Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Optimization Methods for Secure and Energy Efficient MIMO System

Anshu Mukherjee

Published Jan 1, 2023• 0 citations
OpenAlex

Abstract

In this modern era, wireless networks continue to expand worldwide and play a pronounced role in daily life. However, wireless communication has become vulnerable to intelligent invaders in the network. Hence, the security aspects of wireless communication have been the center of modern-era research. Physical Layer Security (PLS) schemes have been proven to be very effective against potential eavesdroppers thanks to their low-complexity nature compared to encryption techniques. Therefore, this thesis is dedicated to the physical layer security of multiple-input multiple-output (MIMO) systems. More specifically, we provide efficient optimization techniques to maximize the achievable secrecy rate of MIMO Wiretap channels (WTC). First, we present novel numerical approaches to finding the achievable secrecy rate of the MIMO WTC subject to multiple power constraints. There is a challenge in deriving computationally efficient solutions to the secrecy rate maximization problem because the secrecy rate is expressed as a difference of convex functions (DC) of the transmit covariance matrix. However, to deal with it, we provided local optimization methods, such as the accelerated DC algorithm (ADCA) and a numerical method named partial best response algorithm (PBRA) based on the concave-convex equivalent reformulation of the secrecy capacity problem. Along with it, we provide a convex reformulation for degraded channels. Interestingly, our experiments have shown that the local optimization methods achieve global optimality for the secrecy rate maximization problem under power constraints. In this regard, we provide analytical proof with Karush-Kuhn-Tucker (KKT) conditions to validate our observation. Motivated by our finding, we propose an accelerated gradient projection algorithm with adaptive momentum parameters (AGPnc) that directly solves the secrecy rate maximization problem rather than the equivalent convex-concave form. We then extend the scope of our research of physical layer security enhancement by considering the Intelligent reflecting surface (IRS)-aided MIMO WTC system, as the IRS has achieved enormous attention recently for enhancement in spectral efficiency. In this context, we provided a numerical solution for the problem of maximizing the achievable secrecy rate of the considered system by jointly optimizing the input covariance matrix and the IRS phase shifts. The proposed is an iterative method based on block successive maximization (BSM) where each iteration is done in closed form. We then extend our research further toward maximizing the secrecy energy efficiency (SEE) of the (IRS)-aided MIMO WTC system. In relation to the SEE maximization problem, we provide a low-complexity based local optimization method. The challenge of this considered problem is the coupling between design variables in both the objective and the constraint. The main idea of the proposed method is to apply a penalty dual decomposition based alternating gradient projection (PDDAPG) method to obtain an efficient solution. Thanks to the low complexity nature of proposed methods, these methods can scale favourably with large-scale wireless systems, which has been an inherent issue of the existing studies. In this way, the contributions of this thesis will significantly impact future research in the field of physical layer security, especially toward the future generation of wireless communication.

Related papers

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

Anshu Mukherjee, Björn Ottersten, Le‐Nam Tran

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.

arXiv:2012.05667•IEEE Transactions on Communications•Jul 21, 2021•24 citations

Efficient Numerical Methods for Secrecy Capacity of Gaussian MIMO Wiretap Channel

Anshu Mukherjee, Björn Ottersten, Le‐Nam Tran

This paper presents two different low-complexity methods for obtaining the secrecy capacity of multiple-input multiple-output (MIMO) wiretap channel subject to a sum power constraint (SPC). The challenges in deriving computationally efficient solutions to the secrecy capacity problem are due to the fact that the secrecy rate is a difference of convex functions (DC) of the transmit covariance matrix, for which its convexity is only known for the degraded case. In the first method, we capitalize on the accelerated DC algorithm, which requires solving a sequence of convex subproblems. In particular, we show that each subproblem indeed admits a water-filling solution. In the second method, based on the equivalent convex-concave reformulation of the secrecy capacity problem, we develop a so-called partial best response algorithm (PBRA). Each iteration of the PBRA is also done in closed form. Simulation results are provided to demonstrate the superior performance of the proposed methods.

