We present techniques for privacy-preserving computation of multidimensional aggregates on data partitioned across multiple clients. Data from different clients is perturbed (randomized) in order to preserve privacy before it is integrated at the server. We develop formal notions of privacy obtained from data perturbation and show that our perturbation provides guarantees against privacy breaches. We develop and analyze algorithms for reconstructing counts of subcubes over perturbed data. We also evaluate the tradeoff between privacy guarantees and reconstruction accuracy and show the practicality of our approach.