Multiple-input Multiple-output Wiretap Channel Research Articles

K-Receiver Wiretap Channel: Optimal Encoding Order and Signaling Design

The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -receiver wiretap channel is a channel model where a transmitter broadcasts <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> independent messages to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> intended receivers while keeping them secret from an eavesdropper. The capacity region of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -receiver multiple-input multiple-output (MIMO) wiretap channel has been characterized by using dirty-paper coding and stochastic encoding. However, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> factorial encoding orders may need to be enumerated to evaluate the capacity region, which makes the problem intractable. In addition, even though the capacity region is known, the optimal signaling to achieve the capacity region is unknown. In this paper, we determine one optimal encoding order to achieve every point on the capacity region, and thus reduce the encoding complexity <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> factorial times. We prove that the optimal decoding order for the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -receiver MIMO wiretap channel is the same as that for the MIMO broadcast channel without secrecy. To be specific, the descending weight ordering in the weighted sum-rate (WSR) maximization problem determines the optimal encoding order. Next, to reach the border of the secrecy capacity region, we form a WSR maximization problem and apply the block successive maximization method to solve this nonconvex problem and find the input covariance matrices corresponding to each message. Numerical results are used to verify the optimality of the encoding order and to demonstrate the efficacy of the proposed signaling design.

Secure communication over RIS-aided underlay CR MIMO wiretap channel without Eve’s CSI

This paper investigates a reconfigurable intelligent surface (RIS)-aided underlay cognitive radio (CR) multiple-input multiple-output (MIMO) wiretap channel where the secondary transmitter (ST) communicates with primary user (PU) and secondary user (SU) in the absence of the eavesdropper’s (Eve’s) channel state information (CSI). To enhance the secrecy performance in CR MIMO wiretap channel, the power of useful signal is minimized at ST, and then the residual power is further utilized to design artificial noise (AN) based on statistical CSI at ST. Specifically, we first optimize the transmit covariance matrix at ST and the diagonal phase-shifting matrix at RIS jointly leveraging large-system approximation results. Then the power allocation for SU is optimized to obtain the minimum transmit power of useful information at ST. Besides, we further design AN with the residual power by aligning it into the null space of the SU channel and thus avert the harmful effects of AN to improve the secure communication quality of SU. Finally, through numerical simulations, we illustrate the effectiveness of the proposed algorithm and validate the existence of a trade-off between the quality-of-service (QoS) at SU and secrecy rate.

Joint Precoding and Artificial Noise Design for MU-MIMO Wiretap Channels

Secure precoding superimposed with artificial noise (AN) is a promising transmission technique to improve security by harnessing the superposition nature of the wireless medium. However, finding a jointly optimal precoding and AN structure is very challenging in downlink multi-user multiple-input multiple-output wiretap channels with multiple eavesdroppers. The major challenge in maximizing the secrecy rate arises from the non-convexity and non-smoothness of the rate function. Traditionally, an alternating optimization framework that identifies beamforming vectors and AN covariance matrix has been adopted; yet this alternating approach has limitations in maximizing the secrecy rate. In this paper, we put forth a novel secure precoding algorithm that jointly and simultaneously optimizes the beams and AN covariance matrix for maximizing the secrecy rate when a transmitter has either perfect or partial channel knowledge of eavesdroppers. To this end, we first establish an approximate secrecy rate in a smooth function. Then, we derive the first-order optimality condition in the form of the nonlinear eigenvalue problem (NEP). We present a computationally efficient algorithm to identify the principal eigenvector of the NEP as a suboptimal solution for secure precoding. Simulations demonstrate that the proposed methods improve secrecy rate significantly compared to the existing methods.

On the Optimality of the Stationary Solution of Secrecy Rate Maximization for MIMO Wiretap Channel

To achieve perfect secrecy in a multiple-input multiple-output (MIMO) Gaussian wiretap channel (WTC), we need to find its secrecy capacity and optimal signaling, which involves solving a difference of convex functions program known to be non-convex for the non-degraded case. To deal with this, a class of existing solutions have been developed but only local optimality is guaranteed by standard convergence analysis. Interestingly, our extensive numerical experiments have shown that these local optimization methods <i>indeed achieve global optimality</i>. In this letter, we provide an analytical proof for this observation. To achieve this, we show that the Karush-Kuhn-Tucker (KKT) conditions of the secrecy rate maximization problem admit a unique solution for <i>both degraded and non-degraded cases</i>. Motivated by this, we also propose a low-complexity algorithm to find a stationary point. Numerical results are presented to verify the theoretical analysis.

