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

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.