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IEEE Transactions on Information Theory

The Degraded Poisson Wiretap Channel

Amine Laourine, Aaron B. Wagner

Published Sep 3, 2012• 19 citations
Publisher

Abstract

Optical and near-optical band communication systems are known to be intrinsically more secure than comparable RF channels, due to their narrow beamwidths and, in some cases, their high atmospheric absorption. The use of coding against wiretapping for such channels is investigated. For the degraded Poisson wiretap channel model, the secrecy capacity is determined exactly. Moreover, a complete characterization of the rate-equivocation region is presented. For achievability, an optimal code is constructed explicitly by using a code designed by Wyner for the Poisson channel. The converse is proved in two ways: the first method leverages the low-SNR nature of the channel and relies only on simple properties of conditional expectation and classical information inequalities. The second method uses a link recently established between minimum mean square error estimation and mutual information over Poisson channels.

DOI
10.1109/tit.2012.2217191

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Optical and near-optical band communication systems are known to be intrinsically more secure than comparable RF channels, due to their narrow beamwidths and, in some cases, their high atmospheric absorption. The use of coding against wiretapping for such channels is investigated. For the degraded Poisson wiretap channel model, the secrecy capacity is determined exactly. Moreover, a complete characterization of the rate-equivocation region is presented. For achievability, an optimal code is constructed explicitly by using a code designed by Wyner for the Poisson channel. The converse is proved in two ways: the first method leverages the low-SNR nature of the channel and relies only on simple properties of conditional expectation and classical information inequalities. The second method uses a link recently established between minimum mean square error estimation and mutual information over Poisson channels.
Providing security guarantees for wireless communication is critically important for today's applications. While previous work in this area has concentrated on radio frequency (RF) channels, providing security guarantees for RF channels is inherently difficult because they are prone to rapid variations due small scale fading. Wireless optical communication, on the other hand, is inherently more secure than RF communication due to the intrinsic aspects of the signal propagation in the optical and near-optical frequency range. In this paper, secure communication over wireless optical links is examined by studying the secrecy capacity of a direct detection system. For the degraded Poisson wiretap channel, a closed-form expression of the secrecy capacity is given. A complete characterization of the general rate-equivocation region is also presented. For achievability, an optimal code is explicitly constructed by using the structured code designed by Wyner for the Poisson channel. The converse is proved in two different ways: the first method relies only on simple properties of the conditional expectation and basic information theoretical inequalities, whereas the second method hinges on the recent link established between minimum mean square estimation and mutual information in Poisson channels.
In this work, we focus on the classical optical channel having Poissonian statistical behavior and propose a novel secrecy coding-based physical layer protocol. Our protocol is different but complementary to both (computationally secure) quantum immune cryptographic protocols and (information theoretically secure) quantum cryptographic protocols. Specifically, our (information theoretical) secrecy coding protocol secures classical digital information bits at photonic level exploiting the random nature of the Poisson channel. It is known that secrecy coding techniques for the Poisson channel based on the classical one-way wiretap channel (introduced by Wyner in 1975) ensure secret communication only if the mutual information to the eavesdropper is smaller than that to the legitimate receiver. In order to overcome such a strong limitation, we introduce a two-way protocol that always ensures secret communication independently of the conditions of legitimate and eavesdropper channels. We prove this claim showing rigorous comparative derivation and analysis of the information theoretical secrecy capacity of the classical one-way and of the proposed two-way protocols. We also show numerical calculations that prove drastic gains and strong practical potential of our proposed two-way protocol to secure information transmission over optical channels.
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.
Providing security guarantees for the information “traveling” through the air is of critical importance for today's applications. Previous research concentrated on studying this problem when the medium used for communication is radio frequency (RF) channels. However, providing security guarantees on RF channels is a complicated task as RF channels' conditions are prone to rapid variations due small scale fading and at a given moment it is difficult to predict which of the main or wiretapper's channel has a better quality. Wireless optical communication is inherently more secure than regular RF communication. Despite this fact, optical communication is not completely immune against wiretapping and signal interception. The purpose of this paper is to study the secrecy capacity of a direct detection optical communication system. For the degraded Poisson wiretap channel, we give a complete characterization of the secrecy capacity in terms of the system parameters.
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)Elog⁡log⁡1E(\alpha_{B}-\alpha_{E})\mathcal{E}\log\log\frac{1}{\mathcal{E}}(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.
Secrecy issues of free-space optical links realizing information theoretically secure communications and high transmission rates are discussed. We numerically study secrecy communication rates of optical wiretap channel based on on-off keying (OOK) modulation under typical conditions met in satellite-ground links. It is shown that, under reasonable degraded conditions on a wiretapper, information theoretically secure communications should be possible in a much wider distance range than a range limit of quantum key distribution, enabling secure optical links between geostationary Earth orbit satellites and ground stations with currently available technologies. We also provide the upper bounds on the decoding error probability and the leaked information to estimate a necessary code length for given required levels of performances. This result ensures that a reasonable length of wiretap channel code for our proposed scheme must exist.