The Rate-Equivocation Region of the Degraded Discrete-Time Poisson Wiretap Channel

This paper addresses the degraded discrete-time Poisson wiretap channel (DT-PWC) in an optical wireless communication system based on intensity modulation and direct detection (IM-DD). Subject to nonnegativity, average-intensity, and bandwidth constraints, we find that the secrecy capacity and the entire boundary of the rate-equivocation region are attained by discrete distributions with a countably infinite number of mass points, but with finitely many mass points in any bounded interval. Additionally, we shed light on the asymptotic behavior of the secrecy capacity in the regimes where the average intensity constraint either tends to zero (low-intensity) or tends to infinity (high-intensity). In the low-intensity regime, we observe that: when the channel gains of the legitimate receiver and the eavesdropper are identical, the secrecy capacity scales linearly in the average-intensity epsilon; whereas when the channel gains are different, the secrecy capacity scales, to within a constant, like (alpha(B) - alpha(E))epsilon log log 1/epsilon, where alpha(B) and alpha(E) are the legitimate receiver's and the eavesdropper's channel gains, respectively. In the high-intensity regime, we establish that the secrecy capacity does not scale with the average intensity constraint.