On the Secrecy Capacity of MIMO Wiretap Channels: Convex Reformulation and Efficient Numerical Methods

This paper presents novel numerical approaches to finding the secrecy capacity of the multiple-input multiple-output (MIMO) wiretap channel subject to multiple linear transmit covariance constraints, including sum power constraint, per antenna power constraints and interference power constraint. An analytical solution to this problem is not known and existing numerical solutions suffer from slow convergence rate and/or high per-iteration complexity. Deriving computationally efficient solutions to the secrecy capacity problem is challenging since the secrecy rate is expressed as a difference of convex functions (DC) of the transmit covariance matrix, for which its convexity is only known for some special cases. In this paper we propose two low-complexity methods to compute the secrecy capacity along with a convex reformulation for degraded channels. In the first method we capitalize on the accelerated DC algorithm which requires solving a sequence of convex subproblems, for which we propose an efficient iterative algorithm where each iteration admits a closed-form solution. In the second method, we rely on the concave-convex equivalent reformulation of the secrecy capacity problem which allows us to derive the so-called partial best response algorithm to obtain an optimal solution. Notably, each iteration of the second method can also be done in closed form. The simulation results demonstrate a faster convergence rate of our methods compared to other known solutions. We carry out extensive numerical experiments to evaluate the impact of various parameters on the achieved secrecy capacity.

Robust Secure Energy Efficiency Optimization in the MIMO Wiretap Channel with Energy Harvesting

This work considers a secure multiple-input multiple-output (MIMO) channel where the energy receiver (ER) is a potential eavesdropper (Eve). Specifically, we aim to maximize the secrecy energy efficiency (SEE) via jointly designing the transmit precoding matrix and the artificial noise (AN) covariance at the base station (BS), as well as the power splitting (PS) ratio at the desired receiver (DR), subject to the constraints of the transmit power budget and harvested energy threshold. To handle the formulated highly nonconvex fractional problem, we apply the successive convex approximation (SCA) method to transform the objective and constraint into a tractable form. Then, a penalty-based iterative algorithm is proposed. Finally, simulation results validate the performance of the proposed design.

Secure Transmission in MIMO-NOMA Networks

This letter focuses on the physical layer security over two-user multiple-input multiple-output (MIMO) non-orthogonal multiple access (NOMA) networks. A linear precoding technique is designed to ensure the confidentiality of the message of each user from its counterpart. This technique first splits the base station power between the two users and, based on that, decomposes the secure MIMO-NOMA channel into two MIMO wiretap channels, and designs the transmit covariance matrix for each channel separately. The proposed method substantially enlarges the secrecy rate compared to existing linear precoding methods and strikes a balance between performance and computation cost. Simulation results verify the effectiveness of the proposed method.

Secure Degrees of Freedom of MIMO Two-Way Wiretap Channel With no CSI Anywhere

This article considers a two-way multiple-input multiple-output (MIMO) Rayleigh block fading wiretap channel with two full-duplex nodes (Alice and Bob) exchanging messages simultaneously and wiretapped by a $ {N}_{ {e}}$ -antennas eavesdropper (Eve). The channel state remains unchanged over a coherence interval T and is unknown to all terminals, including Alice, Bob, and Eve, at the beginning of the coherence interval. We first show the expression of the optimal signal waveform for Alice and Bob, which maximizes the sum secrecy rate, should be the products of independent random diagonal matrices and isotropically distributed unitary matrices. Then, conditioned on large T and small $ {N}_{ {e}}$ , the upper-bound for the secure degree of freedom (s.d.o.f.) of the two-way wiretap channel is derived, and a constant-norm-signaling (CNS) scheme is presented to achieve this theoretical bound. Finally, we formulate an optimization problem of the s.d.o.f. achieved by the CNS scheme for general cases with arbitrary T and $ {N}_{ {e}}$ , and solve this problem by exploring the monotonicity of the achievable s.d.o.f.. Our result shows that the optimal s.d.o.f. can be independent of coherence time T and becomes the product of the numbers of legitimate nodes’s transmitter antennas (NLNTA) for large $ {N}_{ {e}}$ and small NLNTA.

