Encyclopedia of Wireless Networks

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Artificial Noise Schemes Based on MIMO Technology in Secure Cellular Networks

  • Yi-Liang LiuEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-32903-1_74-1
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Keywords

MIMO Technology Physical Layer Security Average Secrecy Rate Instantaneous Secrecy Capacity Private Capacity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Synonyms

Definition

Cellular communications and networks are particularly vulnerable to eavesdropping attacks due to the opened and broadcasting nature of wireless channels. Due to the rapid development of cellular networks and wireless business, the security issue has attracted a lot of attention. Physical layer security takes the advantages of channel randomness nature of transmission media to achieve communication confidentiality, which is the most important and interesting topic of privacy communication technologies. Artificial noise (AN) or jamming signals with MIMO technologies are supposed to implement physical layer security and enhance secrecy capacities in cellular networks, where AN signals along with confidential data confuse potential eavesdroppers via utilizing orthogonal spaces provided by transmit antennas. In these AN-based schemes, message streams were sent in a multiplexing mode via all eigen-subchannels (positive eigenvalue channels) at desired directions, and the AN signals can be transmitted to the null spaces (zero eigenvalue channels) of desired directions, so that they do not affect the desired user, while eavesdropper channels are degraded with a high probability. This chapter introduces a physical layer security review, covering the information theory and physical layer security schemes based on MIMO technologies, and provides an overview on the state-of-the-art works on the AN-based scheme along with conclusions and future research directions.

Historical Background

The origin of physical layer security research can be traced back to Wyner’s definition of an information-theoretic secrecy capacity (Wyner, 1975), which is a maximum message transmission rate in confidential communications. Compared to conventional cryptography that works to ensure all involved entities to load proper and authenticated cryptographic information, physical layer security technologies perform confidentiality functions without considering about how those security protocols are executed. In other words, it does not require to implement any extra security schemes or algorithms on other layers above the physical layer. In addition, the physical layer has the same confidentiality level with the one-time pad, which ensures its security performance. The concept of physical layer security has become more popular with the help of MIMO technologies, because these emerging technologies can improve the secrecy capacity massively via utilizing extra orthogonal spaces provided by multiple antennas.

Information Theory

Wyner implied that the intrinsic and unpredictable elements of physical channels, such as noises, interferences, and wireless fading, play central roles in protecting secure messages (Wyner, 1975) and claimed that there exists a randomized codebook maximizing transmission rates that can achieve prefect secrecy where any eavesdroppers cannot get any information. The maximizing secrecy rate is defined as a secrecy capacity Cs as
$$\displaystyle \begin{aligned} C_s=\max_{V\rightarrow X\rightarrow YZ}I(V;Y)-I(V;Z), \end{aligned} $$
(1)
where I(⋅;⋅) denotes mutual information between V and Y and V is an auxiliary input variable with a joint auxiliary input distribution p(v)p(x|v). Given a discrete memoryless channel PY Z|X, wiretap codes achieve the secrecy capacity via maximization over the choices of the joint distribution PY Z|X, such that the Markov chain V → X → YZ holds, which has been expected to completely address the underlying problem of that traditional cryptographic algorithms are vulnerable to quantum attacks.

In the last two decades, researchers have developed a significant amount of mathematical theories, such as secrecy capacity characterization, wiretap coding designs, wireless fading managements, etc. And the advancements in cellular technologies have improved physical layer security significantly, by exploring spatial diversities and multiplexing gains with the help of multi-antennas technologies.

