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

Abstract

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.

Efficient Numerical Methods for Secrecy Capacity of Gaussian MIMO Wiretap Channel

“…1) X-update: To compute X n as in Line 3 of Algorithm 2, we need to solve (12) which is a convex problem. Since the projection onto X can be done in closed form, we can apply an accelerated projected gradient method (APGM) [16] to solve it efficiently, which is described as follows.…”

“…, σr ] where r is the rank of X, and τ is the unique number such that r i=1 max( σi + − τ, 0) = P 0 . The APGM for solving (12) is outlined in Algorithm 3. Note that a proper step size can be found by a backtracking line search as done in Lines ( 4)- (7).Starting from the step size of the previous iteration, the idea of the backtracking line search is to reduce it by a factor of θ until (7) is met.…”

“…To lighten the notation, we will drop the subscript n onwards. Now, let Ψ 12 Ψ † 12 = U Ψ ΣΨ U † Ψ be the eigenvalue decom-Algorithm 3 Accelerated projected gradient method for solving (12) 1: Input: Y…”

“…In this paper we consider the problem of finding the secrecy capacity-achieving input covariance for Gaussian MIMO WTC subject to a sum power constraint (SPC). The case of general linear transmit covariance constraints such as per-antenna power constraint is studied in [12]. In particular, we develop two low-complexity methods to solve the secrecy-capacity problem for the Gaussian MIMO WTC.…”

“…We skip the details here for the sake of brevity. The convergence proof of Algorithm 1 is provided in [12]. The idea is to show that the sequence {γ n } increasing and the feasible set X is compact and convex.…”