Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution

TL;DR

This paper analyzes the secrecy capacity of an n-dimensional Gaussian wiretap channel under peak-power constraints, characterizing optimal input distributions and their properties in low amplitude and asymptotic regimes.

Contribution

It determines the largest peak-power constraint for which a spherical input distribution is optimal and characterizes the asymptotic behavior as dimension grows.

Findings

Identifies the low amplitude regime where spherical distributions are optimal.

Provides a complete asymptotic characterization of the peak-power constraint as dimension increases.

Shows the secrecy-capacity-achieving distribution is discrete with finitely many points in the scalar case.

Abstract

This paper studies secrecy-capacity of an $n$-dimensional Gaussian wiretap channel under a 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 low 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. Several numerical examples are provided, such as the example of the secrecy-capacity-achieving distribution beyond the low amplitude regime. Furthermore, for the scalar case $(n=1)$ we show that the secrecy-capacity-achieving input distribution is discrete with finitely many points at most of the order of $\frac{\mathsf{R}^2}{\sigma_1^2}$, where $\sigma_1^2$ is the variance of the Gaussian noise over the legitimate channel.