Physical Layer Security Schemes Based on MIMO Technologies

From the traditional communication viewpoints, improving the link quality of the main channel is one of the most feasible approaches to improve the secrecy capacities with existing installations of cellular networks, because most of the cellular wireless communication technologies can improve the channel spectrum utilization, and if the eavesdropping channel spectrum utilization is unchanged, these traditional wireless communication technologies will undoubtedly improve the secrecy capacities. However, this viewpoint has a great security risk, i.e., the secrecy capacities are small if the eavesdropper has a better channel quality than the desired users. Therefore, current research insists on another viewpoint that jointly uses more degree of freedom of main channels and artificial randomness to protect messages while reduces the quality of eavesdropping channels. Prospectively, the randomness of multipath fading of a MIMO channel is stronger than a single channel. Multiplexing technologies of MIMO are promising ways to enhance the degree of freedom of main channels. And AN signals can interfere with eavesdroppers while they are sent into the null spaces of desired directions.

We briefly summarize MIMO-based physical layer security techniques of recent research in four categories, as in Fig. 1, including secure unitary beamforming, zero-forcing (ZF) beamforming, convex optimization-based precoding, and AN-based precoding. It is emphasized that a MIMO-based physical layer security scheme of cellular networks can combine multiple techniques.
Fig. 1

Illustrative diagrams of four basic secure multi-antenna technologies. For simplicity, this sets two-dimensional diagrams and a single message stream. (a) Secure unitary beamforming, (b) Zero-forcing precoding, (c) CVX-based precoding, (d) AN-based precoding (Liu et al., 2017a)

As in Fig. 1a, unitary beamforming techniques use unitary matrices, semi-unitary matrices, or unitary vectors as transmit precoding, which is similar with traditional beamforming technologies of cellular networks. Unitary beamforming techniques send a message stream via multi-antennas to make it as close to the main channel direction as possible. ZF beamforming is shown in Fig. 1b, which is a secure beamforming technology based on ZF precoding, where the message stream is transmitted to a desired receiver via a shifted beamforming direction, which is orthogonal to the eavesdropper’s channel. The illustrative diagram of CVX-based precoding can be seen in Fig. 1c. The problem of optimizing secrecy capacities is usually non-convex in MIMO systems, but Newton methods or Lagrangian dual transformations can be used to trap it in a local maximum problem and address the transmit covariance matrix optimization based on convex optimization tools. AN signals can be transmitted to the null spaces of desired directions, so that they do not affect the desired user, while an eavesdropper’s channel is degraded with a high probability, as in Fig. 1d. This AN-based precoding exploits a fraction of the transmit power to send artificially generated noise signals, so the power for transmitting messages will be reduced.

AN-based schemes have been seen as promising methods because this type of techniques has two advantages. Firstly, there are no requirements on the condition of better main channels. Secondly, when the number of transmit antennas is larger the number of the eavesdroppers, the main channel state information (CSI) and precoding matrices of security schemes can be broadcasted to both legitimate receivers and eavesdroppers. We do not need to worry about the leakage of key precoding messages. More details of the second advantage can be seen in Liu et al. (2017b).

Artificial Noise Schemes

The section will introduce a general AN-based model, which is first provided in Liu et al. (2017b). Assuming t is the number of transmit antennas, messages are encoded in s (which is a variable) strongest eigen-subchannels based on ordered eigenvalues of Wishart matrices (HH or HH, here, H is the main channel CSI matrix), while AN signals are generated in remaining t − s eigen-subchannels. This scheme treats the number of eigen-subchannels for message streams, i.e., s, as an optimization objective that can be leveraged to optimize secrecy capacities.

General AN-Based Model

Let us consider a MIMO communication system in the presence of correlated Rayleigh fading at receiver and eavesdropper sides. The system consists of a transmitter (Alice) with t transmit antennas, a legitimate receiver (Bob) with r receive antennas, and an eavesdropper (Eve) with e receive antennas, as shown in Fig. 2. Define \(m=\max (t,r)\) and \(n=\min (t,r)\). In general, the main channel between Alice and Bob and the wiretap channel between Alice and Eve are defined by receiver-side correlated complex Gaussian matrices \(\mathbf {H}\in \mathbb {C}^{r\times t}\) and \({\mathbf {H}}_e\in \mathbb {C}^{e\times t}\), whose elements obey distribution \(\mathscr {CN}(0,1)\). \({\mathbf {R}}_r\in \mathbb {C}^{r\times r}\) and \({\mathbf {R}}_e\in \mathbb {C}^{e\times e}\) are the receiver-side correlated channel matrices from Bob and Eve, respectively. Also assume that Alice knows full CSI of Bob via a broadcast feedback channel from Bob, including H and Rr, but only knows the channel distribution information (CDI) of Eve and Re. Eve knows the CSI of all channels, including H, He, Rr, and Re, where it is assumed that t > e.
Fig. 2

