A scheme that publishes aggregate information about sensitive data must resolve the trade-off between utility to information consumers and privacy of the database participants. Differential privacy is a well-established definition of privacy--this is a universal guarantee against all attackers, whatever their side-information or intent. In this paper, we present a universal treatment of utility based on the standard minimax rule from decision theory (in contrast to the utility model in, which is Bayesian). In our model, information consumers are minimax (risk-averse) agents, each possessing some side-information about the query, and each endowed with a loss-function which models their tolerance to inaccuracies. Further, information consumers are rational in the sense that they actively combine information from the mechanism with their side-information in a way that minimizes their loss. Under this assumption of rational behavior, we show that for every fixed count query, a certain geometric mechanism is universally optimal for all minimax information consumers. Additionally, our solution makes it possible to release query results at multiple levels of privacy in a collusion-resistant manner.
Random perturbation is a popular method of computing anonymized data for privacy preserving data mining. It is simple to apply, ensures strong privacy protection, and permits effective mining of a large variety of data patterns. However, all the existing studies with good privacy guarantees focus on perturbation at a single privacy level. Namely, a fixed degree of privacy protection is imposed on all anonymized data released by the data holder. This drawback seriously limits the applicability of random perturbation in scenarios where the holder has numerous recipients to which different privacy levels apply. Motivated by this, we study the problem of multi-level perturbation, whose objective is to release multiple versions of a dataset anonymized at different privacy levels. The challenge is that various recipients may collude by sharing their data to infer privacy beyond their permitted levels. Our solution overcomes this obstacle, and achieves two crucial properties. First, collusion is useless, meaning that the colluding recipients cannot learn anything more than what the most trustable recipient (among the colluding recipients) already knows alone. Second, the data each recipient receives can be regarded (and hence, analyzed in the same way) as the output of conventional uniform perturbation. Besides its solid theoretical foundation, the proposed technique is both space economical and computationally efficient. It requires O (n+m) expected space, and produces a new anonymized version in O (n + log m) expected time, where n is the cardinality of the original dataset, and m the number of versions released previously. Both bounds are optimal under the realistic assumption that n » m.