arXiv:2102.10396•IEEE Vehicular Technology Conference•Apr 1, 2021•3 citations

Methods for Optimal Transmission over MIMO Wiretap Channels

Kaiya Li

We discuss the architecture of wireless multiple-input multiple-output (MIMO) channels with additive white Gaussian noise. The transmission optimization problem over covariance matrix for achieving MIMO channel capacity is reviewed under the assumption that the channel state information (CSI) is available from both the transmitter and the receiver. We also discuss several transmission strategies such as beamforming and water-filling algorithm. The information-theoretic approach to physical-layer security for wireless communications system is discussed. Physical-layer security which is different from cryptographic approach security is achieved by exploiting the physical-layer properties of wireless channel. The principal results and concepts in this field such as equivocation rate and secrecy capacity are reviewed. The wiretap channel model is considered. We mainly discussed the Gaussian MIMO wiretap channel and the methods for obtaining its secrecy capacity. The optimization problem for the secrecy capacity of Gaussian MIMO wiretap channel under total transmit power constraint is discussed. CVX, which is a modeling toolbox for solving convex optimization problems, is reviewed. The precision variable of CVX is also considered. The performances of CVX and Monte Carlo optimization are compared. Differential Evolution algorithm is used to compute the secrecy capacity of Gaussian MIMO wiretap channel. The case of weak eavesdropper is considered and the closed form solution based on the necessary KKT conditions is validated. The algorithm for general cases based on the linear approximation and reformulation of the original optimization problem is also investigated and validated. It has been verified that the lower bound and the upper bound of secrecy capacity can be obtained by such algorithm.

•Jan 1, 2014•0 citations

Energy Harvesting in Secure MIMO Systems

Wei Qiang Wu, Xueqi Zhang, Baoyun Wang

The problems of energy harvesting in wireless com- munication systems have recently drawn much attention. In this paper, we focus on the investigation of energy harvesting maximization (EHM) in the important secrecy multi-input multi- output (MIMO) systems where little research has been done due to their complexity. Particularly, this paper studies the resource allocation strategies in MIMO wiretap channels, wherein we attempt to maximize the harvested energy by one or multiple multi-antenna energy receivers (ERs) (potential eavesdropper) while guaranteeing the secure communication for the multi- antenna information receiver (IR). Two types of IR, with and without the capability to cancel the interference from energy signals, are taken into account. In the scenario of single energy receiver (ER), we consider the joint design of the transmis- sion information and energy covariances for EHM. Both of the optimization problems for the two types of IR are non- convex, and appear to be difficult. To circumvent them, the combination of first order Taylor approximation and sequential convex optimization approach is proposed. Then, we extend our attention to the scenario with multiple ERs, where the artificial noise (AN) aided weighted sum-energy harvesting maximization (WS-EHM) problem is considered. Other than the approaches adopted in solving the EHM problems, an algorithm conducts in an alternating fashion is proposed to handle this problem. In particular, we first perform a judicious transformation of the WS- EHM problem. Then, a block Gauss-Seidel (GS) algorithm based on logarithmic barrier method and gradient projection (GP) is derived to obtain the optimal solution of the reformulation by solving convex problems alternately. Furthermore, the resulting block GS method is proven to converge to a Karush-Kuhn-Tucker (KKT) point of the original WS-EHM problem...

arXiv:1601.03468•Jan 14, 2016•1 citation

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

Anshu Mukherjee, Vaibhav Kumar, Eduard Axel Jorswieck +2 more

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 methodsindeed achieve global optimality. 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 forboth degraded and non-degraded cases. Motivated by this, we also propose a low-complexity algorithm to find a stationary point. Numerical results are presented to verify the theoretical analysis.

arXiv:2105.11415•IEEE Wireless Communications Letters•Nov 16, 2021•6 citations

Secrecy rate maximization for information and energy transfer in MIMO beamforming networks

Jens Steinwandt, Sergiy A. Vorobyov, Martin Haardt

Consider a multiple-input multiple-output (MIMO) beamforming system, where a multi-antenna base station transmits information and energy simultaneously to a multi-antenna information receiver (IR) and a number of multi-antenna energy receivers (ERs). The ERs are assumed to possess dual functionality such that they can also decode information. This gives rise to the physical layer security issue of providing confidential information to the IR while ensuring efficient energy harvesting for the ERs. Hence, we employ an artificial noise (AN)-aided secrecy approach at the transmitter, where the AN improves the security by interfering with the information decoding at the ERs while, at the same time, providing them with wireless energy to harvest. In this paper, we address the problem of jointly optimizing the beamforming matrix for the IR and the covariance matrix of the AN such that the secrecy rate is maximized subject to energy harvesting constraints and a total power constraint. The corresponding optimization problem is a difference of convex functions (DC) programming problem, which is generally non-convex. Nevertheless, we propose an alternating optimization (AO) strategy to tackle this problem for the cases of a square and a non-square IR channel. The performance of the proposed algorithms is demonstrated by simulation results.

