/very strong interference regimes [2], [3]. In [4], the broadcast and IC with independent confidential messages are considered and the achievable scheme is based on random binning techniques. The work in [5] demonstrates that with the help of an independent interferer, the secrecy capacity region of the wiretap channel can be enhanced. Intuitively, although the use of an independent interferer increases the interference at both the legitimate receiver and the eavesdropper, the benefit from the latter outweighs the rate loss due to the former. Some more results on the IC under different eavesdropper settings can be found in [6, 7, 8].

The effect of cooperation on secrecy has been explored in [9, 10, 11]. In [9], the effect of user cooperation on the secrecy capacity of the multiple access channel with generalized feedback is analyzed, where the messages of the senders need to be kept secret from each other. In [10], the role of user cooperation on the secrecy of broadcast channel (BC) with relaying, where the receivers can cooperate with each other, is considered. The achievable scheme uses a combination of Marton’s coding scheme for the BC and a compress and forward scheme for the relay channel. The role of a relay in ensuring secrecy under different wireless network settings has been studied in [12, 13, 14].

A linear deterministic model for relay network was introduced in [15], which led to insights on the achievable schemes in Gaussian relay networks. The deterministic model has subsequently been used for studying the achievable rates with the secrecy constraints in [16, 17, 18]. In [16], secret communication over the IC is analyzed with two types of secrecy constraints: in the first case, the secrecy constraint is specific to the agreed-upon signaling strategy, and in the second case, the secrecy constraint takes into account the fact that the other users may deviate from the agreed-upon strategy. The deterministic model has also been studied under different eavesdropper settings in [19, 17, 18].

It is known that limited-rate cooperation between the transmitters or receivers can significantly increase the rate achievable in the 2-user IC without secrecy constraints [1, 20]. In general, the Gaussian IC with transmitter cooperation is more difficult to analyze than Gaussian IC with receiver cooperation, even when there is no secrecy constraints at the receivers. For example, when the receivers can cooperate through a link of infinite capacity, the model reduces to a Gaussian MIMO multiple access channel (MAC). When the transmitters cooperate through a link of infinite capacity, the model reduces to a MIMO BC. The capacity region of the general MAC was characterized in 1970s [21, 22]. In the MAC, the boundary of the rate region can be achieved if the receiver performs MMSE decoding and successive interference cancelation of the input data streams. However, it took a long time for researchers to find a precoding strategy which achieves the boundary of the BC rate region [23, 24]. Similarly, the IC with cooperative receivers is easier to analyze than the IC with cooperative transmitters [20, 1]. Further, when there are secrecy constraints at the receivers, the following difficulties arise in analyzing the system with rate-limited transmitter cooperation.

  1. There are a number of ways in which the transmitters can use the cooperative link for encoding their transmission. The cooperation can involve the exchange of data bits, random bits or any combination of the two.

  2. It is difficult to obtain tractable outer bounds, since the encoded messages are no longer independent due to the cooperation between the transmitters. In addition to providing carefully selected side-information to receivers, the secrecy constraints at the receivers need to be exploited in a judicious manner to obtain tighter outer bounds as compared to the outer bounds that do not use the secrecy constraints at the receivers.

To the best of the authors’ knowledge, the role of limited transmitter cooperation in a -user IC on interference management and secrecy has not been explored and is therefore focus of this work.

Contributions: In order to make headway into this problem, first, the problem is addressed in the linear deterministic setting. For the SLDIC with cooperating transmitters and secrecy constraints at the receivers, achievable schemes and outer bounds on the secrecy rate are derived for all possible parameter settings. This gives useful insights for the achievable schemes and outer bounds in the Gaussian setting. Next, the schemes are adapted to the Gaussian case. The proposed transmission/coding strategy in the Gaussian setting uses a superposition of a non-cooperative private codeword and a cooperative private codeword. For the non-cooperative private part, stochastic encoding is used [25], and for the cooperative private part, Marton’s coding scheme is used [26, 1]. The auxiliary codewords corresponding to the cooperative private part are chosen such that the interference caused by the cooperative private auxiliary codeword of the other user is completely canceled out. This approach is different from the one used in [1], where the interference caused by the unwanted auxiliary codeword is approximately canceled. Further, one of the users transmits dummy information to enhance the achievable secrecy rate. The major contributions of this work can be summarized as follows:

  1. One of the key techniques used in the derivation of the outer bounds for the SLDIC is the proposed partitioning of the encoded messages and outputs depending on the value of . This partitioning of the encoded messages/outputs reveals what side-information needs to be provided to the receivers for canceling negative entropy terms. In addition, partitioning helps to bound or simplify entropy terms which are not easy to evaluate due to the dependence between the encoded messages at the transmitters. Also, the partitioning of the encoded messages/outputs provides a convenient handle for using the secrecy constraints at the receivers efficiently in deriving the outer bounds. The outer bounds are stated as Theorems 1-4 in Sec. III.