Enhancing Secure MIMO Transmission via Intelligent Reflecting Surface

In this article, we consider an intelligent reflecting surface (IRS) assisted Guassian multiple-input multiple-output (MIMO) wiretap channel (WTC), and focus on enhancing its secrecy rate. Due to MIMO setting, all the existing solutions for enhancing the secrecy rate over multiple-input single-output WTC completely fall to this work. Furthermore, all the existing studies are simply based on an ideal assumption that full channel state information (CSI) of eavesdropper (Ev) is available. Therefore, we propose numerical solutions to enhance the secrecy rate of this channel under both full and no Ev's CSI cases. For the full CSI case, we propose a barrier method and one-by-one (OBO) optimization combined alternating optimization (AO) algorithm to jointly optimize the transmit covariance R at transmitter (Tx) and phase shift coefficient Q at IRS. For the case of no Ev's CSI, we develop an artificial noise (AN) aided joint transmission scheme to enhance the secrecy rate. In this scheme, a bisection search (BS) and OBO optimization combined AO algorithm is proposed to jointly optimize R and Q. Such scheme is also applied to enhance the secrecy rate under a special scenario in which the direct link between Tx and receiver (Rx)/Ev is blocked due to obstacles. In particular, we propose a BS and minorization-maximization (MM) combined AO algorithm with slightly faster convergence to optimize R and Q for this scenario. Simulation results have validated the monotonic convergence of the proposed algorithms, and it is shown that the proposed algorithms for the IRS-assisted design achieve significantly larger secrecy rate than the other benchmark schemes under full CSI. When Ev's CSI is unknown, the secrecy performance of this channel also can be enhanced by the proposed AN aided scheme, and there is a trade-off between increasing the quality of service at Rx and enhancing the secrecy rate.

Secure MIMO Transmission via Intelligent Reflecting Surface

In this letter, we consider an intelligent reflecting surface (IRS) assisted Guassian multiple-input multiple-output (MIMO) wiretap channel in which a multi-antenna transmitter communicates with a multi-antenna receiver in the presence of a multi-antenna eavesdropper. To maximize the secrecy rate of this channel, an alternating optimization (AO) algorithm is proposed to jointly optimize the transmit covariance R at transmitter and phase shift coefficient Q at IRS by fixing the other as a constant. When Q is fixed, existing numerical algorithm is used to search for global optimal R . When R is fixed, three sucessive approximation to the objective function to surrogate lower bound is applied and minorization-maximization (MM) algorithm is proposed to optimize the local optimal Q . Simulation results have be provided to validate the convergence and performance of the proposed AO algorithm.

On the Secure Degrees of Freedom for Two-User MIMO Interference Channel With a Cooperative Jammer

We investigate the secure degrees of freedom (SDoF) for the two-user multiple-input multiple-output (MIMO) interference channel with a cooperative jammer, where each transmitter is equipped with $M$ antennas, each receiver is equipped with $N$ antennas, and the jammer is equipped with $K$ antennas. For each one of the four regions of parameters $(M,N)$ , i.e., $1\leq \frac {N}{M} , $\frac {N}{M} \geq 2$ , $\frac {1}{2}\le \frac {N}{M} , and $\frac {N}{M} , we provide both upper and lower bounds on the SDoF of the system which depend on the value of $K$ . In particular, the upper bound is the union of three upper bounds that are based on the DoF of the MIMO interference channel, the SDoF of the MIMO wiretap channel with a jammer, and the Secrecy Penalty Lemma and the Role of A Helper Lemma for the interference channel, respectively. For the lower bound, we provide achievability schemes that make use of signal alignment, zero forcing, cooperative jamming, and multi-slot transmission. Our results indicate that the SDoF of the system under consideration improves with respect to the no jammer case in all regimes, especially when $N>2M$ . Moreover, we quantify the SDoF gap of the network and reveal it as a function of the number of the antennas. For large $K$ , the upper and lower bounds coincide in all regimes, and the exact SDoF can be achieved.

Blind MIMO Cooperative Jamming: Secrecy via ISI Heterogeneity Without CSIT

We investigate the secure degrees of freedom (SDoF) of the multiple-input multiple-output (MIMO) wiretap channel with intersymbol interference (ISI) in the presence of a multi-antenna cooperative jammer. We focus on the practically relevant setting with no channel state information at the transmitters (CSIT). More specifically, the legitimate transmitter and the cooperative jammer only have statistical knowledge of the channel in terms of the effective number of ISI channel taps, i.e., channel impulse response (CIR) lengths, toward the legitimate receiver and the eavesdropper. Our main contribution is to show that in the absence of CSIT, positive SDoF can be achieved by carefully exploiting: 1) the heterogeneity of the CIR lengths toward both receivers and 2) the relative number of antennas at the four terminals. To achieve secrecy, we propose a scheme in which the transmitter strategically sends a mixture of information and artificial noise symbols in a way that exploits the heterogeneity of CIR lengths. Additionally, the cooperative jammer transmits artificial noise symbols in a way that completely masks all the information symbols that are received at the eavesdropping node. Under the proposed scheme, positive SDoF can be achieved, even when the number of antennas at the legitimate receiver is less than the number of antennas at the eavesdropper.