Asilomar Conference on Signals, Systems and Computers (ACSSC)•Nov 1, 2014•4 citations
In this modern era, wireless networks continue to expand worldwide and play a pronounced role in daily life. However, wireless communication has become vulnerable to intelligent invaders in the network. Hence, the security aspects of wireless communication have been the center of modern-era research. Physical Layer Security (PLS) schemes have been proven to be very effective against potential eavesdroppers thanks to their low-complexity nature compared to encryption techniques. Therefore, this thesis is dedicated to the physical layer security of multiple-input multiple-output (MIMO) systems. More specifically, we provide efficient optimization techniques to maximize the achievable secrecy rate of MIMO Wiretap channels (WTC). First, we present novel numerical approaches to finding the achievable secrecy rate of the MIMO WTC subject to multiple power constraints. There is a challenge in deriving computationally efficient solutions to the secrecy rate maximization problem because the secrecy rate is expressed as a difference of convex functions (DC) of the transmit covariance matrix. However, to deal with it, we provided local optimization methods, such as the accelerated DC algorithm (ADCA) and a numerical method named partial best response algorithm (PBRA) based on the concave-convex equivalent reformulation of the secrecy capacity problem. Along with it, we provide a convex reformulation for degraded channels. Interestingly, our experiments have shown that the local optimization methods achieve global optimality for the secrecy rate maximization problem under power constraints. In this regard, we provide analytical proof with Karush-Kuhn-Tucker (KKT) conditions to validate our observation. Motivated by our finding, we propose an accelerated gradient projection algorithm with adaptive momentum parameters (AGPnc) that directly solves the secrecy rate maximization problem rather than the equivalent convex-concave form. We then extend the scope of our research of physical layer security enhancement by considering the Intelligent reflecting surface (IRS)-aided MIMO WTC system, as the IRS has achieved enormous attention recently for enhancement in spectral efficiency. In this context, we provided a numerical solution for the problem of maximizing the achievable secrecy rate of the considered system by jointly optimizing the input covariance matrix and the IRS phase shifts. The proposed is an iterative method based on block successive maximization (BSM) where each iteration is done in closed form. We then extend our research further toward maximizing the secrecy energy efficiency (SEE) of the (IRS)-aided MIMO WTC system. In relation to the SEE maximization problem, we provide a low-complexity based local optimization method. The challenge of this considered problem is the coupling between design variables in both the objective and the constraint. The main idea of the proposed method is to apply a penalty dual decomposition based alternating gradient projection (PDDAPG) method to obtain an efficient solution. Thanks to the low complexity nature of proposed methods, these methods can scale favourably with large-scale wireless systems, which has been an inherent issue of the existing studies. In this way, the contributions of this thesis will significantly impact future research in the field of physical layer security, especially toward the future generation of wireless communication.
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.
This paper presents two different low-complexity methods for obtaining the secrecy capacity of multiple-input multiple-output (MIMO) wiretap channel subject to a sum power constraint (SPC). The challenges in deriving computationally efficient solutions to the secrecy capacity problem are due to the fact that the secrecy rate is a difference of convex functions (DC) of the transmit covariance matrix, for which its convexity is only known for the degraded case. In the first method, we capitalize on the accelerated DC algorithm, which requires solving a sequence of convex subproblems. In particular, we show that each subproblem indeed admits a water-filling solution. In the second method, based on the equivalent convex-concave reformulation of the secrecy capacity problem, we develop a so-called partial best response algorithm (PBRA). Each iteration of the PBRA is also done in closed form. Simulation results are provided to demonstrate the superior performance of the proposed methods.