  2. For the SLDIC, the achievable scheme is based on interference cancelation, transmission of jamming signal (random bits) and relaying of the other user’s data bits. The novelty in the proposed scheme lies in determining how to combine these techniques to achieve rates that are far superior to that achievable individually by these methods. To the best of authors’ knowledge, exchanging a combination of data bits and random bits between the transmitters for the purpose of precoding has not been used in the literature. The details of the achievable scheme can be found in Sec. IV.

  3. Outer bounds on the secrecy rate in the Gaussian setting are derived and stated as Theorems 6-8 in Sec. V. As the partitioning used in deriving the outer bounds for the deterministic case cannot be directly used in the Gaussian case, either analogous quantities as side-information need to be found to mimic the partitioning of the encoded messages/outputs or the bounding steps need to be modified taking cue from the deterministic model. This is one of the key steps in deriving the outer bounds on the secrecy rate.

  4. Using the intuition gained from the SLDIC, achievable schemes for the Gaussian case are proposed, which use a combination of stochastic encoding and Marton’s coding scheme along with dummy message transmission by one of the users. However, in the high interference regime, stochastic encoding alone cannot ensure secrecy of the non-cooperative private message, as cross links are stronger than the direct links. Hence, in addition to stochastic encoding, dummy message transmission is used by one of the users to ensure secrecy of the non-cooperative private message at the unintended receiver. In the Marton’s coding scheme, the codeword carrying the cooperative private message is precoded such that it is completely canceled at the unintended receiver. The details of the achievable scheme can be found in Sec. VI.

  5. Many of the results derived in this paper extend to the asymmetric case also, and these are mentioned as remarks after corresponding theorems, where applicable.

It is shown that with limited-rate transmitter cooperation, it is possible to achieve a nonzero secrecy rate under all parameter settings except for the case. In particular, for the very high interference regime , it is possible to achieve non-zero secrecy rate for both the model as compared to the non-cooperating case. In case of SLDIC, it is found, surprisingly, that in some nontrivial cases, the achievable secrecy rate equals the capacity of the same system without the secrecy constraints. Thus, the proposed schemes allow one to get secure communications for free, in these cases. It is also observed that the proposed outer bounds for the SLDIC with cooperation are strictly tighter than the best existing outer bound without the secrecy constraint [1] in all interference regimes, except for the weak interference regime, where the bounds match. The idea of using a common randomness to improve the achievable rates is an important upcoming theme in multiuser information theory, and the proposed schemes based on sharing random bits between the transmitters is in the same flavor. Thus, the results in this paper provide a deep and comprehensive understanding of the benefit of transmitter cooperation in achieving high data rate in the IC, while also ensuring secrecy. Parts of this work have appeared in [27] and [28].

Notation: Lower case or upper case letters represent scalars, lower case boldface letters represent vectors, and upper case boldface letters represent matrices.

Organization: Section II presents the system model. In Secs. III and IV, the outer bounds and the achievable schemes for the SLDIC are presented, respectively. The outer bounds and achievable results for the GSIC can be found in Secs. V and  VI, respectively. In Sec. VII, some numerical examples are presented to offer a deeper insight into the bounds, to contrast the performance of the various schemes, and to benchmark against known results. Concluding remarks are offered in Sec. VIII. The proofs of the theorems and lemmas are presented in the Appendices.

Ii System Model

   

Fig. 2: \subreffig:sysmodel1 GSIC and \subreffig:sysmodel2 SLDIC with transmitter cooperation.

Consider a -user Gaussian symmetric IC (GSIC) with cooperating transmitters. The signals at the receivers are modeled as [1]:

(1)

where is the Gaussian additive noise, distributed as . The input signals are required to satisfy the power constraint: . Here, and are the channel gains of the direct and cross links, respectively. The transmitters cooperate through a noiseless and secure link of finite rate denoted by . The equivalent deterministic model of (1) at high SNR is as follows [1]:

(2)

where and are binary vectors of length , is a downshift matrix with elements if and otherwise, and stands for modulo- addition (XOR operation).

