Worst- and Average-Case Privacy Breaches in Randomization Mechanisms

Abstract

Abstract. In a variety of contexts, randomization is regarded as an effective technique to conceal sensitive information. We model randomization mechanisms as information-theoretic channels. Our starting point is a semantic notion of security that expresses absence of any privacy breach above a given level of seriousness , irrespective of any background information, represented as a prior probability on the secret inputs. We first examine this notion according to two dimensions: worst vs. average case, single vs. repeated observations. In each case, we characterize the security level achievable by a mechanism in a simple fashion that only depends on the channel matrix, and specifically on certain measures of "distance" between its rows, like norm-1 distance and Chernoff Information. We next clarify the relation between our worst-case security notion and differential privacy (dp): we show that, while the former is in general stronger, the two coincide if one confines to background information that can be factorised into the product of independent priors over individuals. We finally turn our attention to expected utility, in the sense of Ghosh et al., in the case of repeated independent observations. We characterize the exponential growth rate of any reasonable utility function. In the particular case the mechanism provides -dp, we study the relation of the utility rate with : we offer either exact expressions or upper-bounds for utility rate that apply to practically interesting cases, such as the (truncated) geometric mechanism.

Searching secrets rationally

“…[12,11,3,30,4,5,21] and references therein. In the last few years, various extensions of the basic model have been considered: to mention but a few, [6] considers the case of one-try (min-entropy) attacks under repeated observations, while the worst-vs. average-case dimension is examined in [9]. Especially relevant to the present paper is the issue of partial secret disclosure [7] and, more generally, of leakage characterization in terms of adversary's (generic) gain, or g-leakage [1].…”

Towards a Systematic Analysis of Privacy Definitions

“…Work that considers the privacy of multiple pieces of sensitive information against attackers with a wide variety of prior beliefs or background knowledge (e.g., [23,41,42,8,37,54,7,26,5,29,51]) is of particular relevance to this paper. The methodology we present here is complementary to such work and seeks to answer the inverse question, that of identifying the specific attackers and the specific pieces of information that are hidden from them by a privacy definition.…”

“…but, if one accepts the Axiom of Choice, linear constraints are much more complicated and are generally defined via finitely additive measures [59]. 8 On the other hand, in constructive mathematics, 9 more complicated linear constraints cannot be proven to exist ([59], Sections 14.77, 23.10, and 27.45; and [44]). Therefore we only consider the types of linear constraints shown in Equation 1.…”

“…For this reason, there is also interest in more systematic analyses with generalizable results. This include fundamental limits on accuracy of data generated by sanitizing algorithms [24,18,25,36,22,13] and analyses of privacy with respect to a wide variety of Bayesian attackers as well as non-Bayesian attackers with different kinds of background knowledge [23,41,42,8,37,54,7,26,5,29,51].…”

“…Of particular interest are Bayesian approaches such as [37,54,26,8,42,51] that consider the safety of different types of sensitive information with respect to a wide variety of attackers. In this paper, we consider the inverse of this problem: given a privacy definition, who are the attackers and what types of information are being protected from them?…”