Illustration of an artificial noisy MIMO wiretap channel model with receiver-side correlated Rayleigh fading. Alice has t transmit antennas, Bob has r receive antennas, and Eve has e receive antennas. Assumed that Alice, Bob, and Eve have uniformly linear array antennas with du antenna spacing. θ represents AoA between a scattered path and the antenna array, which can be viewed as a random variable with enough scatters, and θ follows a Gaussian distribution

In this AN-based scheme, there are s (s ≤ t) message-sending eigen-subchannels, which are selected by Alice based on the CSI feedback from Bob. More specifically, Alice performs SVD of \({\mathbf {H}}^\dagger \mathbf {H}\in \mathbb {C}^{t\times t}\) in a preprocessor, whose output is a unitary matrix \(\mathbf {U}\in \mathbb {C}^{t\times t}\), its Hermitian transpose form \({\mathbf {U}}^{\dagger }\in \mathbb {C}^{t\times t}\), and a diagonal matrix \(\boldsymbol {\varLambda }\in \mathbb {R}^{t\times t}\), which consists of positive and zero eigenvalues of HH. Then, Alice generates a message precoding matrix \(\mathbf {B} \in \mathbb {C}^{t \times s}\), whose columns are the eigenvectors corresponding to the first to the sth largest eigenvalues of HH, and an AN precoding matrix \(\mathbf {Z} \in \mathbb {C}^{ t\times d}\) (s + d = t), whose columns are the eigenvectors of the remaining eigenvalues of HH.

Alice transmits Bw + Zv via t transmit antennas. It means that each antenna transmits a combination of message components and AN components, but the AN components can be eliminated by the preprocessor at Bob. In this way, we create a capacity difference between the main channels and wiretap channels. Note that B and Z are fixed semi-unitary matrices derived from H. Hence, we have BB = Is and ZZ = Id. An example of AN precoding scheme with s = 2 is illustrated in Fig. 3.
Fig. 3

An example of this AN precoding scheme with s = 2

Theorem 1

The AN signal can be eliminated at Bob if and only if [HB]HZ = 0 (Liu et al.,2017b).

Theorem 2

The AN signal cannot be eliminated at Eve if and only if t > e, [HB]HeZ ≠ 0, and [HeB]HeZ ≠ 0 (Liu et al.,2017b).