The parameters and are related to the GSIC as while the capacity of the cooperative link is . The quantity captures the amount of coupling between the signal and the interference, and is central to characterizing the achievable rates and outer bounds in case of the SLDIC and GSIC. A schematic representation of the GSIC and SLDIC with transmitter cooperation is shown in Fig. 2. The figure also shows the convention followed in this paper for denoting the bits transmitted over the SLDIC, which is the same as that in [1]. The bits denote the information bits of transmitters and , respectively, sent on the level, with the levels numbered starting from the bottom-most entry.

The transmitter has a message , which should be decodable at the intended receiver , but needs to be kept secret from the other, unintended receiver , . In the case of the SLDIC, the encoded message is a function of its own data bits, the bits received through the cooperative link, and possibly some random data bits. The encoding at the transmitter should satisfy the causality constraint, i.e., it cannot depend on future cooperative bits. The decoding is based on solving the linear equations in (2) at each receiver. For secrecy, it is required to satisfy in the case of the SLDIC [29]. The details of the encoding and decoding scheme for the Gaussian case can be found in Sections VI-A and VI-B. In contrast to the SLDIC, the notion of weak secrecy is considered for the Gaussian case [25]. Also, it is assumed that the transmitters trust each other completely and that they do not deviate from the agreed scheme, for both the models.

The results derived in the paper for the deterministic and Gaussian models under the symmetric assumption can be extended to the asymmetric setting in many cases, and these are indicated as remarks in the following sections. There are two ways in which the model considered in the paper can be asymmetric: (a) when , where is the capacity of the cooperative link from transmitter  to transmitter  . This is termed as cooperation asymmetry. (b) The two direct channel gains and two cross channel gains need not be equal to each other; this is termed as channel asymmetry. In this case, the channel is parameterized by in the deterministic case and in the Gaussian case. In the sequel, the phrase asymmetry is used to account for both channel and cooperation asymmetry.

Iii SLDIC: Outer Bounds

In this section, four outer bounds on the symmetric rate for the -user SLDIC with cooperation between transmitters and perfect secrecy constraints at the receivers are stated as Theorems 1-4. Theorem 1 is valid for all , while Theorems 2, 3, and 4 are valid for , , and , respectively.

In the derivation of the outer bounds, the following difficulties arise:

  1. Due to cooperation between the transmitters, the encoded messages are no longer independent. Most existing outer bounding techniques (e.g.: [5, 4]) require the independence of the encoded messages to simplify the entropy terms, hence are not applicable in this case.

  2. Determining when and how to use the secrecy constraints at the receivers along with the reliability criteria is crucial in deriving a tractable outer bound.

To meet these challenges, a novel partitioning of the encoded messages and outputs depending on the value of is proposed. This partitioning of the encoded messages/outputs reveals what side-information needs to be provided to the receivers and helps to bound or simplify entropy terms which are not easy to evaluate due to the dependence between the encoded messages at the transmitter. This partitioning also reveals how to judiciously exploit the secrecy constraints at the receivers in deriving the outer bounds.

The following relation is repeatedly used in the derivation of these outer bounds: conditioned on the cooperative signals, denoted by , the encoded signals and the messages at the two transmitters are independent [30, 1]. This is represented as the following Markov chain relationship:

(3)

Finally, the overall outer bound on the symmetric secrecy rate is obtained by taking the minimum of these outer bounds. The best performing outer bound depends on the value of and the maximum possible rate, i.e., per user, where is the indicator function, equal to if is true, and equal to otherwise.

In the derivation of the first outer bound, the encoded message is partitioned into two parts: one part () which causes interference to the unintended receiver, and another part () which is not received at the unintended receiver. Partitioning the message in this way helps to obtain an outer bound on , which leads to an outer bound on the symmetric secrecy rate. The following theorem gives the outer bound on the symmetric secrecy rate.

