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.
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.
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.
This chapter assumes that a limited amount of secret key and reliable communication are available for use in encoding and transmitting a stochastic signal. The chapter starts at the beginning, with a more general proof of Shannon's “key must be as large as the message” result that holds even for stochastic encoders. Three directions are explored. First, for lossless compression and perfect secrecy, variable length codes or key regeneration allow for a tradeoff between efficiency of compression and efficiency of secret key usage. Second, the relaxation to imperfect secrecy is studied. This is accomplished by measuring the level of secrecy either by applying a distortion metric to the eavesdropper's best possible reconstruction or by considering the eavesdropper's ability to guess the source realization. Finally, an additional relaxation is made to allow the compression of the source to be lossy. The chapter concludes by showing how the oft-used equivocation metric for information theoretic secrecy is a particular special case of the rate–distortion theory contained herein. Introduction Source coding is the process of encoding information signals for transmission through digital channels. Since efficiency is a primary concern in this process, the phrase “source coding” is often used interchangeably with “data compression.” The relationship between source coding and channel coding is that channel coding produces digital resources from natural resources (e.g., a physical medium), and source coding consumes digital resources to accomplish a task involving information, often simply moving it from one point to another. The channel coding side of information theoretic security is referred to as physical layer security. This usually involves designing a communication system for a physical wiretap channel, introduced by Wyner in [1], which produces a provably secure digital communication link. Another important challenge in physical layer security is the production of a secret key based on common observations, such as channel fading parameters or prepared quantum states. Although a key agreement protocol does not necessarily involve a channel, it is consistent with the spirit of channel coding in that the objective is the production of digital resources.
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.
Recently, a secrecy measure based on list-reconstruction has been proposed, in which a wiretapper is allowed to produce a list of 2mRLreconstruction sequences and the secrecy is measured by the minimum distortion over the entire list. In this paper, we show that this list secrecy problem is equivalent to the one with secrecy measured by a new quantity lossy equivocation, which is proved to be the minimum optimistic one-achievable source coding rate (the minimum coding rate needed to reconstruct the source within target distortion with positive probability for infinitely many blocklengths) of the source with the wiretapped signal as two-sided information, and also can be seen as a lossy extension of conventional equivocation. Upon this (or list) secrecy measure, we study source-channel secrecy problem in the discrete memoryless Shannon cipher system with noisy wiretap channel. Two inner bounds and an outer bound on the achievable region of secret key rate, list rate, wiretapper distortion, and distortion of legitimate user are given. The inner bounds are derived by using uncoded scheme and (operationally) separate scheme, respectively. Thanks to the equivalence between lossy-equivocation secrecy and list secrecy, information spectrum method is leveraged to prove the outer bound. As special cases, the admissible region for the case of degraded wiretap channel or lossless communication for legitimate user has been characterized completely. For both these two cases, separate scheme is proved to be optimal. Interestingly, however, separation indeed suffers performance loss for other certain cases. Besides, we also extend our results to characterize the achievable region for Gaussian communication case. As a side product, optimistic lossy source coding has also been addressed.
Recently, a secrecy measure based on list-reconstruction has been proposed [2], in which a wiretapper is allowed to produce a list of 2mRL reconstruction sequences and the secrecy is measured by the minimum distortion over the entire list. In this paper, we show that this list secrecy problem is equivalent to the one with secrecy measured by a new quantity \emph{lossy-equivocation}, which is proven to be the minimum optimistic 1-achievable source coding rate (the minimum coding rate needed to reconstruct the source within target distortion with positive probability for \emph{infinitely many blocklengths}) of the source with the wiretapped signal as two-sided information, and also can be seen as a lossy extension of conventional equivocation. Upon this (or list) secrecy measure, we study source-channel secrecy problem in the discrete memoryless Shannon cipher system with \emph{noisy} wiretap channel. Two inner bounds and an outer bound on the achievable region of secret key rate, list rate, wiretapper distortion, and distortion of legitimate user are given. The inner bounds are derived by using uncoded scheme and (operationally) separate scheme, respectively. Thanks to the equivalence between lossy-equivocation secrecy and list secrecy, information spectrum method is leveraged to prove the outer bound. As special cases, the admissible region for the case of degraded wiretap channel or lossless communication for legitimate user has been characterized completely. For both these two cases, separate scheme is proven to be optimal. Interestingly, however, separation indeed suffers performance loss for other certain cases. Besides, we also extend our results to characterize the achievable region for Gaussian communication case. As a side product optimistic lossy source coding has also been addressed.
Recently, a secrecy measure based on list-reconstruction has been proposed [1], in which a wiretapper is allowed to produce a list of 2mRLreconstruction sequences and secrecy is measured by the minimum distortion over the entire list. In this paper, we show that this list secrecy problem is equivalent to one with secrecy measured by a new quantity lossy-equivocation, which is proven to be the minimum optimistic 1-achievable source coding rate of the source with the wirtapped signal as two-sided information, and also can be seen as an extension of conventional equivocation to lossy case. Upon this (or list) secrecy measure, we study source-channel secrecy problem in the discrete memoryless Shannon cipher system with noisy wiretap channel. Two inner bounds and an outer bound on the achievable region of secret key rate, list rate, wiretapper distortion, and distortion of legitimate user are given. The inner bounds are derived by using an uncoded scheme and a (operationly) separate scheme, respectively. Thanks to the equivalence between lossy-equivocation secrecy and list secrecy, the information spectrum method is leveraged to prove the outer bound. As special cases, the admissible region for the cases of degraded wiretap channel or lossless communication for legitimate user has been characterized completely. For both these two cases, separate scheme is proven to be optimal. Interestingly, however, separation indeed suffers performance loss for other certain cases. Besides, we also extend our results to characterize the achievable region for Gaussian communication case. As a side product optimistic lossy source coding has also been addressed.