Increasing Security Degree of Freedom in Multi-user and Multi-eve Systems

The growing interest in security at the physical layer of wireless communications has sparked a resurgence of research in secure communication. In the early works on information theoretic security, Wyner introduces the wiretap channel model, in which the eavesdropper’s channel is defined to be a degraded version of the legitimate receiver’s channel [1]. It is shown that a non-zero secrecy capacity can be obtained only if the eavesdropper’s channel is of lower quality than that of the intended recipient. Csiszár and Körner extend this problem to a general non-degraded channel condition in which a common message is transmitted to the two receivers and the confidential message to only one of them [2]. Another main assumption in the aforementioned works is that the eavesdropper’s channel is known at the transmitter [3, 4], and then the generalized singular value decomposition (GSVD) ¹¹1The generalized singular value decomposition (GSVD) of an $m\times n$ matrix $\mathbf{A}$ and a $p\times n$ matrix $\mathbf{B}$ is given by the pair of factorizations  $\mathbf{A}=\mathbf{U}\sum_{1}[\mathbf{0,R}]\mathbf{Q}^{T}$ and $\mathbf{B}=\mathbf{V}\sum_{2}[\mathbf{0,R}]\mathbf{Q}^{T}$, where $\mathbf{U}$, $\mathbf{V}$ and $\mathbf{Q}$ are orthogonal matrices, $R$ is an $r\times r$ upper triangular nonsingular matrix, $\sum_{1}$ and $\sum_{2}$ are nonnegative diagonal matrice satisfying $\sum_{1}^{T}\sum_{1}+\sum_{2}^{T}\sum_{2}=\mathbf{I}$. method can be used to transmit the signal to the null space of the channel from transmitter to eavesdropper. Clearly, these assumptions are usually impractical and unreasonable, particularly for passive eavesdroppers. In this paper, we overcome this problem and propose a scheme without using any CSI of eavesdroppers.

In order to achieve secure communication, even when the receiver’s channel is worse than the eavesdropper’s channel, or the absence of eavesdroppers’ channel state information (CSI), various physical-layer techniques have been proposed. One of the most common techniques is the use of cooperative interference or artificial noise to confuse the eavesdropper. The cooperative interference method can be divided into two categories: (i) the trust-friend model, in which two base stations connected by a high-capacity backbone such as optical fiber, and one base station can continuously transmit an interfering signal to secure the uplink communication for the other base station [5, 6]; (ii) the helper-relay model, where the secrecy level can be increased by having the cooperative interferer [7] or relay [8] to send codewords independent to the source message, which can be canceled at the intended receiver.

Another major techniques for secure communication is the use of multiple antennas. When multiple antennas are equipped at the transmitter, it is possible for the transmitter to simultaneously transmit both the information-bearing signal and artificial noise to achieve secrecy in a fading environment [9]-[11], which may replace the role of the cooperative interference method in [5]-[8]. In the design of secure communication with artificial noise, the transmit power allocation between the information signal and the artificial noise is an important issue, which has not been discussed in [9, 10, 11]. A suboptimal power allocation strategy is considered in [12], which aims to meet an ideal signal-to-interference-and-noise ratio (SINR) at the intended receiver to satisfy a quality of service requirement. The secure communication with artificial noise is also discussed in [11], in which the closed-form expression of achievable rate and the optimal power allocation has been obtained, however only single-receiver and single-antenna at receiver was considered.

Most of the previous papers focus on the single-user systems. However most practical communication systems have more than one user and the eavesdroppers may not appear alone as well, and they may choose to cooperate or not [11]. In addition, each terminal may be equipped with multiple antennas, which is representative, for example, of downlink transmission in LTE systems and wireless local area networks. This is the so called Multi-User and Multi-Eve (MUME) MIMO systems, which have been seldom investigated before. In this paper, we will focus on investigating the MUME systems.

It’s also worth noting that the achievable secrecy rate of MUME systems is different from that of single-user and single-antenna systems studied before, which must make sure that any legitimate user will not be wiretapped by any eavesdropper. The authors in [13] put forward an MUME model in which all other users are viewed as potential eavesdroppers by the targeted user. They also give a definition of the achievable secrecy rate of multiuser wiretap model in terms of secrecy sum rate. The authors in [14] give another definition of achievable secrecy rate in Gaussian MIMO multi-receiver Wiretap channel, which is named secrecy capacity region. Besides, the authors in [15, 16] considers the compound wire-tap channel, which is based on the classical wire-tap channel with channel from the source to the destination and the channel from the source to the wiretapper taking a number of states respectively. It can be viewed as the multicast wire-tap channel with multiple destinations and multiple wire-tappers with the same massage transmitted to different destinations, which is slightly different from the broadcast wire-tap channels in this paper. They also give another significant definition of the achievable secrecy rate in terms of absolute secrecy rate, which idea is to take the security of the poorest-performance receive-wiretap pair into consideration. If the poorest-performance pair can meet the quality of service requirement, then all other pairs can do. Therefore this definition of achievable secrecy rate may be more reasonable and constructive in the practical secure communication systems. The authors in [16, 17] discussed the realization of the achievable secrecy rate of multiple users (multiple eavesdroppers) with artificial noise separately, in which, however, the system model is compound wiretap channel but broadcast MUME wiretap channel [18].