Improving physical layer security jointly using full‐duplex jamming receiver and multi‐antenna jammer in wireless networks

The issues of the physical layer security are studied, where one transmitter (Alice), one full-duplex (FD) legitimate receiver (Bob) and one multi-antenna jammer (Charlie) are considered to defend against multiple passive and non-colluding eavesdroppers (Eves), for multiple-input multiple-output wiretap channels. With the assumption that the complete channel state information (CSI) for main channels and the statistical CSI for Eves' channels are available, an improved secrecy transmission design is proposed to minimise the secrecy outage probability (SOP), under the minimum secrecy rate requirement. In addition to integrating artificial noise aided transmit beamforming and multi-antenna cooperative jammer to impair Eves' channels, which is widely used in many existing works, a FD legitimate receiver is applied together to further enhance physical layer security. Based on the basic probability theory and convex optimisation, the accurate closed-form expression of the SOP is derived, and the impacts on secrecy performance are described, including the qualities of main channels, the Alice's transmit power and the number of Eves. Numerical results show that the proposed scheme using the FD Bob can significantly enhance the system security, as compared with that employing the half-duplex Bob.

AN-Aided Secure Beamforming in Power-Splitting-Enabled SWIPT MIMO Heterogeneous Wireless Sensor Networks

In this paper, we investigate the physical layer security in a two-tier heterogeneous wireless sensor network (HWSN) depending on simultaneous wireless information and power transfer (SWIPT) approach for multiuser multiple-input multiple-output wiretap channels with artificial noise (AN) transmission, where a more general system framework of HWSN only includes a macrocell and a femtocell. For the sake of implementing security enhancement and green communications, the joint optimization problem of the secure beamforming vector at the macrocell and femtocell, the AN vector, and the power splitting ratio is modeled to maximize the minimal secrecy capacity of the wiretapped macrocell sensor nodes (M-SNs) while considering the fairness among multiple M-SNs. To reduce the performance loss of the rank relaxation from the SDR technique while solving the non-convex max–min program, we apply successive convex approximation (SCA) technique, first-order Taylor series expansion and sequential parametric convex approximation (SPCA) approach to transform the max–min program to a second order cone programming (SOCP) problem to iterate to a near-optimal solution. In addition, we propose a novel SCA-SPCA-based iterative algorithm while its convergence property is proved. The simulation shows that our SCA-SPCA-based method outperforms the conventional methods.

Robust Secure Beamforming in MIMO Wiretap Channels With Deterministically Bounded Channel Errors

In this paper, a multiple-input multiple-output wiretap channel model is studied, where the information packet sent from the transmitter to the desired receiver is wiretapped by a potential eavesdropper. When ideal channel state information (CSI) is available, we design the secure beamforming using a proposed iterative closed-form updating scheme based on majorization–minimization approach. Numerical results demonstrate that the proposed scheme acquires the same secrecy rate as the existing algorithms whereas it has much lower complexity. When ideal CSI is not available and channel errors are modeled by deterministically bounded model, we investigate robust secure beamforming (RSB) optimization problem whose goal is worst-case secrecy rate maximization. In the formulated optimization problem, the objective function includes the expression of a logarithm of a determinant ( $\log \det$ ) minus $\log \det$ . Instead of approximating the $\log \det$ expression as a trace, we propose to linearize two $\log \det$ terms. After linearization, epigraph reformulation is used to deal with deterministically bounded channel errors and semi-infinite constraints are recast as linear matrix inequalities. Finally, an alternating optimization algorithm is utilized to find the RSB design. After obtaining the RSB design, we derive the worst-case secrecy rate. Numerical results illustrate that the proposed RSB design is superior to the nonrobust one.