Theorem 1

The symmetric rate of the -user SLDIC with limited-rate transmitter cooperation and secrecy constraints at the receivers is upper bounded as:

(4)
{proof}

The proof is provided in the Appendix -A. Remarks:

  • Note that when , the outer bound increases with increasing for a given value of . However, it is intuitive to think that the achievable secrecy rate should decrease with increase in the value of , i.e., the outer bound is loose in the high interference regime. Interestingly, it is found that the achievable secrecy rate also improves with increase in the value in the initial part of the high interference regime, i.e., for , even when . This will be discussed in Sec. VII-B.

  • The outer bound stated above can be extended to obtain an outer bound on for the asymmetric setting. Using a similar approach as used in the proof of this theorem, one can also obtain an outer bound on . Note that, these outer bounds are applicable over all the interference regimes. The outer bounds are as follows:

    (5)

The next outer bound, stated as Theorem 2, focuses on the very high interference regime, i.e., for . In the derivation of the bound, the encoded message at each transmitter is partitioned into three parts, as shown in Fig. 3. The partitioning is based on whether (a) the bits are received at the intended receiver, and are received at the other receiver without interference, (b) the bits are not received at the desired receiver, and received without interference at the other receiver, and (c) the bits are not received at the intended receiver, and are received with interference at the other receiver. To motivate the development of the following outer bound, first consider the case. If receiver  can decode sent by transmitter , then receiver  can decode as well, since it gets these data bits without any interference. Hence, it is not possible to send any data bits securely on those levels. Data transmitted at the remaining levels are not received by receiver , so they cannot be used for secure data transmission either. Now, suppose a genie provides receiver  with the part of the signal sent by transmitter  that is received without any interference at receiver , i.e., . Then, by using the secrecy constraint for the receiver , it is possible to bound the rate of user  by . When , it is possible to show that . When , by using the above mentioned approach and the relation in (3), an outer bound on the symmetric secrecy rate is derived for , and is stated as the following theorem.

    

Fig. 3: \subreffig:veryhighouter SLDIC with and and \subreffig:highouter SLDIC with and : Illustration of partitioning of the encoded message/output.
Theorem 2

In the very high interference regime, i.e., for , the symmetric rate of the -user SLDIC with limited-rate transmitter cooperation and secrecy constraints at the receivers is upper bounded as: .

{proof}

The proof is provided in Appendix -B. Remarks:

  • The outer bound in Theorem  can be extended to the asymmetric case under the following condition

    (6)

    and the outer bound becomes

    (7)
  • Theorem 2 implies that, for , it is not possible to achieve a rate greater than , regardless of and . In particular, when , i.e., without cooperation, it is not possible to achieve a nonzero rate. However, in the other interference regimes, it is possible to achieve rates greater than (See Figs. 12 and 13).

The third outer bound, stated as Theorem 3 below, is applicable in the high interference regime, i.e., . The derivation of the outer bound involves partitioning of the output and the encoded message based on whether the bits are received with interference at the intended receiver, or causes interference to the other receiver, as shown in Fig. 3. The outer bound on the symmetric secrecy rate for the high interference regime is stated in the following theorem.

Theorem 3

In the high interference regime, i.e., for , the symmetric rate of the -user SLDIC with limited-rate transmitter cooperation and secrecy constraints at the receivers is upper bounded as: .

The following theorem gives the outer bound on the symmetric secrecy rate for the case. In this case, both the receivers see the same signal. Hence, it is possible for receiver  decode any message that receiver  is able to decode, and vice-versa. Therefore, it is not possible to achieve a nonzero secrecy rate, irrespective of . A similar reasoning also holds for the Gaussian case, even though the receivers see independent noise instantiations.

Theorem 4

When , the symmetric rate of the -user SLDIC with limited-rate transmitter cooperation and secrecy constraints at the receivers is upper bounded as: .

{proof}

The proof is provided in Appendix -D.

A consolidated expression for the outer bound, obtained by taking minimum of the outer bounds in Theorems 1-4, is stated as the following corollary. In particular, the minimum of the outer bounds in Theorems 1 and 3 is taken for the high interference regime, and the minimum of the outer bounds in Theorems 1 and 2 is taken in the very high interference regime.

Corollary 1

An outer bound on the symmetric secrecy rate of the SLDIC, obtained by taking the minimum of the outer bounds derived in this work, is given by:

(8)

where .

Remarks:

  • Under cooperation asymmetry, all the outer bounds developed in the deterministic model still hold. This requires replacing with in the expression for the outer bound. This is due to the fact that the entropy term can be upper bounded by .

