Jamming-Aided Secure Communication in Massive MIMO Rician Channels

Jamming-Aided Secure Communication
in Massive MIMO Rician Channels

\authorblockNJue Wang, Member, IEEE, Jemin Lee, Member, IEEE,
 Fanggang Wang, Member, IEEE, and Tony Q. S. Quek, Senior Member, IEEE
Manuscript received December 19, 2014, revised April 21, 2015. The editor coordinating the review of this paper and approving it for publication was Z. Han.Jue Wang, Jemin Lee, and Tony Q. S. Quek are with Singapore University of Technology and Design, Singapore 487372. Email: {jue_wang, jemin_lee, tonyquek}@sutd.edu.sgJue Wang is also with School of Electronic and Information Engineering, Nantong University, Nantong 226019, China.Fanggang Wang is with State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China. Email: wangfg@bjtu.edu.cnThe contact author is J. Lee.This research was partly supported by the Temasek Research Fellowship, the National Natural Science Foundation 61401240, 61201201, U1334202, the Key Grant Project of Chinese Ministry of Education (No. 313006), the Fundamental Research Funds for the Central Universities (2015JBM112), and the State Key Lab of Rail Traffic Control and Safety (No. RCS2014ZT09).Part of this work has been presented in IEEE Global Commun. Conf., Workshop on Massive MIMO: From theory to practice, Austin, TX, Dec. 2014.
Abstract

In this paper, we investigate the artificial noise-aided jamming design for a transmitter equipped with large antenna array in Rician fading channels. We figure out that when the number of transmit antennas tends to infinity, whether the secrecy outage happens in a Rician channel depends on the geometric locations of eavesdroppers. In this light, we first define and analytically describe the secrecy outage region (SOR), indicating all possible locations of an eavesdropper that can cause secrecy outage. After that, the secrecy outage probability (SOP) is derived, and a jamming-beneficial range, i.e., the distance range of eavesdroppers which enables uniform jamming to reduce the SOP, is determined. Then, the optimal power allocation between messages and artificial noise is investigated for different scenarios. Furthermore, to use the jamming power more efficiently and further reduce the SOP, we propose directional jamming that generates jamming signals at selected beams (mapped to physical angles) only, and power allocation algorithms are proposed for the cases with and without the information of the suspicious area, i.e., possible locations of eavesdroppers. We further extend the discussions to multiuser and multi-cell scenarios. At last, numerical results validate our conclusions and show the effectiveness of our proposed jamming power allocation schemes.

Jamming, massive MIMO, outage, security.

I Introduction

Massive multiple-input multiple-output (MIMO) systems, where an enormous number of antennas are deployed at the base station, have become a hot research area in recent years [1, 2]. As the number of antennas goes to infinity, the effect of uncorrelated interferences and noises can tend to zero asymptotically by using only simple linear transmit/receive techniques [3], leading to intensive growth in spectrum and power efficiency [4]. When used for beamforming, massive MIMO leads to sharp beam patterns as well as low power leakage to unintended directions [5]. Due to these attractive properties, massive MIMO becomes a promising technique for future communication systems such as the fifth generation cellular system [6]. In the meanwhile, it can be anticipated that massive MIMO will also become crucial in security related applications.

Secure communication in wiretap channels has been studied for decades since the seminal work [7]. Corresponding studies have been further extended to different type of wiretap channels [8, 9], fading channels [10, 11], MIMO channels [12, 13, 14, 15], and networks [16, 17]. The research topics span a wide range from information-theoretical contributions such as secrecy capacity analysis and rate region characterization to practical transmission design issues including precoding, user scheduling, and artificial noise (AN)-aided jamming. For a complete review of the most lately approaches, see [18, 19]. Regarding the communication secrecy, the emergence of the massive MIMO technique brings new opportunities and challenges. Recently, physical layer security techniques using massive MIMO have drawn increasing attentions in the literature. In [20, 21, 22, 23], the secrecy rate in massive MIMO systems has been analyzed using large system analysis and secure precoding schemes were designed. In [24, 25], it has been shown that massive MIMO can benefit the detection of active eavesdropper who performs attacking on the channel training phase [26]. Note that the above-mentioned approaches require either the channel state information (CSI) of eavesdroppers can be known, or their existence can be detected. For the scenarios that eavesdroppers are completely passive and their CSI is unknown, AN-aided jamming [27] can be a feasible solution. Only recently, the AN-aided jamming approach has been applied for massive MIMO systems in [28, 29] and was shown to be beneficial for communication secrecy.

