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IEEE Conference on Communications and Network Security

On the fading Gaussian wiretap channel with statistical channel state information at transmitter

Pin-Hsun Lin, Eduard Axel Jorswieck

Published Oct 1, 2014• 6 citations
Publisher

Abstract

In this paper we partially characterize the relation between the ergodic secrecy capacity of fast fading wiretap channels and the stochastic orders including the usual stochastic orders, the convex orders, and the increasing convex orders, of the main and eavesdropper's channels. We assume that there is only statistical channel state information of both channels at transmitter and both fading channels can be with arbitrary distributions. We derive the sufficient conditions for the existence of the positive ergodic secrecy capacity under these three orders. For more general orders we also derive capacity lower and upper bounds. To achieve it, we apply the concept of channel enhancement to the continuous fading wiretap channel and derive an upper bound. Motivated by the upper bound, we also derive the lower bound of the ergodic secrecy capacity by the layered signaling. Finally numerical results show that under Nakagami-m channels, the proposed achievable scheme outperforms Gaussian signaling in several cases. More specifically, the proposed scheme may result in positive secrecy rate while Gaussian signaling only lead to zero. Thus when we apply the proposed scheme to the wireless network, the connectivity can be significantly improved.

DOI
10.1109/cns.2014.6997476

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In this paper we partially characterize the relation between the ergodic secrecy capacity of fast fading wiretap channels and the stochastic orders including the usual stochastic orders, the convex orders, and the increasing convex orders, of the main and eavesdropper's channels. We assume that there is only statistical channel state information of both channels at transmitter and both fading channels can be with arbitrary distributions. We derive the sufficient conditions for the existence of the positive ergodic secrecy capacity under these three orders. For more general orders we also derive capacity lower and upper bounds. To achieve it, we apply the concept of channel enhancement to the continuous fading wiretap channel and derive an upper bound. Motivated by the upper bound, we also derive the lower bound of the ergodic secrecy capacity by the layered signaling. Finally numerical results show that under Nakagami-m channels, the proposed achievable scheme outperforms Gaussian signaling in several cases. More specifically, the proposed scheme may result in positive secrecy rate while Gaussian signaling only lead to zero. Thus when we apply the proposed scheme to the wireless network, the connectivity can be significantly improved.
In this paper, we investigate the ergodic secrecy capacity of the fast fading Gaussian wiretap channel when only the statistics of the channel state information are known at the transmitter. We derive conditions for the existence of degradedness and a positive ergodic secrecy capacity under the usual stochastic order, the convex order, and the increasing convex order between the legitimate and eavesdropper channels. For more general orders, we prove the secrecy capacity of layered erasure wiretap channels and propose a layered signaling for the achievable scheme, and we derive an upper bound on the capacity for fast fading Gaussian wiretap channels. Finally, the numerical results show that under Nakagami-m fast fading channels, the proposed layered signaling outperforms the Gaussian codebook in several cases. In particular, in certain cases, the Gaussian codebook can achieve only a zero secrecy rate, whereas the proposed scheme achieves positive secrecy rates. Therefore, the connectivity of wireless networks can be significantly improved by the proposed scheme.
The ergodic secrecy capacity of the wiretap channel is known when the main channel (between the transmitter and the legitimate receiver) state information (CSI) is perfect at the transmitter and the coherence period is sufficiently large to enable random coding arguments in each block. In a fast fading scenario, when the codeword length spans many coherence periods, the secrecy capacity is still not known. In this paper, we present a framework that characterizes this secrecy capacity under imperfect main channel estimation at the transmitter. Inner and outer bounds on the ergodic secrecy capacity are derived for a class of independent identically distributed (i.i.d.) fading channels. The achievable rate is a simple on-off scheme using a Gaussian input. The upper bound is obtained using an appropriate correlation scheme of the main and the eavesdropper channels. The upper and the lower bounds coincide with recently derived ones in the perfect main CSI extreme. Furthermore, the lower bound matches the upper bound in no main CSI extreme, where the secrecy capacity is equal to zero. Numerical results are provided for independent identically distributed (i.i.d.) Rayleigh fading channels.
