Secret-Key Agreement with Public Discussion subject to an Amplitude Constraint

TL;DR

This paper investigates secret-key agreement over channels with a peak amplitude constraint, proving the optimal input distribution is discrete, and provides bounds and schemes that approach capacity under various amplitude regimes.

Contribution

It establishes the discreteness of the optimal input distribution under amplitude constraints and introduces suboptimal schemes with near-capacity performance.

Findings

Optimal input distribution is discrete with finite support.

Secret-key capacity converges to the average power constrained capacity as A increases.

Capacity asymptotically matches the legitimate user's capacity when A approaches zero.

Abstract

This paper considers the problem of secret-key agreement with public discussion subject to a peak power constraint $A$ on the channel input. The optimal input distribution is proved to be discrete with finite support. The result is obtained by first transforming the secret-key channel model into an equivalent Gaussian wiretap channel with better noise statistics at the legitimate receiver and then using the fact that the optimal distribution of the Gaussian wiretap channel is discrete. To overcome the computationally heavy search for the optimal discrete distribution, several suboptimal schemes are proposed and shown numerically to perform close to the capacity. Moreover, lower and upper bounds for the secret-key capacity are provided and used to prove that the secret-key capacity converges for asymptotic high values of $A$, to the secret-key capacity with an average power constraint $A^2$. Finally, when the amplitude constraint A is small ($A \to 0$), the secret-key capacity is proved to be asymptotically equal to the capacity of the legitimate user with an amplitude constraint A and no secrecy constraint.