- freely available
Information 2016, 7(1), 15; doi:10.3390/info7010015
1.1. Our Model and Main Contributions
- We study lower and upper bounds of . The lower bound, in particular, establishes a multiplicative bound on for any optimal privacy filter. Specifically, we show that for a given and there exists a channel such that and
- We propose an information-theoretic setting in which appears as a natural upper-bound for the achievable rate in the so-called "dependence dilution" coding problem. Specifically, we examine the joint-encoder version of an amplification-masking tradeoff, a setting recently introduced by Courtade  and we show that the dual of upper bounds the masking rate. We also present an estimation-theoretic motivation for the privacy measure . In fact, by imposing , we require that an adversary who observes Z cannot efficiently estimate , for any function f. This is reminiscent of semantic security  in the cryptography community. An encryption mechanism is said to be semantically secure if the adversary’s advantage for correctly guessing any function of the privata data given an observation of the mechanism’s output (i.e., the ciphertext) is required to be negligible. This, in fact, justifies the use of maximal correlation as a measure of privacy. The use of mutual information as privacy measure can also be justified using Fano’s inequality. Note that can be shown to imply that and hence the probability of adversary correctly guessing X is lower-bounded.
- We also study the rate of increase of at and show that this rate can characterize the behavior of for any provided that . This again has connections with the results of . Letting
- Finally, we generalize the rate-privacy function to the continuous case where X and Y are both continuous and show that some of the properties of in the discrete case do not carry over to the continuous case. In particular, we assume that the privacy filter belongs to a family of additive noise channels followed by an M-level uniform scalar quantizer and give asymptotic bounds as for the rate-privacy function.
2. Utility-Privacy Measures: Definitions and Properties
2.1. Mutual Information as Privacy Measure
2.2. Maximal Correlation as Privacy Measure
2.3. Non-Trivial Filters For Perfect Privacy
3. Operational Interpretations of the Rate-Privacy Function
3.1. Dependence Dilution
3.2. MMSE Estimation of Functions of Private Information
4. Observation Channels for Minimal and Maximal
4.1. Conditions for Minimal
- Y is uniformly distributed,
- is constant for all .
4.2. Special Observation Channels
4.2.1. Observation Channels With Symmetric Reverse
- for .
- The initial efficiency of privacy-constrained information extraction is
4.2.2. Erasure Observation Channel
5. Rate-Privacy Function for Continuous Random Variables
5.1. General Properties of the Rate-Privacy Function
- There exist constants , and bounded function such that
- and are both finite,
- the differential entropy of satisfies ,
- , where denotes the largest integer ℓ such that .
5.2. Gaussian Information
Conflicts of Interest
Appendix A. Proof of Lemma 19
Appendix B. Completion of Proof of Theorem 25
Appendix C. Proof of Theorems 28 and 29
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