Statistical matching in the presence of anonymization and obfuscation: Non-asymptotic results in the discrete case

Many popular applications use traces of user data to tune services to their users but come with a significant risk to user privacy. In particular, even if user traces are anonymized, statistical matching of these traces to prior user behavior can be used to identify the user and their behavior. Because of this threat, there has been significant recent work exploring the theoretical foundations of this problem in the limit of a large number of users and/or observations, where the asymptotic nature of the approaches allows for clean analytical results. In this paper, we turn attention to exact performance analysis for a finite number of users and observations. We consider the case where a user is distributed over a discrete set of states according to a probability distribution drawn at random, which we assume is known to the adversary based on his/her analysis of past user behavior. The finite-length traces are then anonymized and obfuscated at a cost in user utility. We analyze the ability of the adversary to correctly identify user data samples as a function of the rate of anonymization and degree of obfuscation, and we arrive at complicated yet readily numerically evaluated expressions. These results allow us to investigate interesting questions left open by the asymptotic nature of previous work.