Forgot your password: Correlation dilution

We consider the problem of diluting common randomness from correlated observations by separated agents. This problem creates a new framework to study statistical privacy, in which a legitimate party, Alice, has access to a random variable X, whereas an attacker, Bob, has access to a random variable Y dependent on X drawn from a joint distribution p_X,Y. Alice´s goal is to produce a non-trivial function of her available information that is uncorrelated with (has small correlation with) any function that Bob can produce based on his available information. This problem naturally admits a minimax formulation where Alice plays first and Bob follows her. We define dilution coefficient as the smallest value of correlation achieved by the best strategy available to Alice, and characterize it in terms of the minimum principal inertia components of the joint probability distribution p_X,Y. We then explicitly find the optimal function that Alice must choose to achieve this limit. We also establish a connection between differential privacy and dilution coefficient and show that if Y is ε-differentially private from X, then dilution coefficient can be upper bounded in terms of ε. Finally, we extend to the setting where Alice and Bob have access to i.i.d. copies of (X_i, Y_i), i = 1, ..., n and show that the dilution coefficient vanishes exponentially with n. In other words, Alice can achieve better privacy as the number of her observations grows.