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arXiv:1806.05589

Inducing information stability and applications thereof to obtaining information theoretic necessary conditions directly from operational requirements

Eric Graves, Tan F. Wong

Published Jun 14, 2018• 0 citations
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Abstract

This work constructs a discrete random variable that, when conditioned upon, ensures information stability of quasi-images. Using this construction, a new methodology is derived to obtain information theoretic necessary conditions directly from operational requirements. In particular, this methodology is used to derive new necessary conditions for keyed authentication over discrete memoryless channels and to establish the capacity region of the wiretap channel, subject to finite leakage and finite error, under two different secrecy metrics. These examples establish the usefulness of the proposed methodology.

DOI
10.48550/arxiv.1806.05589

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This work constructs a discrete random variable that, when conditioned upon, ensures information stability of quasi-images. Using this construction, a new methodology is derived to obtain information theoretic necessary conditions directly from operational requirements. In particular, this methodology is used to derive new necessary conditions for keyed authentication over discrete memoryless channels and to establish the capacity region of the wiretap channel, subject to finite leakage and finite error, under two different secrecy metrics. These examples establish the usefulness of the proposed methodology.
This work presents a new methodology for obtaining information theoretic necessary conditions directly from general operational requirements. This methodology is based on the construction of a discrete random variable that, when conditioned upon, ensures information stability of quasi-images. The induced information stability allows a more direct way to develop information theoretic necessary conditions from operational requirements beyond using Fano's inequality. That is, while Fano's inequality uses the probability of error to establish an upper bound on the entropy of a random variable given its estimator, the proposed new methodology can be applied to arbitrary operational requirements to obtain corresponding conditions on information theoretic quantities. To demonstrate its power, this new methodology is employed, to derive new necessary conditions for keyed authentication over a discrete memoryless channels and to establish the capacity region subject to finite leakage and finite error of the wiretap channel under two different secrecy metrics. These examples establish the usefulness of the proposed methodology.
In many information-theoretic channel coding problems, adding an input cost constraint to the operational setup amounts to restricting the optimization domain in the capacity formula. This paper shows that, in contrast to common belief, such a simple modification does not hold for the cost-constrained (CC) wiretap channel (WTC). The secrecy-capacity of the discrete memoryless (DM) WTC without cost constraints is described by a single auxiliary random variable. For the CC DM-WTC, however, we show that two auxiliaries are necessary to achieve capacity. Specifically, we first derive the secrecy-capacity formula, proving the direct part via superposition coding. Then, we provide an example of a CC DM-WTC whose secrecy-capacity cannot be achieved using a single auxiliary. This establishes the fundamental role of superposition coding over CC WTCs.
Secret-key agreement based on biometric or physical identifiers is a promising security protocol for authenticating users or devices with small chips and has been extensively studied recently. Kittichokechai and Caire (2015) investigated the optimal trade-off in a secret-key agreement model with physical identifiers, where the structure of the authentication channels is similar to the wiretap channels, from information theoretic approaches. Later, the model was extended by Günlü et al. (2018) introducing noise in the enrollment phase and cost-constrained actions at the decoder. The results of these studies show that two auxiliary random variables are involved in the expressions of the optimal rate regions of secret-key, storage, and privacy-leakage rates. However, with these two auxiliary random variables, the complexity of computing the rate region may be prohibitively high. Due to this problem, we are interested in exploring classes of authentication channels that need only one auxiliary random variable in the capacity region expression for discrete source settings. The result shows for the class of degraded and less noisy authentication channels, a single auxiliary random variable is sufficient to express the capacity region of the model. As an example, we also derive the capacity region of secret-key, storage, and privacy-leakage rates for binary sources. Furthermore, the capacity region for scalar Gaussian sources is derived under Gaussian authentication channels.
In this paper, a novel message authentication model using the same key over wiretap channel is proposed to achieveinformation-theoretic security. Specifically, in the proposed model, there is a discrete memoryless channelW1:X→Ybetween transmitter Alice and receiver Bob, while an attacker Oscar is connected with Alice via discrete memoryless channelW2:X→Z. Alice encodes messageMto codeword (S,Xn), using an encoding function with secret keyK. Then,Sis sent to Bob over a one-way noiseless channel (fully controlled by Oscar), andXnis sent over the wiretap channel, sayX→(Y,Z). Building on this model, a new message authentication scheme is proposed. The scheme incorporates a secure channel coding, which uses random coding techniques to detect man-in-the-middle (MITM) attacks. The authentication channel capacity is studied in a specific channel model whenW2is not less noisy thanW1. We theoretically demonstrate that the authentication channel capacity is much larger than the secrecy capacity, since Bob does not need to recover information transmitted over the noisy channel.
We investigate an approach to physical-layer security based on the premise that the coding mechanism for secrecy over noisy channels is fundamentally tied to the notion of resolvability. Instead of considering capacity-based constructions, which associate to each message a sub-code whose rate approaches the capacity of the eavesdropper’s channel, we consider resolvability-based constructions, which associate to each message a sub-code whose rate is beyond the resolvability of the eavesdropper’s channel. We provide evidence that resolvability is a more powerful and perhaps more fundamental coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels. Specifically, we show that, at least for binary symmetric wiretap channels, random capacity-based constructions fail to approach the strong secrecy capacity while resolvability-based constructions achieve it. We then obtain the secrecy-capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics, which generalizes existing results. Finally, we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information, compound and mixed channels, as well as the secret-key capacity of source models for secret-key agreement. By tying secrecy to resolvability, we obtain achievable rates for stronger secrecy metrics and with simpler proofs than previously derived. Index Terms information-theoretic security, wiretap channel, secret-key agreement, information-spectrum, resolv-ability, wireless channels.