  • There are cases where it is non-trivial to extend these bounds to the asymmetric scenario (e.g.: Theorem 3). One of the key techniques used in the derivation of these outer bounds is the partitioning of the encoded messages/outputs and careful selection of the side-information to be provided to the receiver. This partitioning and side-information does not easily generalize to the asymmetric scenario.

Next, the achievable schemes for the SLDIC are presented.

Iv SLDIC: Achievable Schemes

Iv-a Weak interference regime

In this regime, the proposed scheme uses interference cancelation. It is easy to see that data bits transmitted on the lower levels remain secure, as these data bits do not cause interference at the unintended receiver. Hence, it is possible to transmit bits securely, when , as shown in Fig. 4. However, with cooperation , it is possible to transmit on the top levels by appropriately xoring the data bits with the cooperative bits in the lower levels prior to transmission. These cooperative bits are precoded (xored) with the data bits at the levels to cancel interference caused by the data bits sent by the other transmitter. When , it can be shown that the proposed scheme achieves the maximum possible rate of  bits. When , bits can be discarded and cooperative bits can be used for encoding as above, to achieve  bits. Hence, in the sequel, it will not be explicitly mentioned that . The proposed encoding scheme achieves the following symmetric secrecy rate:

(9)

A high level description of the achievable scheme is illustrated in Fig. 4. The details of the encoding scheme and the derivation of (9) can be found in [27].

Fig. 4: SLDIC: \subreffig:sldic-modified-weak1 , , and , \subreffig:sldic-modified-weakmod1 and , and .

Remarks:

  1. In this regime, the proposed achievable scheme meets the symmetric capacity of the SLDIC without secrecy constraints [1] for all values of (See Figs. 12 and 13). Thus, the secrecy constraints at the receivers do not reduce the symmetric capacity region of the SLDIC.

  2. In this regime, the proposed scheme does not involve transmission of a jamming signal (or random bits), even when . In the next subsection, it will be seen that the transmission of the jamming signal improves the achievable secrecy rate, when the capacity of the cooperative link is not sufficient to cancel interference at the unintended receiver.

Iv-B Moderate interference regime

In this regime, the proposed scheme uses interference cancelation along with the transmission of random bits. Without transmitter cooperation, it is possible to transmit at least bits securely, as in the weak interference regime. Depending on the value of and , with the help of transmission of random bits, it is possible to send additional data bits on the higher levels by carefully placing data bits along with zero bits and random bits.

The proposed scheme achieves the following symmetric secrecy rate:

(10)

where , , , and .

In the above equation, the first term corresponds to the number of data bits transmitted securely without using random bits transmission or cooperation. The term corresponds to the number of data bits that can be securely transmitted using the help of random bits transmission. The last term represents the gain in rate achievable due to cooperation.

A high level description of the achievable scheme is illustrated in Fig. 4. The details of the encoding scheme and the derivation of (10) can be found in [27].

Remark: In this regime, it is possible to transmit data bits securely in the higher levels by intelligently choosing the placement of data and random bits, in addition to interference cancelation.

Iv-C Interference is as strong as the signal

In this case, from Theorem 4, it is not possible to achieve a nonzero secrecy rate.

Iv-D High interference regime

The achievable scheme is similar to that proposed for the moderate interference regime, but it differs in the manner the encoding of the message is performed at each transmitter. The proposed scheme achieves the following secrecy rate:

  1. When :

    (11)

    where , , , and .

  2. When :

    (12)

    where , , , , , and .

The details of the encoding scheme and some illustrative examples can be found in Appendix -E.

Remarks:

  1. When and , the proposed scheme is capacity achieving. The outer bound in Theorem 3 helps to establish this.

  2. One can note that the achievable schemes for the moderate (Sec. IV-B) and high interference regime (Sec. IV-D) use a combination of interference cancelation and transmission of a jamming signal (random bits transmission). When precoding is done using the other user’s signal, it cancels the interference and also ensures secrecy. In the technique based on random bits transmission, the transmitter self-jams its own receiver, so that the receiver cannot decode the other user’s data. But, in this process, transmitter causes interference to the other receiver, thereby adversely impairing the achievable rate of secure communication. Thus, self jamming in that form only helps if the benefit to the secrecy rate due to the interference caused at the own receiver outweighs the negative impact of the interference caused at the other receiver. However, when the jamming signal can be canceled at an unintended receiver by transmission of the same random bits by the other transmitter, its adverse impact is completely alleviated, leading to larger achievable rates.

