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On Secrecy Capacity of the Helper-Assisted Gaussian Wiretap Channel with Multi-Antennas

Lingxiang Li, Zhi Chen, Duo Zhang, Jun Fang

Published Dec 1, 2014• 0 citations
Publisher

Abstract

We investigate the secrecy capacity of Gaussian wiretap channel with a source, an external helper, an eavesdropper and a single-antenna legitimate receiver where the former three terminals are equipped with Na, Njand Neantennas, respectively. Generally, an analytical form of the secrecy capacity in this scenario is difficult to obtain. Instead, we recast the original nonconvex secrecy rate maximization (SRM) problem into a sequence of convex optimization problems. In doing so, the secrecy capacity can be obtained using a combination of convex optimization and a one-dimensional search. On the other hand, to gain more insight into how the secrecy capacity behaves, we study the secure degrees of freedom (s.d.o.f.) and quantify its connection with system parameters, where our result proves that the s.d.o.f. equal to 1 can be achieved if and only if Nea+Nj-1. As a by-product, we give a suboptimal but closed-form solution to the original SRM problem for the scenario where Nea+Nj-1.Numerical results are presented to validate the theoretical findings and illustrate the efficacy of the proposed schemes.

DOI
10.1109/glocom.2014.7417056

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We investigate the secrecy capacity of Gaussian wiretap channel with a source, an external helper, an eavesdropper and a single-antenna legitimate receiver where the former three terminals are equipped with Na, Njand Neantennas, respectively. Generally, an analytical form of the secrecy capacity in this scenario is difficult to obtain. Instead, we recast the original nonconvex secrecy rate maximization (SRM) problem into a sequence of convex optimization problems. In doing so, the secrecy capacity can be obtained using a combination of convex optimization and a one-dimensional search. On the other hand, to gain more insight into how the secrecy capacity behaves, we study the secure degrees of freedom (s.d.o.f.) and quantify its connection with system parameters, where our result proves that the s.d.o.f. equal to 1 can be achieved if and only if Nea+Nj-1. As a by-product, we give a suboptimal but closed-form solution to the original SRM problem for the scenario where Nea+Nj-1.Numerical results are presented to validate the theoretical findings and illustrate the efficacy of the proposed schemes.
We investigate the secrecy capacity of Gaussian wiretap channel with a source, an external helper, an eavesdropper and a single-antenna legitimate receiver where the former three terminals are equipped with NaN_aNa​, NjN_jNj​ and NeN_eNe​ antennas, respectively. Generally, an analytical form of the secrecy capacity in this scenario is difficult to obtain. Instead, we recast the original nonconvex secrecy rate maximization (SRM) problem into a sequence of convex optimization problems. In doing so, the secrecy capacity can be obtained using a combination of convex optimization and a one-dimensional search. On the other hand, to gain more insight into how the secrecy capacity behaves, we study the secure degrees of freedom (s.d.o.f.) and quantify its connection with system parameters, where our result proves that the s.d.o.f. equal to 1 can be achieved if and only if N_e<N_a+N_j-1. As a by-product, we give a suboptimal but closed-form solution to the original SRM problem for the scenario where N_e<N_a+N_j-1.Numerical results are presented to validate the theoretical findings and illustrate the efficacy of the proposed schemes.
We study the secrecy capacity of a helper-assisted Gaussian wiretap channel with a source, a legitimate receiver, an eavesdropper and an external helper, where each terminal is equipped with multiple antennas. Determining the secrecy capacity in this scenario generally requires solving a nonconvex secrecy rate maximization (SRM) problem. To deal with this issue, we first reformulate the original SRM problem into a sequence of convex subproblems. For the special case of single-antenna legitimate receiver, we obtain the secrecy capacity via a combination of convex optimization and one-dimensional search, while for the general case of multi-antenna legitimate receiver, we propose an iterative solution. To gain more insight into how the secrecy capacity of a helper-assisted Gaussian wiretap channel behaves, we examine the achievable secure degrees of freedom (s.d.o.f.) and obtain the maximal achievable s.d.o.f. in closed-form. We also derive a closed-form solution to the original SRM problem which achieves the maximal s.d.o.f.. Numerical results are presented to illustrate the efficacy of the proposed schemes.
