Manipulating Statistical Di erence

We give several eecient transformations for manipulating the statistical diierence (variation distance) between a pair of probability distributions. The eeects achieved include increasing the statistical diierence, decreasing the statistical diierence, \polarizing" the statistical relationship, and \reversing" the statistical relationship. We also show that a boolean formula whose atoms are statements about statistical diierence can be transformed into a single statement about statistical diierence. All of these transformations can be performed in polynomial time, in the sense that, given circuits which sample from the input distributions, it only takes polynomial time to compute circuits which sample from the output distributions. By our prior work (see FOCS 97), such transformations for manipulating statistical diierence are closely connected to results about SZK, the class of languages possessing statistical zero-knowledge proofs. In particular, some of the transformations given in this paper are equivalent to the closure of SZK under complement and under certain types of Turing reductions. This connection is also discussed brieey in this paper.