According to CSI matrix H and the preprocessing method, the received signals at Bob and Eve can be expressed as
$$\displaystyle \begin{aligned} &\mathbf{y}=\mathbf{HBw}+\mathbf{H}\mathbf{Zv}+\mathbf{n},{} \end{aligned} $$
(2)
$$\displaystyle \begin{aligned} & {\mathbf{y}}_e={\mathbf{H}}_e\mathbf{Bw}+{\mathbf{H}}_e\mathbf{Zv}+{\mathbf{n}}_e, {} \end{aligned} $$
(3)
respectively. Here, w is a transmitted signal of the desired user, and v is a random AN signal. Both w and v are circularly symmetric complex Gaussian vectors with zero means and its covariance matrices PtIs and PtId, respectively, where P is the average transmit power constraint. For simplification, we distribute total power to each antenna evenly. n and ne are additive white Gaussian noise (AWGN) vectors with their covariance matrices Ir and Ie, respectively.
Bob can eliminate the AN signal v by preprocessing ([HB]HZ = 0) the received signal y as
$$\displaystyle \begin{aligned} \tilde{\mathbf{y}}=\big[\mathbf{H}\mathbf{B}\big]^{\dagger}\mathbf{y}= \boldsymbol{\varLambda}_s\mathbf{w}+\tilde{\mathbf{n}}, \end{aligned} $$
(4)
where \(\tilde {\mathbf {n}}=[\mathbf {H}\mathbf {B}]^{\dagger }\mathbf {n} \in \mathbb {C}^{s\times 1}\) is an AWGN vector with its distribution \(\mathscr {CN}(\mathbf {0},\boldsymbol {\varLambda }_s)\). \(\boldsymbol {\varLambda }_s\in \mathbb {R}^{s\times s}\) is a diagonal matrix formed by the first to the sth eigenvalues of HH. In the elimination process, the received signal multiplied by a fixed matrix will not change its capacity. Even if we consider the worst case that Eve has the knowledge of H, He, B, and Z, the AN signal still degrades Eve’s channel because Eve cannot eliminate the AN signal because t > e, [HB]HeZ ≠ 0, and [HeB]HeZ ≠ 0, as in Theorem 1.

Secrecy Metric

The secrecy capacity and secrecy rate are important measures of AN-based schemes. Here, four expressions are provided to represent the instantaneous secrecy capacity, the average secrecy capacity, the instantaneous secrecy rate, and the average secrecy rate, respectively. These measures are fit in with different conditions as in Table 1
Table 1

Secrecy metrics with their conditions

Measures

Symbols

Conditions

Instantaneous secrecy capacity

Cs

He is available and optimal input distribution.

Average secrecy capacity

\(\tilde {C}_s\)

He is unavailable and optimal input distribution

Instantaneous secrecy rate

Rs

He is available and Gaussian distribution input

Average secrecy rate

\(\tilde {R}_s\)

He is unavailable and Gaussian distribution input

Instantaneous Secrecy Capacity

In a MIMO wiretap channel model, while Bob has the knowledge of H, He, Rr, and Re, the instantaneous secrecy capacity is
$$\displaystyle \begin{aligned} C_s=\max_{p(\mathbf{w}),p(\mathbf{v})}\big\{I(\mathbf{w};\mathbf{y})-I(\mathbf{w};{\mathbf{y}}_e)\big\}, \end{aligned} $$
(5)
where I(w;y) is the mutual information between information variables w and received vector y at Bob and I(w;ye) is the mutual information between information variables w and received vector ye at Eve. And the maximization is taken over all possible input distributions of p(w) and p(v). However, it is hard to begin the optimization process when He is unavailable.

Average Secrecy Capacity

Assuming that the communication lasts longer enough to experience all channel states, to average out the randomness of Cs when only CDI of He is available, we remark that Cs is a function of He, and then, the average secrecy capacity is
$$\displaystyle \begin{aligned} \tilde{C}_s=\max_{p(\mathbf{w}),p(\mathbf{v})}\big\{I(\mathbf{w};\mathbf{y})-I(\mathbf{w};{\mathbf{y}}_e|{\mathbf{H}}_e)\big\}, \end{aligned} $$
(6)
where \(I(\mathbf {w};\mathbf {y}|{\mathbf {H}}_e)=\text{E}_{{\mathbf {H}}_e}[I(\mathbf {w};{\mathbf {y}}_e)]\) is the expected value of the conditional mutual information between w and ye for given He. And the maximization is taken over all possible input distributions of p(w) and p(v).