Iv-E Very high interference regime

In this case, when , it is not possible to achieve nonzero secrecy rate as established by the outer bound in Theorem 2. However, with cooperation , it is possible to achieve nonzero secrecy rate. The proposed scheme uses interference cancelation, time sharing, and relaying the other user’s data bits. In contrast to the achievable schemes for other interference regimes, the transmitters exchange data bits, random bits, or both, depending on the capacity of the cooperative link. The proposed scheme achieves the following secrecy rate:

(a) Random bits sharing: .         (b) Data bits sharing: .

Fig. 5: SLDIC with , and .
  1. When is even:

    (13)
  2. When is odd:

    (14)

where , , , , , , and , .

The details of the achievable scheme can be found in Appendix -F.

Remarks:

  1. When , the capacity achieving scheme involves exchanging only random bits through the cooperative links. This is useful in scenarios where the transmitters trust each other to follow the agreed-upon scheme, but are not allowed to share their data bits through the cooperative link. The outer bound in Theorem  establishes the optimality of the proposed scheme. The achievable scheme is illustrated for random bits sharing and data bits sharing for C = 1 in Figs. 5(a) and 5(b), respectively.

  2. When and is even (or odd) valued, the proposed scheme shares a combination of random bits and data bits through the cooperative links. In Fig. 6, a schematic representation of the achievable scheme for and , with bits is shown for the first time slot. In the second time slot, the encoding for transmitters  and is reversed. In the second time slot, users  and achieve a rate of and , respectively. Hence, a symmetric rate of is achievable.

Fig. 6: SLDIC with , and : and is achievable in the first time slot. In the second time slot, the role of transmitters  and is reversed and users  and achieve a rate of and , respectively.

Interestingly, it turns out that the symmetric capacity region of the SLDIC does not change if the perfect secrecy constraint at the receiver is replaced with the strong or the weak notion of secrecy, when the proposed scheme is capacity achieving. This result is stated as the following theorem.

Theorem 5

The symmetric secrecy capacity region of the deterministic SLDIC with transmitter cooperation satisfies the following relationship, when the proposed scheme is capacity achieving:

(15)

where , and correspond to the capacity region with the perfect, strong and weak notions of secrecy, respectively.

{proof}

Any communication scheme satisfying the perfect secrecy condition will automatically satisfy the strong and weak secrecy condition. Similarly, a communication scheme satisfying strong secrecy will automatically satisfy the weak secrecy condition. Hence, the following holds

(16)

The achievable results in Sec. IV are obtained under perfect secrecy constraints at the receivers. It is not difficult to show that the outer bounds on the secrecy rate in Theorems 1-4 do not change if the perfect secrecy is replaced with the weak notion of secrecy. When the achievable rates meet the corresponding outer bounds, the relation in (15) holds.

Finally, this section is concluded with the following remarks:

  1. When , i.e., the cooperative link is as strong as the interference, and when , the proposed scheme achieves the maximum possible rate of .

  2. In [31], it is shown that the proposed outer bound in Theorem  in Sec. III is tight for and , when . Hence, the secrecy capacity is characterized for these regimes of also. However, the symmetric secrecy capacity region of the -user SLDIC, when and , remains an open problem.

  3. It is possible to extend the achievable scheme based on interference cancelation (involving exchange of data bits between the transmitters) as well as the scheme based on transmission of random bits to the asymmetric case for the deterministic model. However, it is not straightforward to extend the achievable schemes which rely on the exchange of both data and random bits to the asymmetric case. The extension requires a careful re-working of a scheme for sharing random bits and data bits in the asymmetric setting.

In the following section, outer bounds for the GSIC are presented.

V GSIC: Outer Bounds

In this section, the outer bounds on the secrecy rate for the GSIC with limited-rate transmitter cooperation are stated as Theorems 6-8. The extension of the outer bounds from the deterministic model to the Gaussian model is non-trivial, because of the following well known differences between the models:

  1. In the deterministic model, interference or superposition of signals is modeled using the XOR operation. Hence, the levels do not interact with each other.

  2. In the deterministic model, noise is modeled using truncation.

  3. In the Gaussian model, due to finite rate cooperation between the transmitters, the differential entropy terms contain discrete as well as continuous random variables. This makes the derivation of the outer bounds more difficult in the Gaussian case.