We study the secrecy capacity of fast fading channels under imperfect main channel (between the transmitter and the legitimate receiver) estimation at the transmitter. Lower and upper bounds on the ergodic secrecy capacity are derived for a class of independent identically distributed (i.i.d.) fading channels. The achievable rate follows from a standard wiretap code in which a simple on-off power control is employed along with a Gaussian input. The upper bound is obtained using an appropriate correlation scheme of the main and eavesdropper channels and is the best known upper bound so far. The upper and lower bounds coincide with recently derived ones in case of perfect main CSI. Furthermore, the upper bound is tight in case of no main CSI, where the secrecy capacity is equal to zero. Asymptotic analysis at high and low signal-to-noise ratio (SNR) is also given. At high SNR, we show that the capacity is bounded by providing upper and lower bounds that depend on the channel estimation error. At low SNR, however, we prove that the secrecy capacity is asymptotically equal to the capacity of the main channel as if there were no secrecy constraint. Numerical results are provided for i.i.d. Rayleigh fading channels.
We consider the fast-fading Gaussian wiretap channel with single antenna nodes and without channel state information at the transmitter (CSIT), where the fading processes of the two links are arbitrary and independent of each other. We derive an upper bound to the secrecy capacity for this channel and an achievable rate as well. Subsequently, we identify a class of channel statistics for which the outer bound and the achievable rates are identical thereby characterizes the exact secrecy capacity of the channel. The class of channels with such channel statistics are called stochastically degraded in this paper. Many practical wireless settings including the Rayleigh fading environment fall under this stochastically degraded class. We illustrate our results by computing explicit expression for the secrecy capacity for Rayleigh distributed wiretap channels.
We investigate the relation between different stochastic orders and the degradedness of a fast fading wiretap channel with statistical channel state information at the transmitter (CSIT). In particular, we derive sufficient conditions to identify the ergodic secrecy capacities for both single and multiple antenna cases even though there is only statistical CSIT. Introduction To design a wiretap code with higher secrecy rate, channel state information at the transmitter of the legitimate and eavesdropper's channels1 should be known to a certain degree. When there is perfect CSIT of both channels, the secrecy capacity of a Gaussian wiretap channel can be achieved by a Gaussian input. However, due to several practical issues such as (1) being a malicious user, Eve is by no means intending to feed back the correct CSI to Alice; (2) limited feedback bandwidth; (3) the delay caused by channel estimation; (4) the speed of channel variation, etc., it is more reasonable to consider cases where perfect CSIT is unavailable. In this chapter we consider the cases when only statistical CSIT of both channels from Alice to Bob and Alice to Eve, respectively, are available. One possible way to get such information from Eve virtually is as follows. For some bounded space like indoor parking lots or malls, it is possible to collect the channel statistics offline of most of the positions within that space, where the channel variation may be due to the movement of people or cars, etc. When Alice wants to transmit, she can use the known statistics measured offline of the channel between her and the unknown user closest to her as Eve's channel information to design her wiretap code for the worst-case scenario. Although the secrecy capacity formula for discrete memoryless non-degraded wiretap channels was proved in [1], the optimal selection of the auxiliary random variable and channel prefixing are still unknown for additive white Gaussian noise (AWGN) channels with partial CSIT in general. Only a few capacity results under the considered scenario are known, and are summarized as follows.
The characterization of secrecy capacities of Gaussian wiretap channels depends on the knowledge of channel state information at the transmitter (CSIT). When there is only statistical CSIT, the secrecy capacity is unknown in general. In this paper we investigate the relation between the usual multivariate stochastic order and the degradedness among the legitimate receiver's and eavesdropper's channels under fast fading with only statistical CSIT. The considered transmitter has multiple antennas and the legitimate receiver and the eavesdropper both have multiple antennas. Based on the technique of coupling we derive criteria to check the degradedness of the fast fading Gaussian wiretap channel including some commonly used channel models even there is only statistical CSIT. We illustrate one example of a Rician 2 × 2 × 2 channel where the ergodic secrecy capacity can be derived under different K-factors.