Instantaneous Secrecy Rate

Either in the instantaneous expression or in the average expression, it is hard to find optimal distributions of p(w) and p(v) to maximize the secrecy capacity. Here follow the convention in Liu et al. (2017b, 2015), and use the secrecy rate instead of the secrecy capacity with Gaussian input alphabets and Gaussian AN, i.e., both w and v are circularly symmetric complex Gaussian vectors. In this case, the instantaneous secrecy rate can be expressed as
$$\displaystyle \begin{aligned} R_s=&[C_m-C_w]^+, \end{aligned} $$
(7)
where \([x]^+=\max (x,0)\). Cm and Cw are
$$\displaystyle \begin{aligned} C_m=&\log_2\det({\mathbf{I}}_r+(P/t){\mathbf{H}}_1{\mathbf{H}}_1^{\dagger}), \\ \notag C_w=&\log_2\text{det}\bigg({\mathbf{I}}_e+\frac{(P/t){\mathbf{H}}_2{\mathbf{H}}_2^{\dagger}}{(P/t){\mathbf{H}}_3{\mathbf{H}}_3^{\dagger}+{\mathbf{I}}_e}\bigg), \end{aligned} $$
(8)
respectively. Here, \({\mathbf {H}}_1=\mathbf {HB}\in \mathbb {C}^{r\times s}\), \({\mathbf {H}}_2={\mathbf {H}}_e\mathbf {B}\in \mathbb {C}^{e\times s}\), and \({\mathbf {H}}_3={\mathbf {H}}_e\mathbf {Z}\in \mathbb {C}^{e\times d}\). However, the instantaneous secrecy capacity is hard to be calculated by Eq. (7) in the absence of Eve’s CSI.

Average Secrecy Rate

Assuming the CDI of He, the average secrecy rate can be expressed as
$$\displaystyle \begin{aligned} \tilde{R}_s=&\text{E}_{{\mathbf{H}}_e}[C_m-C_w]^+\\ \notag \geq &C_m-\text{E}_{{\mathbf{H}}_e}[C_w]. \end{aligned} $$
(9)
Equation (9) gets an equality if and only if the secrecy rates are always positive over all channel states. Since AN signals are used to jam the wiretap channels, the average secrecy rate is always nonnegative. From Lemma 1 in the Appendix, H2, H3, and H4 are complex Gaussian matrices with their distributions
$$\displaystyle \begin{aligned} {\mathbf{H}}_2\sim \mathscr{CN}_{e,s}(\mathbf{0},{\mathbf{R}}_e\otimes {\mathbf{I}}_s), \end{aligned} $$
(10)
and
$$\displaystyle \begin{aligned} {\mathbf{H}}_3\sim \mathscr{CN}_{e,d}(\mathbf{0},{\mathbf{R}}_e\otimes {\mathbf{I}}_d), \end{aligned} $$
(11)
respectively. Hence, the expected value of average secrecy rates can be calculated by Monte Carlo simulations with Eve’s CDI.

Correlated Matrices

The correlated matrices Rr and Re can affect all the secrecy metrics of an AN-based system. As in Bolcskei et al. (2003), a correlated matrix R (generalized for Rr and Re) is a function of AoA (defined by θ) distribution. Here consider an AoA model with respect to a uniform antenna array based on 3GPP 3GP (2003), as shown in Fig. 4, where the AoA model belongs to a Gaussian distribution, the mean AoA of θ is \(\bar {\theta }\), and the RAS (variance) of θ is δ. Then, we get the following simple model for the correlated coefficient [R]u,v[5], which is
$$\displaystyle \begin{aligned}{}[\mathbf{R}]_{u,v}=&\exp\big\{-j2\pi(u-v)d_u\cos\bar{\theta}\big\} \\ \notag \times &\exp\big\{-\frac{1}{2}\big(2\pi \delta (u-v)d_u\sin\bar{\theta}\big)^2\big\}, \end{aligned} $$
(12)
where u ∈{1, …, r} and v ∈{1, …, r} are the receive antenna index numbers. According to the parameters from 3GPP standard 3GP (2003), \(\bar {\theta }\) is set in the range [0, 100] and δ in the range [5, 35], respectively. Assume that all antennas form a uniformly linear antenna array with \(d_u=d_{\min }/\omega \), where \(d_{\min }\) is the normalized minimum distance and ω is the wavelength. In the wiretap channel model, Rr and Re have the same structure as R. Here define the mean AoA at Bob and Eve as \(\bar {\theta }_r\) and \(\bar {\theta }_e\), respectively, and define the RAS at Bob and Eve as δr and δe, respectively.
Fig. 4

AoA model with respect to a uniform antenna array

Simulations

Simulation results are provided to investigate joint impacts of the number of antennas and the number of selected message-sending eigen-subchannels on the average secrecy rates.

