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arXiv:2301.12526 · IEEE Transactions on Information Forensics and Security

Outer Bounds on the CEO Problem With Privacy Constraints

Vamoua Yachongka, Hideki Yagi, Hideki Ochiai

Published Dec 25, 2024• 3 citations
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Abstract

We investigate the rate-distortion-leakage region of the Chief Executive Officer (CEO) problem, considering the presence of a passive eavesdropper and privacy constraints. We start by examining the region where a general distortion measure quantifies the distortion. While the inner bound of the region is derived from previous work, this paper newly develops an outer bound. To derive the outer bound, we introduce a new lemma tailored for analyzing privacy constraints. Next, as a specific instance of the general distortion measure, we demonstrate that the tight bound for discrete and Gaussian sources is obtained when the eavesdropper has no side information, and the distortion is quantified by the log-loss distortion measure. We further investigate the rate-distortion-leakage region for a scenario where the eavesdropper has side information, and the distortion is quantified by the log-loss distortion measure and provide an outer bound for this case. The derived outer bound differs from the inner bound by only a minor quantity that appears in the constraints associated with the privacy-leakage rates, and these bounds match when the distortion is large.

DOI
10.1109/tifs.2024.3522775

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We investigate the rate-distortion-leakage region of the Chief Executive Officer (CEO) problem, considering the presence of a passive eavesdropper and privacy constraints. We start by examining the region where a general distortion measure quantifies the distortion. While the inner bound of the region is derived from previous work, this paper newly develops an outer bound. To derive the outer bound, we introduce a new lemma tailored for analyzing privacy constraints. Next, as a specific instance of the general distortion measure, we demonstrate that the tight bound for discrete and Gaussian sources is obtained when the eavesdropper has no side information, and the distortion is quantified by the log-loss distortion measure. We further investigate the rate-distortion-leakage region for a scenario where the eavesdropper has side information, and the distortion is quantified by the log-loss distortion measure and provide an outer bound for this case. The derived outer bound differs from the inner bound by only a minor quantity that appears in the constraints associated with the privacy-leakage rates, and these bounds match when the distortion is large.
We study a lossy source coding problem with secrecy constraints in which a remote information source should be transmitted to a single destination via multiple agents in the presence of a passive eavesdropper. The agents observe noisy versions of the source and independently encode and transmit their observations to the destination via noiseless rate-limited links. The destination should estimate the remote source based on the information received from the agents within a certain mean distortion threshold. The eavesdropper, with access to side information correlated to the source, is able to listen in on one of the links from the agents to the destination in order to obtain as much information as possible about the source. This problem can be viewed as the so-called CEO problem with additional secrecy constraints. We establish inner and outer bounds on the rate-distortion-equivocation region of this problem. We also obtain the region in special cases where the bounds are tight. Furthermore, we study the quadratic Gaussian case and provide the optimal rate-distortion-equivocation region when the eavesdropper has no side information and an achievable region for a more general setup with side information at the eavesdropper.
A lossy source coding problem with secrecy constraints is considered where a remote information source should be transmitted to a single destination via multiple agents in the presence of an eavesdropper. The agents observe noisy versions of the source and independently encode and transmit their observations to the destination via noiseless rate-limited links. Unbeknownst to the agents, an eavesdropper intercepts one of the links from the agents to the destination to learn as much as possible about the source. The destination should estimate the remote source subject to a mean distortion threshold. This problem can be viewed as the CEO problem with addition of secrecy constraints. We establish inner and outer bounds on the rate-distortion-equivocation region. In addition, we provide the optimal rate-distortion-equivocation region for the quadratic Gaussian case when the eavesdropper has no side information.
In this paper, the problem of function computation with privacy and secrecy constraints is considered. The considered model consists of three legitimate nodes (i.e., two transmitters, Alice and Bob, and a fusion center that acts as the receiver) that observe correlated sources and are connected by noiseless public channels, and an eavesdropper Eve who has full access to the public channels and also has its own source observations. The fusion center would like to compute a function of the distributed sources within a prefixed distortion level under a certain distortion metric. To facilitate the function computation, Alice and Bob will send messages to the fusion center. Different from the existing setups in function computation, we assume that there is a privacy constraint on the sources at Alice and Bob. In particular, Alice and Bob would like to enable the fusion center to compute the function, but at same time, they do not want the fusion center to learn too much information about the source observations. We introduce a quantity to precisely measure the privacy leakage to the fusion center. In addition to this privacy constraint, we also have a secrecy constraint to Eve and use equivocation of sources to measure this quantity. Under this model, we study the tradeoffs among message rates, private information leakage, equivocation, and distortion. We first consider a scenario that has only one transmitter, i.e., the source at Bob is empty, and fully single-letter characterize the corresponding regions. Then, we consider the more general case and provide both outer and inner bounds on the corresponding regions.
We study the secure lossy transmission of a Gaussian vector source to a legitimate user in the presence of an eaves-dropper, where both the legitimate user and the eavesdropper have Gaussian vector side information. The transmitter describes the source to the legitimate user in a way that the legitimate user can reconstruct the source within a certain distortion while the eavesdropper is kept ignorant of the source as much as possible. We obtain an outer bound for the rate, equivocation and distortion region of this secure lossy transmission problem. This outer bound is tight when the transmission rate constraint is removed. In other words, we obtain the maximum equivocation at the eavesdropper when the legitimate user needs to reconstruct the source within a fixed distortion level while there is no constraint on the transmission rate.
We consider the problems of secure lossy source coding with side information in the presence of a passive eavesdropper who has access to the source description. The encoder wishes to compress the source sequence in order to satisfy a distortion criterion at the decoder, while revealing only limited knowledge about the source to the eavesdropper. The side information available to the encoder, the legitimate decoder, or the eavesdropper can be influenced by a cost-constrained action sequence. Three different settings are studied. In the first two settings, we are interested in understanding the influence of the action sequence on the rate-distortion-leakage tradeoff where the action is taken either by the decoder or by the encoder to influence side information at the decoder and eavesdropper. Next, we consider a setting where common action-dependent side information is available securely to both encoder and decoder, and thus can be used for secret key generation. We characterize the optimal rate-distortion-cost-leakage region or the corresponding inner bounds for a discrete memoryless source for above settings. The results are useful in characterizing fundamental limits for example in secure sensor networking and future cyber physical systems.
The problem of secure multiterminal source coding with side information at the eavesdropper is investigated. This scenario consists of a main encoder (referred to as Alice) that wishes to compress a single source but simultaneously satisfying the desired requirements on the distortion level at a legitimate receiver (referred to as Bob) and the equivocation rate—average uncertainty—at an eavesdropper (referred to as Eve). It is further assumed the presence of a (public) rate-limited link between Alice and Bob. In this setting, Eve perfectly observes the information bits sent by Alice to Bob and has also access to a correlated source which can be used as side information. A second encoder (referred to as Charlie) helps Bob in estimating Alice's source by sending a compressed version of its own correlated observation via a (private) rate-limited link, which is only observed by Bob. For instance, the problem at hands can be seen as the unification between the Berger–Tung and the secure source coding setups. Inner and outer bounds on the so-called rate-distortion-equivocation region are derived. The inner region turns to be tight for two cases: 1) uncoded side information at Bob and 2) lossless reconstruction of both sources at Bob—secure distributed lossless compression. Application examples to secure lossy source coding of Gaussian and binary sources in the presence of Gaussian and binary/ternary (respectively) side informations are also considered. Optimal coding schemes are characterized for some cases of interest where the statistical differences between the side information at the decoders and the presence of a nonzero distortion at Bob can be fully exploited to guarantee secrecy.