The Secrecy Capacity of Gaussian MIMO Wiretap Channels Under Interference Constraints

Secure signaling over multiple-input multiple-output (MIMO) wiretap channel (WTC) is studied under interference and transmit power constraints. The classical MIMO WTC model is extended to interference-limited scenarios, so that interference to other users does not exceed a given threshold while ensuring simultaneously no information leakage to an eavesdropper. The operational secrecy capacity of the Gaussian MIMO WTC under interference and transmit power constraints is rigorously established in two forms (a non-convex max problem and a convex–concave max–min problem), to which per-antenna power constraints can be added as well. Optimal signaling directions are characterized in the general case, from which (tight) upper bounds to the rank of optimal transmit covariance matrix are derived. A sufficient condition for the optimality of beamforming and a necessary condition for optimal full-rank signaling are given. Closed-form rank-1 and high-rank solutions are obtained in the case of zero interference constraints. Sufficient and necessary conditions for non-zero secrecy capacity are established. The results are extended to multi-user scenarios. A sufficient and necessary condition for the unbounded growth of the secrecy capacity with transmit power is obtained. The interplay between transmit and interference power constraints is studied, and its significant impact on optimal signaling is demonstrated (so that neither constraint can be absorbed into the other one in general, as was sometimes suggested in the literature). Overall, these results provide insights into fundamental information-theoretic limits and optimal signaling strategies for secure communications under interference constraints.

MIMO Wiretap Channels Based on Generalized Extended Orthogonal STBCs and Feedback

This paper presents a two-phase process consisting of advantage distillation (AD) and information reconciliation for achieving secret sharing over a wiretap channel. By using the feedback bits from the legitimate receiver to transmitter, we propose a novel AD scheme for multiple-input multiple-output (MIMO) wiretap (MIMO-WT) channels. We first generalize the extended orthogonal space-time block codes (EO-STBCs). Then, based on the generalized EO-STBCs (GEO-STBCs) and channel state information (CSI), the feedback bits transmitted from the legitimate receiver to the transmitter are generated to increase the signal-to-noise ratio (SNR) gap between the legitimate receiver and the eavesdropper. Simulation results show that the feedback performance gain of the proposed AD scheme increases rapidly with the number of transmit antennas and the value of SNR. A bit error rate (BER) performance gain up to 5.5 dB is obtained by the legitimate receiver over the eavesdropper at the BER of $10^{-5}$ . Moreover, it is also shown that the secrecy capacity of two-way MIMO-WT channels with feedback grows rapidly in low SNR regime and grows slowly in high SNR regime.

The MIMO Wiretap Channel Decomposed

The problem of sending a secret message over the Gaussian multiple-input multiple-output (MIMO) wiretap channel is studied. While the capacity of this channel is known, it is not clear how to construct optimal coding schemes that achieve this capacity. In this paper, we use linear operations along with successive interference cancellation to attain effective parallel single-antenna wiretap channels. By using independent scalar Gaussian wiretap codebooks over the resulting parallel channels, the capacity of the MIMO wiretap channel is achieved. The derivation of the schemes is based upon joint triangularization of the channel matrices. We find that the same technique can be used to rederive capacity expressions for the MIMO wiretap channel in a way that is simple and closely connected to a transmission scheme. This technique allows to extend the previously proven strong security for scalar Gaussian channels to the MIMO case. We further consider the problem of transmitting confidential messages over a two-user broadcast MIMO channel. For that problem, we find that derivation of both the capacity and a transmission scheme is a direct corollary of the proposed analysis for the MIMO wiretap channel.

Optimal Beamforming for Gaussian MIMO Wiretap Channels With Two Transmit Antennas

A Gaussian multiple-input multiple-output wiretap channel in which the eavesdropper and legitimate receiver are equipped with arbitrary numbers of antennas and the transmitter has two antennas is studied in this paper. The input covariance matrix that achieves the secrecy capacity is determined. In particular, it is shown that the secrecy capacity of this channel can be achieved by linear precoding . Precoding and power allocation schemes that maximize the achievable secrecy rate, and thus achieve the secrecy capacity, are developed. The secrecy capacity is then compared with the achievable secrecy rate of generalized singular value decomposition (GSVD)-based precoding, which is the best previously proposed technique for this problem. Numerical results demonstrate that substantial gain can be obtained in secrecy rate between the proposed and GSVD-based precodings.

Secrecy Rate Improvement Based on Joint Decoding in MIMO Wiretap Channels With a Helping Interferer

In a wiretap channel, a transmitter sends confidential messages to an intended legitimate receiver in the presence of an eavesdropper. An external helper can assist secret communication by transmitting independent artificial interference to confuse the eavesdropper. We consider that the legitimate receiver jointly decodes messages from the transmitter and interference from the helper to boost the secrecy rate. In the discrete-memoryless wiretap channel with a helper, we characterize the achievable secrecy rates by comparing the strength of interference at the legitimate receiver and at the eavesdropper. Based on these secrecy rates, we formulate a secrecy rate maximization problem in Gaussian multiple-input multiple-output (MIMO) wiretap channels with a helping interferer, and present an alternating optimization algorithm. Numerical results show that the proposed joint decoding achieves higher secrecy rates than the conventional separate decoding.