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

This paper studies the secrecy capacity of an n-dimensional Gaussian wiretap channel under a peak power constraint. This work determines the largest peak power constraint (Formula presented.), such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the low-amplitude regime. The asymptotic value of (Formula presented.) 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 to 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 (Formula presented.), we show that the secrecy-capacity-achieving input distribution is discrete with finitely many points at most at the order of (Formula presented.), where (Formula presented.) is the variance of the Gaussian noise over the legitimate channel.