In this paper, we study the secure communication in massive MIMO systems via AN-aided jamming. Differently from [28], we consider the scenario that eavesdroppers are randomly located around a legitimate transmitter equipped with large antenna array, and all channels follow Rician distribution. In this case, the geometric locations (described by both the angle of arrival and distance to the transmitter) of the legitimate receivers and eavesdroppers become essential in the secrecy outage analysis, which highlights the main difference between our work and [28]. The motivation of our paper is based on the following considerations: 1) Since the beam towards the legitimate receiver becomes sharper and the power leakage to other directions becomes trivial in massive MIMO systems, it is doubtful whether jamming is still beneficial for secrecy, and 2) as the number of antennas grows, the dimension of the jamming space increases and jamming power needs to spread over a large number of directions, which makes conventional uniform jamming inefficient with massive MIMO. Regarding these issues, two questions are raised:

  1. Does conventional uniform jamming still benefit the secure communication in massive MIMO systems when goes to infinity?

  2. Is there more efficient scheme rather than uniform jamming in the massive MIMO setup?

In this paper, we will answer these two questions by making the following contributions:

  • For the massive MIMO Rician fading channels, we analytically describe the secrecy outage region (SOR) as geometric locations of eavesdroppers that can induce secrecy outage. The concept of SOR further has been used to characterize the secrecy outage probability (SOP).

  • With the information of the suspicious area where eavesdroppers are possibly located, we derive analytical expression of the SOP in the presence of one legitimate receiver and multiple passive eavesdroppers. After that, it is proved that conventional uniform jamming is still useful in terms of reducing the SOP when any eavesdroppers are located within a certain distance range to Alice, which we call it as the jamming-beneficial range. This conclusion provides an answer to the first question.

  • For uniform jamming, the optimal signal and jamming power allocation is investigated for different scenarios. We further devise practical directional jamming algorithms, either with or without the information of the suspicious area. The proposed directional jamming schemes use the jamming power more efficiently to further and substantially reduce the SOP, which provides answers to the second raised question.

The rest of this paper is organized as follows: Section II provides system model. Section III describes the SOR, further provides an analytical expression of SOP and a jamming-beneficial range. Optimal jamming power allocation is studied for uniform jamming in Section IV, and in Section V, directional jamming algorithms are proposed. In Section VI, the SOR is discussed for multiuser and multi-cell scenarios. Section VII concludes this paper.

Ii System Model

In this section, we first present the network model. As an important concept in subsequent analysis, we further define the normalized crosstalk between two wireless links and introduce its characteristics. Then, the AN-aided secure transmission and the definition of SOP are described.

Ii-a Network Model

Fig. 1: Description of the network layout.

We consider the network shown in Fig. 1, where a transmitter (Alice) equipped with antennas transmits to a single-antenna user (Bob) in the existence of external passive single-antenna eavesdroppers (Eves ). Alice uses beamforming for the data transmission to Bob, while jamming with AN in other spaces (or directions). We define the set of receivers where denotes Bob and denotes Eve . Considering Rician fading, the channel between Alice and receiver is given by

(1)

where is the Rician -factor, is the i.i.d. fast fading part whose elements follow distribution (complex normal distribution with zero mean and unit variance). For uniform linear array with inter-antenna spacing (in wavelength), the line of sight (LOS) component can be written as the steering vector at incident angle :

(2)

where is the LOS angle of receiver . In addition, we consider large scale fading where is the distance from Alice to receiver , and is the path loss coefficient.

(7)

We consider a practical scenario that Eves are uniformly distributed within an angular range and a distance range , where and are functions of , defining two borders of this area. Throughout this paper, we use

(3)

to define the suspicious area. In practice, if Alice has only limited information of and , she can assume the two boundaries are defined by constant values, and . For instance, if Alice knows nothing about , she can set , , indicating that the suspicious area (from Alice’s point of view) spans the entire space with radius . The effectiveness of this assumption, referred to as “constant boundaries” and defined below, depends on that how accurately it can describe the real .

Definition 1 (Constant Boundaries)

To facilitate practical design, it is convenient to set the two boundaries of to be constants such that .

Ii-B Normalized Crosstalk

For , () defined as (1), the following asymptotic results hold as [30]:

(4)
(5)

where denotes the approximation that is asymptotically accurate,111In this paper, we focus on the large antenna regime, and will use equalities instead of approximations for brevity. , and . Stemming from (5), we introduce the following definition.

Definition 2 (Normalized Crosstalk)

Define the normalized crosstalk between nodes as

(6)
Lemma 1

The normalized crosstalk has the following characteristics:

  1. is a sinc-like function composed of one main lobe and multiple side lobes.

