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IEEE Transactions on Information Theory

Rate-Distortion Theory for Secrecy Systems

Curt M. Schieler, Paul Cuff

Published Oct 31, 2014• 89 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2014.2365175

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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.
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.
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.
The use of secret keys in broadcast channels with receiver side information is studied. The particular scenario is analyzed where a transmitter wants to send two confidential messages to two receivers, while keeping an external eavesdropper ignorant. Each receiver has one of the confidential messages as side information available for decoding. In addition to that, the transmitter shares independent secret keys of arbitrary rates with both receivers. The secret keys can be used in different ways: They can act as one-time pads to encrypt the confidential messages or they can be used as randomization resources for wiretap coding. Both approaches are discussed and an achievable rate region based on superposition coding is established for the one-time pad approach. For the wiretap coding approach, the secrecy capacity for degraded channels is derived. In the optimal coding scheme, the available secret keys are used as the randomization part of the wiretap code to keep the eavesdropper ignorant. In establishing the capacity region, a new upper bound on the sum-rate is derived. This bound shows that in an optimal coding scheme, in the degraded case, the total equivocation-rate of the (opposite) secret-keys at the legitimate receivers must equal the equivocation-rate of the secret-keys at the eavesdropper, when informed about the messages.