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arXiv:2501.11636

Characterization of the Arithmetic Complexity of the Secrecy Capacity of Fast-Fading Gaussian Channels

Holger Boche, Andrea Grigorescu, Rafael F. Schaefer, H. Vincent Poor

Published Jan 20, 2025• 0 citations
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Abstract

This paper studies the computability of the secrecy capacity of fast-fading wiretap channels from an algorithmic perspective, examining whether it can be computed algorithmically or not. To address this question, the concept of Turing machines is used, which establishes fundamental performance limits of digital computers. It is shown that certain computable continuous fading probability distribution functions yield secrecy capacities that are non-computable numbers. Additionally, we assess the secrecy capacity's classification within the arithmetical hierarchy, revealing the absence of computable achievability and converse bounds.

DOI
10.48550/arxiv.2501.11636

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This paper studies the computability of the secrecy capacity of fast-fading wiretap channels from an algorithmic perspective, examining whether it can be computed algorithmically or not. To address this question, the concept of Turing machines is used, which establishes fundamental performance limits of digital computers. It is shown that certain computable continuous fading probability distribution functions yield secrecy capacities that are non-computable numbers. Additionally, we assess the secrecy capacity's classification within the arithmetical hierarchy, revealing the absence of computable achievability and converse bounds.
This paper studies the computability of the secrecy capacity of fast-fading wiretap channels from an algorithmic perspective, examining whether it can be computed algorithmically. To address this question, the concept of Turing machines is used, providing the fundamental performance limits of digital computers. It is shown that certain computable continuous fading probability distribution functions yield secrecy capacities that are non-computable numbers. Additionally, we assess the secrecy capacity's classification within the arithmetic hierarchy, revealing absence of computable achievability and converse bounds.
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.
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.
An alternative or supplementary approach named as physical layer security has been proposed to afford an extra security layer on top of the conventional cryptography technique. In this paper, an overview of secrecy performance investigations over the classic Alice-Bob-Eve wiretap fading channels is conducted. On the basis of the classic wiretap channel model, we have comprehensively listed and thereafter compared the existing works on physical layer secrecy analysis considering the small-scale, large-scale, composite, and cascaded fading channel models. Exact secrecy metrics expressions, including secrecy outage probability (SOP), the probability of non-zero secrecy capacity (PNZ), average secrecy capacity (ASC), and secrecy bounds, including the lower bound of SOP and ergodic secrecy capacity, are presented. In order to encompass the aforementioned four kinds of fading channel models with a more generic and flexible distribution, the mixture gamma (MG), mixture of Gaussian (MoG), and Fox's HHH-function distributions are three useful candidates to largely include the above-mentioned four kinds of fading channel models. It is shown that they are flexible and general when assisting the secrecy analysis to obtain closed-form expressions. Their advantages and limitations are also highlighted. Conclusively, these three approaches are proven to provide a unified secrecy analysis framework and can cover all types of independent wiretap fading channel models. Apart from those, revisiting the existing secrecy enhancement techniques based on our system configuration, the on-off transmission scheme, jamming approach (including artificial noise (AN) & artificial fast fading (AFF)), antenna selection, and security region are presented.
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.