Due to the above differences, the partitioning used in the derivation of the outer bounds for the deterministic case is not directly applicable to the Gaussian case. To overcome this problem, either analogous quantities that serve as side-information need to be found to mimic the partitioning of the encoded messages/outputs, or the bounding steps need to be modified taking cue from the deterministic model. This is discussed in detail in this section.

The outer bound derived in Theorem 1 partitions the encoded message into two parts: (received at receiver , ) and (not received at receiver , ). However, it is not possible to partition the message in this way for the Gaussian case. Hence, in the derivation of Theorem 6, is used as a proxy for . In this section, the following notation is used: , and .

Theorem 6

The symmetric rate of the -user GSIC with limited-rate transmitter cooperation and secrecy constraints at the receiver is upper bounded as follows:

(17)

where ,

and represents the determinant of a matrix.

{proof}

The proof is provided in Appendix -G. Remarks:

  1. The outer bound in Theorem 6 for the Gaussian model can be extended to obtain an outer bound on under the asymmetric setting. The outer bound becomes111With a slight abuse of notation, has been used to represent the capacity of the cooperative link from transmitter  to transmitter  for both the deterministic and the Gaussian models in the asymmetric case.

    (18)

    where ,

    where , , , and .

  2. Using a similar approach as used in the proof of Theorem 6, an outer bound on can be obtained.

The outer bound on the secrecy rate presented in the following theorem is based on the idea used in deriving outer bounds in Theorems 2 and 3 for case of the SLDIC. But, in the Gaussian setting, it is not possible to partition the encoded message as was done for the SLDIC. For example, in Theorem 2, a part of the output at receiver  which does not contain the signal sent by transmitter  is provided as side information to receiver . Hence, the approach used in the derivation of the outer bound in case of SLDIC cannot be directly used for the Gaussian case. To overcome this problem, for the Gaussian case, first is provided as side information to receiver ; this eliminates the interference caused by transmitter . Then, the receiver  is provided with as side-information. The outer bound on the symmetric secrecy rate is stated in the following theorem.

Theorem 7

The symmetric rate of the -user GSIC with limited-rate transmitter cooperation and secrecy constraints at the receiver is upper bounded as follows:

(20)
{proof}

The proof is provided in Appendix -H. Remark: The outer bound in Theorem 7 can be extended to the asymmetric setting, and the outer bound becomes

(21)

Using a similar approach as used in the proof of Theorem 7, an outer bound on can be obtained.

The outer bound presented in the following theorem is similar to the outer bound presented in Theorem 4 in case of the SLDIC. This kind of outer bound exists in the literature (see, for example, [18]), but for the sake of completeness, it is presented in the following theorem. Unlike the results in Theorems 6 and 7, this outer bound does not depend on the capacity of the cooperative link.

Theorem 8

The symmetric rate of the -user GSIC with limited-rate transmitter cooperation and secrecy constraints at the receiver is upper bounded as follows:

(22)
{proof}

The proof is provided in Appendix -I. Remark: The outer bound in Theorem 8 can be extended to the asymmetric setting, and the outer bound becomes

(23)

where and .

Using a similar approach as used in the proof of Theorem 8, an outer bound on can be obtained.

V-a Relation between the outer bounds for SLDIC and GSIC

In the following, it is shown that at high SNR and INR, the outer bounds developed for the Gaussian case (Theorems 6 and 7) are approximately equal to the outer bounds for the SLDIC, when .222When , from Fig. 7, it appears that the approximate equivalence of the bounds for the GSIC and SLDIC will still hold. In Fig. 7, the outer bounds on the achievable secrecy rate in Theorems 6-8 are compared as a function of , for and , when  dB and . This validates that the approaches used in obtaining outer bounds in the two models are consistent with each other.

Fig. 7: Comparison of different outer bounds on the secrecy rate for the GSIC with  dB and . In the legend, OB stands for the outer bound.

In the following, for ease of presentation, it is assumed that and are integers. Recall that, the parameters and of the SLDIC are related to the GSIC as and , respectively.

V-A1 Outer bound in Theorem 6

Consider the following bound in the proof of Theorem 6, when :

(24)

where the last equation is obtained for high SNR and INR. Using the above mentioned definitions of and , (24) reduces to:

(25)

The above is the same as the outer bound for the SLDIC in Theorem 1, when .

