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. Subject to nonnegativity, peakand average-intensity as well as bandwidth constraints, we study the secrecy-capacityachieving input distribution of this wiretap channel and prove it to be unique and discrete with a finite number of mass points; one of them located at the origin. Furthermore, we establish that every point on the boundary of the rate-equivocation region of this wiretap channel is also obtained by a unique and discrete input distribution with finitely many mass points. In general, the number of mass points of the optimal distributions are greater than two. This is in contrast with the degraded continuous-time PWC when the signaling bandwidth is not restricted and where the secrecy capacity and the entire boundary of the rate-equivocation region are achieved by binary distributions. Furthermore, we extend our analysis to the case where only an average-intensity constraint is active. For this case, we find that the secrecy capacity and the entire boundary of the rate-equivocation region are attained by discrete distributions with 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 constraints either tend to zero (low-intensity) or tend to infinity (high-intensity). In the low-intensity regime, we observe that: 1) when only the the peak-intensity constraint is active, the secrecy capacity scales quadratically in the peak-intensity; 2) when both peakand average-intensity constraints are active with their ratio held fixed, the secrecy capacity again scales quadratically in the peak-intensity constraint; 3) when both peakand average-intensity constraints are active and the peakintensity is held fixed while the average-intensity tends to zero, the secrecy capacity scales linearly in M. Soltani was with the Department of Electrical and Computer Engineering, University of Idaho, Moscow, Idaho, USA, e-mail: solt8821@vandals.uidaho.edu, and Z. Rezki is with the Department of Electrical and Computer Engineering, University of California Santa Cruz, CA, USA, e-mail:zrezki@ucsc.edu. This work has been supported by the King Abdullah University of Science and Technology (KAUST), under a competitive research grant (CRG) OSR-2016-CRG5-2958-01. Parts of this paper has been presented at the 2019 IEEE International Symposium on Information Theory (ISIT’2019), Paris, France, July 2019. January 12, 2021 DRAFT ar X iv :2 10 1. 03 65 0v 1 [ cs .I T ] 1 1 Ja n 20 21