Worst- and Average-Case Privacy Breaches in Randomization Mechanisms

In a variety of contexts, randomization is regarded as an effective technique to conceal sensitive in-formation. We model randomization mechanisms as information-theoretic channels. Our starting point isa semantic notion of security that expresses absence of any privacy breach above a given level of serious-ness , 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 ob-servations. In each case, we characterize the security level achievable by a mechanism in a simple fashionthat only depends on the channel matrix, and specifically on certain measures of “distance” between itsrows, like norm-1 distance and Chernoff Information. We next clarify the relation between our worst-casesecurity notion and differential privacy (dp): we show that, while the former is in general stronger, the twocoincide if one confines to background information that can be factorised into the product of independentpriors over individuals. We finally turn our attention to expected utility, in the sense of Ghosh et al., in thecase of repeated independent observations. We characterize the exponential growth rate of any reasonableutility function. In the particular case the mechanism provides -dp, we study the relation of the utilityrate with : we offer either exact expressions or upper-bounds for utility rate that apply to practicallyinteresting cases, such as the (truncated) geometric mechanism.