  2. With fixed and random which is uniformly distributed between and , the CDF of can be written as (7) (see top of this page) where the definitions of , , and are referred to Appendix A.

  3. With fixed , the feasible range of is , where is determined by the distribution range of , i.e., .

{proof}

See Appendix A.

Ii-C Secrecy Transmission Scheme

We use linear precoding for data transmission, while AN symbols are sent in the space defined by , to degrade the channels of Eves. For the null space-based jamming [27], it holds that and . The received signal at receiver is given by

(8)

where is the precoder for Bob, is the unit-norm data symbol, and is the additive Gaussian noise. Moreover, and respectively are the powers allocated to Bob and the -th jamming direction, with total power constraint such as where is the total available transmit power. We define the jamming power allocation coefficient as

(9)

For ease of description, we assume that Bob and all Eves share the same noise covariance being . Moreover, we consider maximum ratio transmission (MRT) for precoding of the data symbol , i.e., . In this case, according to (8), the SINR at receiver is given by

(10)

We assume that the Eves are not colluding, but consider the most-capable Eve, which has the maximum receive SINR, to define the secrecy rate as [22]

(11)

where and . We say a secrecy outage occurs if is less than a target rate , hence the SOP is defined as

(12)

Iii Secrecy Outage Analysis

In this section, we first introduce the secrecy outage region (SOR) which describes all possible locations of Eves who can cause secrecy outage. Analytical expression of the SOR is derived for uniform jamming, then the SOP is studied with variant shapes of . At last, a jamming-beneficial range is derived to show that uniform jamming is still useful in reducing the SOP.

(22)

Iii-a Secrecy Outage Region

In the large antenna regime, all fast fading effects are completely averaged out as shown in (4) and (5). Therefore, whether the secrecy outage occurs or not, will be essentially determined by the geometric location of Eve. In this light, we introduce the SOR defined in the following.

Definition 3 (Secrecy Outage Region)

The SOR is defined in terms of polar coordinates as

(13)

Herein, we note that is a function of and .

In the large antenna regime, secrecy outage occurs if there exists at least one Eve within the SOR. If all Eves locate outside of the SOR, the target secrecy rate can be guaranteed. To characterize the SOR, we first evaluate the received SINRs assuming uniform jamming.

Lemma 2

With uniform jamming in , the SINRs at Bob and any Eve can be, respectively, written as222Hereafter, if one notation is applied for any Eve, we will use the subscript instead of for the sake of brevity.

(14)
(15)

where, and hereafter, we use the notations for brevity. Note that in (15), is the normalized crosstalk between Eve and Bob as defined in (6). Considering fixed , we hereafter write as a function of only .

{proof}

Since , jamming causes no interference at Bob. Applying (4) to (10), we get (14). Noting that and , from (10), we have

(16)

By applying (6), the numerator of (16) can be written as . On the other hand, noting that and constitute a complete orthogonal basis of the -dimensional vector space, we have in the denominator of (16). Therefore, (15) can be obtained.

Remark 1

The result in (14) leads to a constraint on (defined in (9)), written as

(17)

which stems from the fact that the jamming power cannot be too large, otherwise, even without Eves, the target rate cannot be guaranteed since the remained signaling power is too small. Unless otherwise specified, we assume (17) can always hold via proper power allocation.

From Lemma 2, we characterize the SOR for the uniform jamming as follows.

Proposition 1

With uniform jamming in and given , the SOR is described as

(18)

where

(19)
(20)
(21)

and is given by (22) shown at the top of this page.

{proof}

Substituting (14) and (15) into (11) and using the definition of SOR in (13), we can obtain the value of in (19). Note that the value of should be positive, this straightforwardly introduces the constraint on the minimum value of such as where can be readily obtained by letting .

From Lemma 1, in (19) is a function with one main lobe and multiple side lobes, resulting in a multi-lobe shaped SOR. In order to gain some insights from this complex shape (as will be shown in the simulations), we focus on several critical security-related metrics as

  • Largest radius of the main lobe () and the -th side lobe (): and can be obtained by replacing in (19) respectively with and (defined as (55) in the proof of Lemma 1), such that

    (23)
    (24)

    Since is much larger than , can be considered as the largest distance of the SOR. For any Eve whose distance to Alice is larger than , we can conclude that it causes no secrecy outage regardless of its LOS direction.

