On the Capacity Achieving Input of Amplitude Constrained Vector Gaussian Wiretap Channel

TL;DR

This paper characterizes the secrecy-capacity of an n-dimensional Gaussian wiretap channel under peak-power constraints, identifying optimal input distributions and their asymptotic behavior as dimension grows, with practical numerical insights.

Contribution

It determines the largest peak-power constraint for which a uniform spherical input is optimal and characterizes the asymptotic behavior of this constraint as dimension increases.

Findings

Optimal input distribution is uniform on a sphere in the small-amplitude regime.

Asymptotic behavior of the peak-power constraint as n approaches infinity.

Secrecy-capacity can be computed explicitly and examples provided.

Abstract

This paper studies secrecy-capacity of an $n$-dimensional Gaussian wiretap channel under the peak-power constraint. This work determines the largest peak-power constraint $\bar{\mathsf{R}}_n$ such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the small-amplitude regime. The asymptotic of $\bar{\mathsf{R}}_n$ as $n$ goes to infinity is completely characterized as a function of noise variance at both receivers. Moreover, the secrecy-capacity is also characterized in a form amenable for computation. Furthermore, several numerical examples are provided, such as the example of the secrecy-capacity achieving distribution outside of the small amplitude regime.