This paper addresses the discrete-time Poisson wiretap channel (DT-PWC) in an optical wireless communications system based on intensity modulation and direct detection. Subject to nonnegativity and peak intensity as well as bandwidth constraints imposed on the channel input, we study the secrecy-capacity-achieving input distribution of this wiretap channel and prove it to be unique and discrete with a finite number of mass points. 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 a finite support. In general, the number of mass point of the optimal distributions are greater than two. This is in contrast with the continuous-time PWC where the secrecy capacity and the entire boundary of the rate-equivocation region are achieved by binary distributions when the signaling bandwidth is not restricted. Additionally, we shed light on the asymptotic behavior of the secrecy capacity in the low intensity regime and observe that the secrecy capacity scales quadratically with the peak intensity constraint. Finally, Our numerical results indicate that there is a tradeoff between the secrecy capacity and the capacity in the sense that both may not be achieved simultaneously.
This paper addresses the discrete-time Poisson wiretap channel (DT-PWC) in an optical wireless communications system based on intensity modulation and direct detection. Subject to nonnegativity and peak intensity as well as bandwidth constraints imposed on the channel input, we study the secrecy-capacity-achieving input distribution of this wiretap channel and prove it to be unique and discrete with a finite number of mass points. 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 a finite support. In general, the number of mass point of the optimal distributions are greater than two. This is in contrast with the continuous-time PWC where the secrecy capacity and the entire boundary of the rate-equivocation region are achieved by binary distributions when the signaling bandwidth is not restricted. Additionally, we shed light on the asymptotic behavior of the secrecy capacity in the low intensity regime and observe that the secrecy capacity scales quadratically with the peak intensity constraint. Finally, Our numerical results indicate that there is a tradeoff between the secrecy capacity and the capacity in the sense that both may not be achieved simultaneously.
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, peak- and average-intensity as well as bandwidth constraints, we study the secrecy-capacity-achieving 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 is 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 \textit{infinite} number of mass points, but with finitely many mass points in any bounded interval.
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 ε; whereas when the channel gains are different, the secrecy capacity scales, to within a constant, like(αB−αE)EloglogE1, where αBand αEare 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.
This paper studies the optical wiretap channel with input-dependent Gaussian noise, in which the main distortion is caused by an additive Gaussian noise whose variance depends on the current signal strength. Subject to nonnegativity and peak-intensity constraints on the channel input, we first present a practical optical wireless communication scenario for which the considered wiretap channel is stochastically degraded. We then study the secrecy-capacity-achieving input distribution of this wiretap channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. Moreover, we show that the entire rate-equivocation region of this wiretap channel is also obtained by discrete input distributions with a finite support. Similar to the case of the Gaussian wiretap channel under a peak-power constraint, here too, we observe that under nonnegativity and peak-intensity constraints, there is a tradeoff between the secrecy capacity and the capacity in the sense that both may not be achieved simultaneously. Furthermore, we prove the optimality of discrete input distributions in the presence of an additional average intensity constraint. Finally, we shed light on the asymptotic behavior of the secrecy capacity in the low- and high-intensity regimes. In the low-intensity regime, the secrecy capacity scales quadratically with the peak-intensity constraint. On the other hand, in the high-intensity regime, the secrecy capacity does not scale with the constraint.
This paper studies the degraded optical wiretap channel with an input-dependent Gaussian noise when the channel input is only constrained by nonnegativity and average-intensity constraints. We consider the rate-equivocation region of this wiretap channel and through solving a convex optimization problem, we establish that discrete input distributions with an infinite number of mass points exhaust the entire rate-equivocation region of the degraded OWC-IDGN with non-negativity and average-intensity constraints. This result implies that when nonnegativity and average-intensity constraints are imposed on the channel input: 1) the secrecy-capacity-achieving input distribution of the degraded OWC-IDGN is discrete with an unbounded support, i.e., the support set of the optimal distribution is countably infinite; 2) the channel capacity (the case with no secrecy constraints) is also achieved by a discrete distribution with an unbounded support set.
We consider the Gaussian wiretap channel with amplitude and variance constraints on the channel input. We first show that the entire rate-equivocation region of the Gaussian wiretap channel with an amplitude constraint is obtained by discrete input distributions with finite support. We prove this result by considering the existing single-letter description of the rate-equivocation region, and showing that discrete distributions with finite support exhaust this region. Our result highlights an important difference between the peak power (amplitude) constrained and the average power (variance) constrained cases. Although, in the average power constrained case, both the secrecy capacity and the capacity can be achieved simultaneously, our results show that in the peak power constrained case, in general, there is a tradeoff between the secrecy capacity and the capacity, in the sense that, both may not be achieved simultaneously. We also show that under sufficiently small amplitude constraints the possible tradeoff between the secrecy capacity and the capacity does not exist and they are both achieved by the symmetric binary distribution. Finally, we prove the optimality of discrete input distributions in the presence of an additional variance constraint.
We consider the Gaussian wiretap channel with an amplitude constraint, i.e., a peak power constraint, on the channel input. We show that the entire rate-equivocation region of the Gaussian wiretap channel with an amplitude constraint is obtained by discrete input distributions with finite support. We prove this result by considering the existing single-letter description of the rate-equivocation region, and showing that discrete distributions with finite support exhaust this region. Our result highlights an important difference between the peak power constraint and the average power constraint cases: Although, in the average power constraint case, both the secrecy capacity and the capacity can be achieved simultaneously, our results show that in the peak power constraint case, in general, there is a tradeoff between the secrecy capacity and the capacity, in the sense that, both may not be achieved simultaneously.