Since the transmitter needs to transmit different message to different receivers in the broadcast MUME wiretap model, there must be considerable co-channel interference (CCI) in the system. In order to limit the CCI from the signals transmitted to other users and mask receivers’ own message signal simultaneously, two practical linear transmission schemes were often used in the early works: (i) the SVD method discussed in [9, 11], which conducts an SVD decomposition on each user’s channel matrix to get a maximum channel gain for their own message but can not suppress the interference from other user’s message; (ii) the ZF beamforming method [19] and its promotion—-the BD method [20, 21], in which all the information is transmitted in the null space of all other receivers’ channels. The SVD method and ZF beamforming method are simple, but of little ideal performance. While the BD method is of somewhat ideal performance but more complicated than the formers.

In view of the drawbacks of the previous schemes, we propose an alternative approach, which takes idea from dirty-paper coding (DPC) [22, 23] and ZF beam-forming [19]. It can directly increase the degree of freedom when designing the transmission precoders, which in return make obvious improvement not only at the achievable secrecy rate, but also at the antenna constraints at transmitter compared with the BD method. What’s more, in our proposed scheme, we choose to map the artificial noise into the null space of co-precoder matrix instead of the null space of legitimate receivers’ channels in the existing schemes, which may offer extra improvement on the secrecy rate in the low SNR region. The performance will be further improved when the water-filling (WF) method is used. Besides, the power allocation between the information signal and the artificial noise are also discussed.

In this paper, ${(\cdot)}^{H}$ and $tr(\cdot)$ denote the Hermitian transpose and trace of a matrix. $E(\cdot)$ denotes expectation, and $\mathbf{I}$ denotes an identity matrix. $I(\cdot,\cdot)$ denotes mutual information. $[x]^{+}$ = max$\{0,x\}$.

In this paper, we consider the broadcast MUME wiretap model as shown in Fig. 1, in which there is one transmitter named Alice, $J$ legitimate users named Bobs and $K$ passive eavesdroppers named Eves. Alice tries to send independent messages to all the legitimate receivers while keeping each of the eavesdropper ignorant of all the messages. All of the terminals are equipped with multiple antennas. $N_{Bj}$ antennas are equipped at the $j$-th Bob, $N_{Ek}$ antennas at the $k$-th Eve, and $N_{A}$ antennas at the single Alice. This scenario is representative, for example, of downlink transmission in the LTE systems and wireless local area networks.

Let the transmit signal ${\bf X}=\sum_{j=1}^{J}{{\bf U}_{j}}+{\bf V}$, where ${\bf U}_{j}$ is the information bearing signal vector for user $j$, and ${\bf V}$ is the artificial noise signal vector to interference Eves. Then the received signals at Bobs and Eves are respectively:

where ${\bf H}_{j}$ is the $N_{Bj}\times N_{A}$ channel matrix between the transmitter and Bob $j$, ${\bf G}_{k}$ is the $N_{Ek}\times N_{A}$ channel matrix between the transmitter and eavesdropper $k$, $\mathbf{N}^{B}_{j}$ and $\mathbf{N}^{E}_{k}$ are respectively the additive white Gaussian noise vectors observed at the $j$-th Bob and $k$-th Eve, which covariance matrices satisfy $E[\mathbf{N}^{B}_{j}{\mathbf{N}^{B}_{j}}^{H}]=\sigma_{Bj}^{2}\mathbf{I}$, and $E[\mathbf{N}^{E}_{k}{\mathbf{N}^{E}_{k}}^{H}]=\sigma_{Ek}^{2}\mathbf{I}$ respectively.

We assume that the channel matrix $\mathbf{H}_{j}$ and ${\bf G}_{k}$ are block-fading, whose entries are complex Gaussian variables with zero-mean and unit-variance. We also assume that perfect channel state information (CSI) of the receiver, i.e., the channel matrices $\mathbf{H}_{j}$, $j=1,2,~{}\ldots~{},J$, are available at Alice, e.g., either through reverse channel estimation in time-division-duplex (TDD) or feedback in frequency-division-duplex (FDD). While the channel matrices $\mathbf{G}_{k}$, $k=1,2,~{}\ldots~{},K$, are unavailable at Alice due to the passive nature of eavesdroppers.

Our objective is to transmit different secret message to the corresponding Bobs. We try to reduce the CCI from the others, and make sure that the underlying Eves can not wiretap any communication between Alice and Bobs. In the following, we provide lower and upper bounds on the achievable secrecy rate of the generalized broadcast MUME wire-tap channel.

