Amplitude-Constrained Gaussian Wiretap Channel: Computation of the Optimal Input Distribution

Abstract

This paper studies a scalar Gaussian wiretap channel where instead of an average input power constraint, we consider a peak amplitude constraint on the input. The goal is to obtain insights into the secrecy-capacity and the structure of the secrecy-capacity-achieving distribution. Capitalizing on the recent theoretical progress on the structure of the secrecycapacity-achieving distribution, this paper develops a numerical procedure, based on the gradient ascent algorithm and a version of the Blahut-Arimoto algorithm, for computing the secrecycapacity and the secrecy-capacity-achieving input and output distributions.

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

“…where W ∼ N (0 n+2 , (σ 2 2 − σ 2 1 )I n+2 ). A calculation related to (33) was erroneously performed in [27]. However, this error does not change the results of [27] as only the sign of the derivative is important and not the value itself.…”

“…To that end, we need Karlin's oscillation theorem presented in Section 2.1. By using (33), the fact that the pdf of a chi-square is a positive defined kernel [26], and Theorem 1, the number of sign changes of Ξ ( x ; P X R ) is upperbounded by the number of sign changes of…”

“…In the case of scalar Gaussian wiretap channels, the secrecy capacity and the optimal input pmf can be estimated via the algorithm described in [33], i.e., a numerical procedure that takes inspiration from the deterministic annealing algorithm sketched in [34]. Let us denote by Ĉs (σ 2 1 , σ 2 2 , R, n) the numerical estimate of the secrecy capacity, and by P X , the estimate of the optimal pmf on the input norm.…”

“…Let us denote by Ĉs (σ 2 1 , σ 2 2 , R, n) the numerical estimate of the secrecy capacity, and by P X , the estimate of the optimal pmf on the input norm. To numerically evaluate Ĉs (σ 2 1 , σ 2 2 , R, n) and P X , we extend to the vector case the algorithm in [33]. Our extension is defined in Algorithm 1.…”

“…In this work, we focus on the secrecy capacity and on the secrecy-capacity-achieving input distribution. However, it is possible to study other points of the rate-equivocation region of the degraded wiretap Gaussian channel by suitably changing the KKT conditions, as reported in [21], Equations (33) and (34). With the due modifications, the proposed optimization algorithm can find the optimal input distribution for any point of the rate-equivocation region.…”