On Perfect Privacy

Abstract

The problem of private data disclosure is studied from an information theoretic perspective. Considering a pair of dependent random variables (X, Y ), where X and Y denote the private and useful data, respectively, the following problem is addressed: What is the maximum information that can be revealed about Y , measured by mutual information I(Y ; U ), in which U denotes the revealed data, while disclosing no information about X, captured by the condition of statistical independence, i.e., X ⊥ ⊥ U , and henceforth called perfect privacy)? We analyze the supremization of utility, i.e., I(Y ; U ) under the condition of perfect privacy for two scenarios: output perturbation and full data observation models, which correspond to the cases where a Markov kernel, called privacy-preserving mapping, applies to Y and the pair (X, Y ), respectively. When both X and Y have a finite alphabet, the linear algebraic analysis involved in the solution provides some interesting results, such as upper/lower bounds on the size of the released alphabet and the maximum utility. Afterwards, it is shown that for the jointly Gaussian (X, Y ), perfect privacy is not possible in the output perturbation model in contrast to the full data observation model. Finally, an asymptotic analysis is provided to obtain the rate of released information when a sufficiently small leakage is allowed. In particular, in the context of output perturbation model, it is shown that this rate is always finite when perfect privacy is not feasible, and two lower bounds are provided for it; When perfect privacy is feasible, it is shown that under mild conditions, this rate becomes unbounded.

On the Privacy-Utility Trade-Off With and Without Direct Access to the Private Data

“…In [9], average total variation is used as a privacy measure and a χ 2 -privacy criterion is considered in [5], where an upper bound and a lower bound on the privacy-utility trade-off have been derived. The problem of privacy-utility trade-off considering mutual information both as measures of utility and privacy given the Markov chain X − Y − U is studied in [10]. Under the perfect privacy assumption it is shown that the privacy mechanism design problem can be reduced to a linear program.…”

“…This has been extended in [11] considering the privacy utility trade-off with a rate constraint on the disclosed data. Moreover, in [10], it has been shown that information can only be revealed if the kernel (leakage matrix) between useful data and private data is not invertible. In [12], we generalize [10] by relaxing the perfect privacy assumption allowing some small bounded leakage.…”

“…Moreover, in [10], it has been shown that information can only be revealed if the kernel (leakage matrix) between useful data and private data is not invertible. In [12], we generalize [10] by relaxing the perfect privacy assumption allowing some small bounded leakage. More specifically, we design privacy mechanisms with a perletter privacy criterion considering an invertible kernel where a small leakage is allowed.…”

“…Our results in this work can be divided into three main parts as follows: Part I (Privacy-utility trade-off with non-zero leakage): In the first part of the paper, we consider the problems studied in [33] and [10]. Furthermore, our setup is related to [14] and [16], where in [16] the problem of secrecy by design is studied.…”

Pointwise Maximal Leakage

“…Recently, the problem of private data analysis has attracted much attention from an information-theoretic perspective. A wide variety of privacy measures, for example, mutual information [1][2][3][4][5], measures based on f -divergences [6,7], probability of correctly guessing [8], information privacy [9,10], and log-lift [11,12] are studied that aim to quantify the amount of information leaking about a (private) random variable X by disclosing a related random variable Y . (See [13] for a recent survey on information-theoretic privacy measures.)…”