  • Largest angle difference of the SOR: For any Eve whose angle difference to the LOS direction of Bob is larger than , we can conclude that it causes no secrecy outage regardless of its distance to Alice. If , we can write

    (25)

Clearly, to have smaller SOR, we expect to reduce both and . To minimize in (23), we need to minimize while maximizing ; however, since , this results in a maximized (correspondingly, a larger ), which is not desired. It is clear that a trade-off in exists in balancing the effects of both and , which can be formulated as jamming power allocation problems, as described in the following sections.

At last, by setting in Proposition 1, we obtain the following corollary:

Corollary 1

Without jamming, i.e., , the SOR can be found as

(26)

where the superscript stands for “no jamming”.

{proof}

The corollary is directly obtained by setting in Proposition 1.

Differently from (18), the constraint on vanishes in (26), indicating that the SOR now is extended to the entire angular domain. Moreover, compared with (23), we see that in no-jamming case,333Hereafter, for the conditions where the corresponding notation can be applied for either the uniform-jamming or no-jamming cases, we ignore the superscript “uj” or “nj” for brevity. on the contrary, is reduced compared to uniform jamming. In conclusion, uniform jamming induces two opposite effects: the beneficial one is that the SOR can be squeezed in angular domain, and the disadvantage is that the SOR is enlarged in Bob’s direction, i.e., the main lobe. Illustration of the SOR changing caused by jamming will be shown later in simulations.

Iii-B SOP Analysis

With a single Eve uniformly distributed in , using the derived SOR, the SOP is given by

(27)

where denotes the area of a certain geometric region. Considering that there are Eves uniformly distributed in , the SOP of the entire network can be written as

(28)

From (27), the SOP is determined by the overlapping area between two geometrical regions. If , zero SOP is achieved. Recalling (3), as well as (23) and (25), two sufficient conditions of can be written as

(29)

where is defined in (23) and is in (25). The physical insight of (III-B) is clear: when an Eve is far away or its angle difference to is large, it does not cause outage.

For the general case with arbitrary shape of , in (27) can be numerically evaluated and further applied to jamming power allocation design. However, due to the non-regular shapes of and , closed-form expressions of as well as the SOP in (28) do not exist for the general case. Yet, by considering constant boundaries of as described in Definition 1, (27) can be written in an integral form as the following proposition.

Proposition 2

With constant boundaries of , i.e., and , and uniform jamming in , the SOP can be given as

(30)

where is defined in (7).

{proof}

For the ease of analytical description, herein we utilize the CDF of the normalized crosstalk in (7). First, rewrite (12) as and recall (11), we have

(31)

where is the CDF of , which is given by since all Eves are independently distributed. Using (15), we have

(32)

where both and are random. Since and are independent, (32) can be presented as

(33)

where is the PDF of , corresponding to the uniform distribution between two boundaries defined by . Then, is directly obtained as (30).

Practically, (30) can be used for jamming power allocation. As stated in Remark 1, Alice can arbitrarily adjust the value of the constant boundaries in the design, based on the information about that she has. Particularly, if Alice knows nothing about (i.e., she assumes and ), minimizing becomes equivalent to minimizing .

Iii-C Jamming-beneficial Range

Based on Proposition 2, we find a jamming-beneficial range defined in (i.e., the larger constant distance boundary of ) as follows.

Proposition 3

A constraint on that makes the uniform jamming beneficial in reducing the SOP is given by

(34)

where is the largest feasible crosstalk value defined in Lemma 1.

{proof}

See Appendix B.

Remark 2

Proposition 3 shows that when Eves are located close enough to Alice, uniform jamming is always beneficial in reducing the SOP. Clearly, this range expands with larger , as well as larger or larger . On the other hand, the range shrinks with larger or .

Moreover, we note that (34) has a similar form of that described for the SOR without jamming, i.e., in (26). Recalling the definitions of the largest distance of SOR in (23) and (24), the physical insight of Proposition 3 can be explained as follows: as long as , there always exists an optimal , with which the SOP can be reduced by uniform jamming, compared with the SOP without jamming. The optimization of is discussed in the next section.

Iv Jamming Power Allocation

In this section, considering uniform jamming, we investigate the optimal jamming power allocation that minimizes the SOP. The problem can be simply described as

(35)

In practice, Alice may have different accuracy levels of information about as follows:

  1. Alice knows nothing about the suspicious area, or only partial information about the suspicious area such as only (or only); and

  2. Alice knows exact information about the suspicious area, i.e., both and .

For these two cases, we respectively investigate the jamming power allocation in the following.

