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IEEE Transactions on Signal Processing

Optimal Power Allocation and Optimal Linear Encoding for Parameter Estimation in the Presence of a Smart Eavesdropper

Erfan Mehdipour Abadi, Çağrı Göken, Cuneyd Ozturk, Sinan Gezici

Published Jan 1, 2022• 7 citations
Publisher

Abstract

In this paper, we consider secure transmission of a deterministic vector parameter from a transmitter to an intended receiver in the presence of a smart eavesdropper. The aim is to determine the optimal power allocation and optimal linear encoding strategies at the transmitter to maximize the estimation performance at the intended receiver under constraints on the estimation performance at the eavesdropper and on the transmit power. First, the A-optimality criterion is adopted by utilizing the Cramér-Rao lower bound (CRLB) as the estimation performance metric, and the optimal power allocation and optimal linear encoding strategies are characterized theoretically. Then, corresponding to the D-optimality criterion, the determinant of the Fisher information matrix is considered as the estimation performance metric. It is shown that the optimal linear encoding and optimal power allocation strategies lead to the same solution for this criterion. In addition, extensions of the theoretical results are provided to cases with statistical knowledge of systems parameters. Numerical examples are provided to investigate the optimal power allocation and optimal linear encoding strategies in different scenarios.

DOI
10.1109/tsp.2022.3198180

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In this paper, we consider secure transmission of a deterministic vector parameter from a transmitter to an intended receiver in the presence of a smart eavesdropper. The aim is to determine the optimal power allocation and optimal linear encoding strategies at the transmitter to maximize the estimation performance at the intended receiver under constraints on the estimation performance at the eavesdropper and on the transmit power. First, the A-optimality criterion is adopted by utilizing the Cramér-Rao lower bound (CRLB) as the estimation performance metric, and the optimal power allocation and optimal linear encoding strategies are characterized theoretically. Then, corresponding to the D-optimality criterion, the determinant of the Fisher information matrix is considered as the estimation performance metric. It is shown that the optimal linear encoding and optimal power allocation strategies lead to the same solution for this criterion. In addition, extensions of the theoretical results are provided to cases with statistical knowledge of systems parameters. Numerical examples are provided to investigate the optimal power allocation and optimal linear encoding strategies in different scenarios.
Optimal power allocation for secure estimation of multiple deterministic parameters is investigated under a total power constraint. The goal is to minimize the Cramér-Rao lower bound (CRLB) at an intended receiver while keeping estimation errors at an eavesdropper above specified target levels. To that end, an optimization problem is formulated by considering measurement models involving linear transformation of the parameter vector and additive Gaussian noise. Although the proposed optimization problem is nonconvex, it is decomposed into convex sub-problems by utilizing the structure of the secrecy constraints. Then, optimal solutions to the sub-problems are characterized via optimization theoretic approaches. An algorithm utilizing that characterization is developed to obtain the optimal solution of the proposed problem.
In this paper, optimal deterministic encoding of a vector parameter is investigated in the presence of an eavesdropper. The objective is to minimize the expectation of the conditional Cramér-Rao bound at the intended receiver, while satisfying an individual secrecy constraint on the mean-squared error of estimating each parameter at the eavesdropper. The eavesdropper is modeled to employ the linear minimum mean-squared error estimator based on the noisy observation of the encoded parameter without being aware of encoding. First, the problem is formulated as a constrained optimization problem in the space of vector-valued functions. Then, two practical solution strategies are developed based on nonlinear individual encoding and affine joint encoding of parameters. Theoretical results on the solutions of the proposed strategies are provided for various scenarios on channel conditions and parameter distributions. Finally, numerical examples are presented to illustrate the performance of the proposed solution approaches.
Estimation theoretic secure transmission of a scalar random parameter is investigated in the presence of an eavesdropper. The aim is to minimize the estimation error at the receiver under a secrecy constraint at the eavesdropper; or, alternatively, to maximize the estimation error at the eavesdropper for a given estimation accuracy limit at the receiver. In the considered setting, the encoder at the transmitter is allowed to use a randomized mapping between two one-to-one and continuous functions and the eavesdropper is fully aware of the encoding strategy at the transmitter. For small numbers of observations, both the eavesdropper and the receiver are modeled to employ linear minimum mean-squared error (LMMSE) estimators, and for large numbers of observations, the expectation of the conditional Cramér-Rao bound (ECRB) metric is employed for both the receiver and the eavesdropper. Optimization problems are formulated and various theoretical results are provided in order to obtain the optimal solutions and to analyze the effects of encoder randomization. In addition, numerical examples are presented to corroborate the theoretical results. It is observed that stochastic encoding can bring significant performance gains for estimation theoretic secrecy problems.
In this paper, optimal deterministic encoding of a scalar parameter is investigated in the presence of an eavesdropper. The aim is to minimize the expectation of the conditional Cramér-Rao bound at the intended receiver while keeping the mean-squared error (MSE) at the eavesdropper above a certain threshold. First, optimal encoding functions are derived in the absence of secrecy constraints for any given prior distribution on the parameter. Next, an optimization problem is formulated under a secrecy constraint and various solution approaches are proposed. Also, theoretical results on the form of the optimal encoding function are provided under the assumption that the eavesdropper employs a linear minimum mean-squared error (MMSE) estimator. Numerical examples are presented to illustrate the theoretical results and to investigate the performance of the proposed solution approaches.
In this letter, optimal deterministic encoding of a uniformly distributed scalar parameter is performed in the presence of an eavesdropper. The objective is to maximize the worst case Fisher information of the parameter at the intended receiver while keeping the mean-squared error (MSE) at the eavesdropper above a certain level. The eavesdropper is modeled to employ the linear minimum MSE estimator based on the encoded version of the parameter. First, the optimal encoding function is derived when there exist no secrecy constraints. Next, to obtain the solution of the problem in the presence of the secrecy constraint, the form of the encoding function that maximizes the MSE at the eavesdropper is explicitly derived for any given level of worst case Fisher information. Then, based on this result, a low-complexity algorithm is provided to calculate the optimal encoding function for the given secrecy constraint. Finally, numerical examples are presented.
We consider the problem of secure communication in wireless fading channels in the presence of non-colluding passive eavesdroppers. The transmitter has multiple antennas and is able to simultaneously transmit an information bearing signal to the intended receiver and artificial noise to the eavesdroppers. We obtain an analytical closed-form lower bound for secrecy capacity, which is used as the objective function to optimize transmit power allocation between the information signal and the artificial noise. Our analytical and numerical results show that equal power allocation is a simple and generic strategy which achieves near optimal capacity performance. We also find that adaptive power allocation based on each channel realization provides no or insignificant capacity improvement over equal power allocation.