We discuss the architecture of wireless multiple-input multiple-output (MIMO) channels with additive white Gaussian noise. The transmission optimization problem over covariance matrix for achieving MIMO channel capacity is reviewed under the assumption that the channel state information (CSI) is available from both the transmitter and the receiver. We also discuss several transmission strategies such as beamforming and water-filling algorithm. The information-theoretic approach to physical-layer security for wireless communications system is discussed. Physical-layer security which is different from cryptographic approach security is achieved by exploiting the physical-layer properties of wireless channel. The principal results and concepts in this field such as equivocation rate and secrecy capacity are reviewed. The wiretap channel model is considered. We mainly discussed the Gaussian MIMO wiretap channel and the methods for obtaining its secrecy capacity. The optimization problem for the secrecy capacity of Gaussian MIMO wiretap channel under total transmit power constraint is discussed. CVX, which is a modeling toolbox for solving convex optimization problems, is reviewed. The precision variable of CVX is also considered. The performances of CVX and Monte Carlo optimization are compared. Differential Evolution algorithm is used to compute the secrecy capacity of Gaussian MIMO wiretap channel. The case of weak eavesdropper is considered and the closed form solution based on the necessary KKT conditions is validated. The algorithm for general cases based on the linear approximation and reformulation of the original optimization problem is also investigated and validated. It has been verified that the lower bound and the upper bound of secrecy capacity can be obtained by such algorithm.
The problems of energy harvesting in wireless com- munication systems have recently drawn much attention. In this paper, we focus on the investigation of energy harvesting maximization (EHM) in the important secrecy multi-input multi- output (MIMO) systems where little research has been done due to their complexity. Particularly, this paper studies the resource allocation strategies in MIMO wiretap channels, wherein we attempt to maximize the harvested energy by one or multiple multi-antenna energy receivers (ERs) (potential eavesdropper) while guaranteeing the secure communication for the multi- antenna information receiver (IR). Two types of IR, with and without the capability to cancel the interference from energy signals, are taken into account. In the scenario of single energy receiver (ER), we consider the joint design of the transmis- sion information and energy covariances for EHM. Both of the optimization problems for the two types of IR are non- convex, and appear to be difficult. To circumvent them, the combination of first order Taylor approximation and sequential convex optimization approach is proposed. Then, we extend our attention to the scenario with multiple ERs, where the artificial noise (AN) aided weighted sum-energy harvesting maximization (WS-EHM) problem is considered. Other than the approaches adopted in solving the EHM problems, an algorithm conducts in an alternating fashion is proposed to handle this problem. In particular, we first perform a judicious transformation of the WS- EHM problem. Then, a block Gauss-Seidel (GS) algorithm based on logarithmic barrier method and gradient projection (GP) is derived to obtain the optimal solution of the reformulation by solving convex problems alternately. Furthermore, the resulting block GS method is proven to converge to a Karush-Kuhn-Tucker (KKT) point of the original WS-EHM problem...
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 methodsindeed achieve global optimality. 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 forboth degraded and non-degraded cases. Motivated by this, we also propose a low-complexity algorithm to find a stationary point. Numerical results are presented to verify the theoretical analysis.
Consider a multiple-input multiple-output (MIMO) beamforming system, where a multi-antenna base station transmits information and energy simultaneously to a multi-antenna information receiver (IR) and a number of multi-antenna energy receivers (ERs). The ERs are assumed to possess dual functionality such that they can also decode information. This gives rise to the physical layer security issue of providing confidential information to the IR while ensuring efficient energy harvesting for the ERs. Hence, we employ an artificial noise (AN)-aided secrecy approach at the transmitter, where the AN improves the security by interfering with the information decoding at the ERs while, at the same time, providing them with wireless energy to harvest. In this paper, we address the problem of jointly optimizing the beamforming matrix for the IR and the covariance matrix of the AN such that the secrecy rate is maximized subject to energy harvesting constraints and a total power constraint. The corresponding optimization problem is a difference of convex functions (DC) programming problem, which is generally non-convex. Nevertheless, we propose an alternating optimization (AO) strategy to tackle this problem for the cases of a square and a non-square IR channel. The performance of the proposed algorithms is demonstrated by simulation results.