Figure 5 shows the relationship between the average secrecy rates and the number of Bob’s antennas. In the case of adequate transmit antennas, i.e., t : r = κ ≥ 2 and e = r, the average secrecy rates increase with an increasing number of Bob’s antennas in both high (P = 30 W) and low SNR (P = 3 × 10−4 W) regions. In the case with a small number of transmit antennas and adequate receive antennas (here we set t : r =≤ 1, s = 1, and e = t − 1), we find that the average secrecy rates converge to a deterministic constant when e becomes large. It means that an increasing number of Bob’s antennas bring in no benefit when Alice uses less antennas than Bob and selects s = 1, i.e., uses a transmit diversity scheme.
Fig. 5

Average secrecy rates in terms of the number of Bob’s antennas

Conclusion and Further Work

Physical layer security will play a critical role in the future security architecture of cellular communications. This research should keep competing with cryptography and quantum communications, which are believed to be the three major security architectures for cellular communication systems. AN-based schemes are seen as promising methods and now have produced a raft of fascinating and important results. However, these AN-based schemes are optimal only under the condition of that the number of transmit antennas is larger than eavesdropper antennas. When the number of transmitted antennas is constrained or even smaller than that of eavesdropper antennas, AN-based schemes cannot get positive secrecy capacities or rates. This motivates us to design a better AN-based scheme. In addition, the secrecy capacity (rate) quantization is a great challenge in AN-based schemes. Without perfect CSI of eavesdroppers, it is hard for encoders to calculate instantaneous secrecy capacities or rates. And the influence of fading does not allow us to use a simple AWGN or Rayleigh channel model to calculate a secrecy rate. It seems that we need to investigate main and wiretap fading channels that belong to other models of cellular communications. At last, power allocation in cellular networks should be investigated further, because power allocation problems are usually non-convex ones with a lot of cellular users.

Appendix

Lemma 1 (Proved in Gupta and Nagar 1999)

Define an r × t matrix \(\mathbf {A}\sim \mathscr {CN}_{r,t}(\mathbf {0},\mathbf {R}\otimes {\mathbf {I}}_t)\) as a receiver-side correlated central complex Gaussian matrix, and establish an independent (t × s) unitary matrix B. We have
$$\displaystyle \begin{aligned} \mathbf{A}\mathbf{B}\sim \mathscr{CN}_{r,s}(\mathbf{0},\mathbf{R}\otimes {\mathbf{I}}_s), \end{aligned} $$
(13)

where \(s\in \mathbb {R}\) and t  s.

Key Applications

Artificial noise (AN) schemes based on MIMO technologies can be applied to various cellular scenarios. For example, Massive MIMO systems have an enormous number of antennas, which offer more degrees of freedom for wireless channels, and a more secure performance in terms of secrecy capacities and the number of AN beamforming. The small call base stations deployed as cooperative jammers in cellular networks can be used to provide well-designed AN signals. In addition, the long-term evolution advanced (LTE-A) system supports device to device (D2D) communications, which is defined as the direct communications between two mobile users via shared radio resources with cellular users. D2D interference caused by the shared radio resources can be seen as AN signals to interfere with the illegitimate eavesdropper.

Cross-References

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Authors and Affiliations

  1. 1.Department of Electronics Information EngineeringHarbin Institute of TechnologyHarbinChina

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