Let $R^{B}_{j}$ denote the mutual information rate between Alice and Bob $j$, and $R^{E}_{k}$ denote that between Alice and Eve $k$ for Bob $j$. Then

In the sequel, the achievable secrecy rate of the receive-wiretap pair $(j,k)$ (for Bob $j$ and Eve $k$) can be denoted by [24]

The achievable secrecy rate of MUME wiretap model is usually noted by secrecy sum rate ($R_{ssr}$) [13] or secrecy rate region [14]. However, in the practical broadcast MUME wireless communication systems, the massages transmitted vary with different users, we should take each user into consideration, once it is chosen according to some criterion. Therefore we must make sure that any user’s communication can not be wiretapped by any eavesdropper. Hence the secrecy rate of the system is determined neither by the best transmission pair nor the total rate gap between Bobs and Eves, but by the poorest-performance transmission pair in reality. Then we propose an alternative definition of the secrecy rate for MUME wire-tap channel, which is called absolute secrecy rate ($R_{asr}$). Obviously, the absolute secrecy rate actually is just the lower bound of the achievable secrecy rate, and can be given by

where $P_{U_{j}}$ is an input distribution. As for the secrecy sum rate $(R_{ssr})$, we have

At Alice, the data for each user is processed before transmission. Then it is launched into the MIMO channel with the random artificial noise. Let $\mathbf{W}_{j}$ be an $N_{A}\times d_{j}$ linear precoder, $\mathbf{u}_{j}$ be a $d_{j}\times 1$ symbol vector for Bob $j$, and $d_{j}$ be the number of parallel data symbols transmitted simultaneously for Bob $j$ [22] satisfying $1\leq d_{j}\leq N_{Bj}$. Let $\mathbf{V}$ be the artificial noise signal vector, $\mathbf{W}$ be the transmission preprocessing matrix, and $\mathbf{v}$ be the symbol vector. They are both used for the artificial noise. Then the transmission signal is

The received signal at Bobs and Eves are respectively

The emphasis of this paper is to design the precoders $\mathbf{W}_{\ell}$ for $\ell=1,2,~{}\ldots~{},J$ and $\mathbf{W}$. In the SVD method all the data is transmitted in their own channel image space, so that each Bob can get the maximum channel gain for the corresponding message. Another one is the block diagonalization (BD) method, in which all the data is transmitted in the null space of all other Bobs, which is the promotion of the ZF beamforming method and can reduce the interference from the other users’ message signal. Here, we propose an alternative approach—-the increasing security degrees of freedom (ISDF) method, which is based on dirty-paper coding (DPC) [22, 23] and ZF beam-forming [19]. In this paper, a single data stream is to be sent to each receiver when $d_{j}=1,~{}\forall j$, and multiple data streams are sent when $d_{j}>1,~{}\forall j$. Note that, only a maximum of $N_{Bj}$ streams can be transmitted simultaneously for user $j$, else the message will not be decoded. Criterion of judging the design is whether the secrecy rate is sufficiently good under the given power constraints, which will be discussed in detail with the design of precoders in the following part.

In MUME scenarios, several co-channel Bobs with multiple antennas aim to communicate with Alice in the same frequency and time slots. In this case, it is necessary to design transmission scheme that is able to suppress the CCI at Bobs. In multi-user wireless security communications with artificial noise, the precoding matrix is usually designed in the null space of channel matrix $\mathbf{H}_{j}$. We may called it precoding selection space as well. Obviously, the smaller the rank of matrix $\mathbf{H}_{j}$ is, the larger the dimension of its corresponding precoding selection space will be. In return, the design of the corresponding precoder has more freedom and the secrecy performance will be better. Therefore we define the dimension of the precoding selection space as the security degrees of freedom (SDF).

In this section, we will analyze the secrecy rate of ISDF1 and ISDF2. Suppose that the variance of the transmit symbol vector $\mathbf{u}_{j}$ is ${\sigma_{u_{j}}^{2}}$, and the complex Gaussian random elements of $\mathbf{v}$ are i.i.d whose variance is ${\sigma_{v}^{2}}$. It is assumed that Alice has a total amount of transmit power budget $P$. Due to the normalization of the noise variance at Bob, we can also refer to $P$ as the transmission SNR. One important parameter should be designed is the power ratio, denoted by $\rho_{j}~{}~{}(0<\rho_{j}<1)$, allocated for the user $j$’s information transmission. We define the power ratio for transmitting artificial noise as $\alpha$ ($0<\alpha<1$). Let

Then we have

and

in which, we have used the following facts

In order to analyze the secrecy rate concisely, Eq. $(\ref{L7 equation 7})$ can be rewritten as:

where we have defined

for $j,i,\ell=1,2,~{}\ldots~{},J,~{}k=1,2,~{}\ldots~{},K$.

The secrecy rate is the maximum transmission rate at which the intended receiver can decode the data with arbitrarily small error, which is bounded by the difference in the capacity between Alice and Bob and that between Alice and Eve [2]. In the following part, the secrecy rate will be given in terms of secrecy sum rate [13] and absolute secrecy rate, where the secrecy sum rate is noted by $R_{ssr}$ and the absolute secrecy rate is noted by $R_{asr}$.