We study the secrecy capacity of Gaussian wiretap channel aided by an external jammer/helper. Both the transmitter and the intended receiver are equipped with a single antenna, while the eavesdropper and the jammer are equipped withMMMandNNNantennas, respectively. Generally, an analytical form of the secrecy capacity in this scenario is difficult to obtain. Instead, we consider a null-space jamming scheme which totally nulls out the jamming signal at the legitimate receiver, and derive lower and upper bounds on its maximal achievable secrecy rateRN{R_N}RN​. The relationship between the average secrecy capacityCˉN{\bar C_N}CˉN​of Gaussian wiretap channel and the average secrecy rateRˉN{\bar R_N}RˉN​achieved by the null-space jamming scheme is investigated, and we prove thatRˉN≤CˉN≤RˉN+1{\bar R_N} \leq {\bar C_N} \leq {\bar R_{N + 1}}RˉN​≤CˉN​≤RˉN+1​. Based on this inequality and the derived lower and upper bounds onRN{R_N}RN​, the upper and lower bounds on the average secrecy capacity of Gaussian wiretap channel aided by an external jammer can be obtained, where our result shows that whenN>MN > MN>M, the average secrecy capacity increases linearly with the total transmit power; while whenN≤M−1N \leq M - 1N≤M−1, there exists a performance ceiling on it.
This paper considers secrecy communication from a signal processing point of view, and studies the maximal achievable secrecy degrees of freedoms (S.D.o.F.) of a helper-assisted Gaussian wiretap channel, consisting of a source, a legitimate receiver, an eavesdropper and an external helper. Each terminal is equipped with multiple antennas. We first propose a cooperative secrecy transmission scheme, and show that it achieves the maximal secrecy degrees of freedom. We then propose a heuristic method, through which, we solve analytically the optimization problem associated with the proposed cooperative secrecy transmission scheme. By this way, we obtain the maximal achievable S.D.o.F. and also the precoding matrices which achieve the maximal S.D.o.F. in closed-form.
We consider a multiple-input multiple-output (MIMO) Gaussian wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance that achieves secrecy capacity subject to a power constraint, which in general leads to a difficult non-convex optimization problem. For the Gaussian multiple-input single-output (MISO) wiretap channel where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper each equipped with one antenna, the optimal input covariance is obtained in closed form. For the general case, we derive the necessary conditions for the optimal input covariance in the form of a set of equations. We show that for MISO case, the derived conditions guarantee the optimal input covariance. Numerical results are presented to illustrate the proposed theoretical findings.
We consider a Gaussian multiple-input multiple-output (MIMO) wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint, which leads to a non-convex optimization problem that is in general difficult to solve. Existing results for this problem address the case in which the transmitter and the legitimate receiver have two antennas each and the eavesdropper has one antenna. For the general cases, it has been shown that the optimal input covariance matrix has low rank when the difference between the Grams of the eavesdropper and the legitimate receiver channel matrices is indefinite or semi-definite, while it may have low rank or full rank when the difference is positive definite. In this paper, the aforementioned non-convex optimization problem is investigated. In particular, for the multiple-input single-output (MISO) wiretap channel, the optimal input covariance matrix is obtained in closed form. For general cases, we derive the necessary conditions for the optimal input covariance matrix consisting of a set of equations. For the case in which the transmitter has two antennas, the derived necessary conditions can result in a closed form solution; For the case in which the difference between the Grams is indefinite and has all negative eigenvalues except one positive eigenvalue, the optimal input covariance matrix has rank one and can be obtained in closed form; For other cases, the solution is proved to be a fixed point of a mapping from a convex set to itself and an iterative procedure is provided to search for it. Numerical results are presented to illustrate the proposed theoretical findings.