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arXiv:1305.3905 · IEEE International Symposium on Information Theory (ISIT)

Rate-distortion theory for secrecy systems

Curt M. Schieler, Paul Cuff

Published Jul 1, 2013• 16 citations
arXivHTMLPDF

Abstract

In this work, secrecy in communication systems is measured by the distortion incurred by the worst-case adversary. The transmitter and receiver share secret key, which they use to encrypt communication and cause distortion at an adversary. In our model, we assume that an adversary not only intercepts the communication between the transmitter and receiver, but also may have access to noisy observations of the system. For example, the adversary may have causal access to a signal that is correlated with the source sequence or with the output of the receiver. Our main contribution is the solution of the optimal tradeoff among communication rate, secret key rate, distortion at the adversary, and distortion at the legitimate receiver. We demonstrate that side information at the adversary plays a pivotal role, and provide a number of examples that motivate and give insight into our results.

DOI
10.1109/isit.2013.6620620

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In this work, secrecy in communication systems is measured by the distortion incurred by the worst-case adversary. The transmitter and receiver share secret key, which they use to encrypt communication and cause distortion at an adversary. In our model, we assume that an adversary not only intercepts the communication between the transmitter and receiver, but also may have access to noisy observations of the system. For example, the adversary may have causal access to a signal that is correlated with the source sequence or with the output of the receiver. Our main contribution is the solution of the optimal tradeoff among communication rate, secret key rate, distortion at the adversary, and distortion at the legitimate receiver. We demonstrate that side information at the adversary plays a pivotal role, and provide a number of examples that motivate and give insight into our results.
Secrecy in communication systems is measured herein by the distortion that an adversary incurs. The transmitter and receiver share secret key, which they use to encrypt communication and ensure distortion at an adversary. A model is considered in which an adversary not only intercepts the communication from the transmitter to the receiver, but also potentially has side information. In particular, the adversary may have causal or noncausal access to a signal that is correlated with the source sequence or the receiver's reconstruction sequence. The main contribution is the characterization of the optimal tradeoff among communication rate, secret key rate, distortion at the adversary, and distortion at the legitimate receiver. It is demonstrated that causal side information at the adversary plays a pivotal role in this tradeoff. It is also shown that measures of secrecy based on normalized equivocation are a special case of the framework.
The secrecy of a communication system in which both the legitimate receiver and an eavesdropper are allowed some distortion is investigated. The secrecy metric considered is the exponent of the probability that the eavesdropper estimates the source sequence successfully within an acceptable distortion level. The problem is first studied when the transmitter and the legitimate receiver do not share any key and the transmitter is not subject to a rate constraint, which corresponds to a stylized model of a side channel and reveals connections to source coding with side information. The setting is then generalized to include a shared secret key between the transmitter and the legitimate receiver and a rate constraint on the transmitter, which corresponds to the Shannon cipher system. A single-letter characterization of the highest achievable exponent is provided, and asymptotically optimal strategies for both the primary user and the eavesdropper are demonstrated.
We consider theoretical limits of partial secrecy in a setting where an eavesdropper attempts to causally reconstruct an information sequence with low distortion based on an intercepted transmission and the past of the sequence. The transmitter and receiver have limited secret key at their disposal but not enough to establish perfect secrecy with a one-time pad. From another viewpoint, the eavesdropper is acting as an adversary, competing in a zero-sum repeated game against the sender and receiver of the secrecy system. In this case, the information sequence represents a sequence of actions, and the distortion function captures the payoff of the game. We give an information theoretic region expressing the tradeoff between secret key rate and max-min distortion for the eavesdropper. We also simplify this characterization to a linear program. As an example, we discuss how to optimally use secret key to hide Bernoulli-p bits from an eavesdropper so that they incur maximal Hamming distortion.
Imperfect secrecy in communication systems is investigated. Instead of using equivocation as a measure of secrecy, the distortion that an eavesdropper incurs in producing an estimate of the source sequence is examined. The communication system consists of a source and a broadcast (wiretap) channel, and lossless reproduction of the source sequence at the legitimate receiver is required. A key aspect of this model is that the eavesdropper's actions are allowed to depend on the past behavior of the system. Achievability results are obtained by studying the performance of source and channel coding operations separately, and then linking them together digitally. Although the problem addressed here has been solved when the secrecy resource is shared secret key, it is found that substituting secret key for a wiretap channel brings new insights and challenges: the notion of weak secrecy provides just as much distortion at the eavesdropper as strong secrecy, and revealing public messages freely is detrimental.
A secrecy system in which both the legitimate receiver and an eavesdropper are allowed some distortion is studied. The eavesdropper is allowed to make one guess, and the considered secrecy metric is the exponent of the probability that the guess is within an acceptable distortion level. A subsequent notion of information leakage is suggested. In the absence of any shared key between the transmitter and the legitimate receiver, a single-letter characterization of the highest achievable exponent is provided. Moreover, asymptotically-optimal universal strategies for both the primary user and the eavesdropper are demonstrated, where universality means independence of the source statistics. When a secret key is shared between the transmitter and the legitimate receiver, and a rate constraint is imposed, upper and lower bounds on the optimal exponent are given. Certain sufficient conditions for the bounds to meet are provided, and examples where they apply are discussed.
Confidentiality of information is a key consideration in many networking applications, including e-commerce, online banking, and intelligence operations. How can information be communicated reliably to the legitimate users, while keeping it secret from eavesdroppers? How does such a secrecy constraint on communication affect the limits on information flow in the network? In this chapter, we study these questions under the information theoretic notion of secrecy, which requires each eavesdropper to obtain essentially no information about the messages sent from knowledge of its received sequence, the channel statistics, and the codebooks used. We investigate two approaches to achieve secure communication. The first is to exploit the statistics of the channel from the sender to the legitimate receivers and the eavesdroppers. We introduce the wiretap channel as a 2-receiver broadcast channel with a legitimate receiver and an eavesdropper, and establish its secrecy capacity, which is the highest achievable secret communication rate. The idea is to design the encoder so that the channel from the sender to the receiver becomes effectively stronger than the channel to the eavesdropper; hence the receiver can recover the message but the eavesdropper cannot. This wiretap coding scheme involves multicoding and randomized encoding. If the channel from the sender to the receiver is weaker than that to the eavesdropper, however, secret communication at a positive rate is not possible. This brings us to the second approach to achieve secret communication, which is to use a secret key shared between the sender and the receiver but unknown to the eavesdropper. We show that the rate of such secret key must be at least as high as the rate of the confidential message. This raises the question of how the sender and the receiver can agree on such a long secret key in the first place. After all, if they had a confidential channel with sufficiently high capacity to communicate the key, then why not use it to communicate the message itself! We show that if the sender and the receiver have access to correlated sources (e.g., through a satellite beaming common randomness to them), then they can still agree on a secret key even when the channel has zero secrecy capacity. We first consider the source model for key agreement, where the sender communicates with the receiver over a noiseless public broadcast channel to generate a secret key from their correlated sources.