V-A2 Outer bound in Theorem 7

When , the outer bound in Theorem 7 reduces to the following, in the high SNR and high INR regime:

(26)

Using the above mentioned definitions of and , (26) reduces to:

(27)

The above is the same as the outer bound for the SLDIC in Theorem 2, when .

In the following section, achievable schemes on the secrecy rate for the GSIC are presented.

Vi GSIC: Achievable Schemes

Vi-a Weak/moderate interference regime

The achievable scheme is based on the approach used in Secs. IV-A and IV-B, for the SLDIC. Again, the achievable scheme proposed for the deterministic model is not directly applicable to the Gaussian case due to the differences between the two models mentioned earlier. In the case of the SLDIC, the achievable scheme used a combination of interference cancelation, transmission of random bits, or both, depending on the value of and . That scheme is extended to the Gaussian setting, as follows.

The message at transmitter is split into two parts: a non-cooperative private part and a cooperative private part . The non-cooperative private message is encoded using stochastic encoding [25], and the cooperative private part is encoded using Marton’s coding scheme [26, 1]. For the SLDIC, data bits transmitted at the lower levels are not received at the unintended receiver. Hence, these data bits remain secure. However, there is no one-to-one analogue of this in the GSIC, so the scheme does not extend directly. In the Gaussian case, for the non-cooperative private part, stochastic encoding is used to ensure secrecy. The transmitter  encodes the non-cooperative part into . A stochastic encoder is specified by a conditional probability density , where and , and it satisfies the following condition:

(28)

where is the probability that is output by the stochastic encoder, when message is to be transmitted.

The cooperative private message and at transmitters and are encoded using Marton’s coding scheme. One of the key aspects of the achievable scheme is in the proposed method for encoding of the cooperative private message, which is chosen to ensure that this part of the message is completely canceled at the non-intended receiver. This corresponds to the scheme used for interference cancelation in the SLDIC. This serves two purposes: it cancels interference over the air, and simultaneously ensures secrecy. The transmitter  sends a dummy message along with the cooperative private message and the non-cooperative private message. Note that stochastic encoding is sufficient to ensure secrecy of the non-cooperative private message. However, the additional dummy message sent by the transmitter  can enhance the achievable secrecy rate, depending on the values of and . In this case, both the receivers treat the dummy message as noise.

Vi-A1 Encoding and decoding

For the non-cooperative private part, transmitter generates i.i.d. sequences of length at random according to

(29)

The codewords in the codebook are randomly grouped into bins, with each bin containing codewords. Any codeword in is indexed as for and . In order to transmit , transmitter  selects a randomly and transmits the codeword .

In order to transmit a dummy message, transmitter  generates i.i.d. sequences of length at random according to

(30)

The codewords in codebook are randomly grouped into bins, with each bin containing codewords (and thus ). Any codeword in is indexed as , where and . During encoding, transmitter  selects and independently at random and sends the codeword .

For the cooperative private message, the transmitter generates the cooperative private vector codeword based on Marton’s coding scheme according to

(31)

where and are auxiliary codewords. The choice of these codewords are discussed in the proof of Theorem 9. Finally, the non-cooperative private codeword and cooperative private codeword are superimposed to form the transmit codeword at the transmitter  and the non-cooperative private codeword, cooperative private codeword and the dummy message codeword are superimposed to form the transmit codeword at the transmitter :

(32)

where is defined in (106) in the proof of Theorem 9.

For decoding, receiver  looks for a unique message tuple such that is jointly typical. Based on the above coding strategy, the following theorem gives the achievable result on the secrecy rate.

Theorem 9

In the weak/moderate interference regime, the following rate is achievable for the GSIC with limited-rate transmitter cooperation and secrecy constraints at the receivers:

(33)

where . The achievable secrecy rate for the user  can be obtained by exchanging the indices and in (33).

{proof}

The proof is provided in Appendix -J. The achievable symmetric secrecy rate for the GSIC is stated in the following Corollary.

Corollary 2

Using the achievable result in Theorem 9 and time-sharing between transmitters, following symmetric secrecy rate is achievable for the GSIC with limited-rate transmitter cooperation:

(34)

where and are the achievable secrecy rates for transmitter  in the first and second time slots, respectively, which are obtained by maximizing over parameters and . The achievable rates for users  and in the first time slot are as follows:

(35)