Iv-a Jamming with None/Partial Information about

When Alice knows nothing about , minimizing the SOP becomes equivalent to minimizing the area of SOR, which can be calculated as

(36)

where is defined in (19). Note that is composed of many side lobes. We use to denote the -th side lobe, and to denote a group of side lobes with indices described by the set . For the case that Alice knows or , we can simplify the problem by minimizing partial, other than the entire area of such as

(37)

where is the set of the concerned side lobe indices, determined by either or . Using (36) (or (37)) along with (19), the areas can be numerically calculated and the optimal can be easily founded via one dimensional linear search. Since it is difficult to derive closed-form expression for , we evaluate the area of in the following corollary for a special case to further provide some discussions.

Corollary 2 (Area of )

With and the path loss coefficient being (which corresponds the free space propagation [32]), can be upper bounded as

(38)

where and are defined in (20) and (21).

{proof}

See Appendix C.

Remark 3

In (38), it is shown that the area of every side lobe is inversely proportional to , indicating that the SOR side lobes can asymptotically vanish with ultimately large . Moreover, it is inversely proportional to , which means that the area of the SOR will rapidly decrease for the side lobes with large indices, i.e., with large angle difference to . This result indicates that Eves from different directions (i.e., within different side lobes) have different significance in causing secrecy outage, hence should be treated differently in the jamming design.

Iv-B Jamming with Exact Information of

With the information of , Alice can calculate and apply the value of in the design (at least numerically),444The calculation requires the knowledge of and . Clearly, uniform jamming is not optimal in this condition. However, for the ease of analysis, we first devise the optimal power allocation for uniform jamming; then, the resulted jamming power can be allocated directionally to further improve efficiency. using either (28) or (30). Although in practice, (35) can be readily solved by one dimensional linear search, it fails to provide the optimal in closed form. In the following corollary, we provide closed-form solutions and discussions for a special case.

Corollary 3 (Jamming power allocation for given )

For constant boundaries of and given , which is equal for all Eves, the optimal can be determined as

(39)

where

(40)
(41)
{proof}

See Appendix D.

Note when , the optimal jamming power decreases with and , whereas it will increase when . The part that dominates the final result in (39) depends on the value of . Detailed discussions will be provided in Section VII along with simulations.

V Directional Jamming

In this section, we propose directional jamming algorithms to allocate jamming power more efficiently than uniform jamming, based on the following facts:

  1. With the information of , Alice can perform jamming only to the suspicious directions instead of the entire null space of .

  2. Without information of , jamming towards different directions also needs to be treated differently, as stated in Remark 3.

At a cost of slightly increasing the implementation complexity compared with uniform jamming, directional jamming is able to substantially reduce the SOP. In following subsections, we present power allocation algorithms for directional jamming with and without the information of .

V-a Directional Jamming with the Information of

When jamming is not uniformly performed, from (10), the SINR at Eve is represented as

(42)

where is the matrix that spans the jamming space, with the -th column vector being , and , where is the power allocated to the -th jamming direction as defined in (8). Correspondingly, the SOR now can be described as

(43)
(44)

where , was defined in (2). The superscript stands for “directional jamming”.

Design directional jamming using (42) induces high complexity especially when is large, since changing any element in the -dimensional vector requires re-calculation of . Hence, we alternatively propose a two-step suboptimal power allocation method for directional jamming in Algorithm 1, which firstly find the optimal jamming power assuming uniform jamming, then reallocate it directionally based on a criterion of jamming subspace selection.

1:Initialization: Update the information of , and at Alice.
2:Assuming null space-based uniform jamming, find
through one dimensional linear search over .
3:Select a -dimensional subspace ( and ) from according to (46), where is defined as (45).
4:Equally allocate to the selected beams in step such as .
Algorithm 1 Directional jamming with the information of

In Step , can be calculated numerically using (28) or (30), depending on the available information of . Note that , and are long term parameters, thus the updating period of Step can be much longer than the computation time required by the other steps.

Since is defined by physical angles, it is necessary to set up a mapping between the jamming space and physical angle to concentrate the jamming power towards . We propose a heuristic subspace selection method for Step . First, map to physical angle as

(45)

After that, is equally reallocated to the beams whose indices are

(46)

The power allocated to each beam is now . In practice, is not necessarily in and an alternative is to find as the column vectors of a -dimensional DFT matrix for the following reasons; 1) selected columns of the DFT matrix can form a good substitute of , as [33]; 2) using pre-defined DFT basis as the jamming space avoids channel inverse calculation, which induces high computation complexity especially when is large [28]; and 3) most importantly, the structure of DFT matrix provides very sharp beam pattern towards the physical angle