Differentially Private Secure Multiplication: Hiding Information in the Rubble of Noise

We consider the problem of private distributed multiparty computation. It is well-established that coding strategies can enable perfect information-theoretic privacy in distributed computation (e.g., the BGW protocol). However, perfect privacy comes at a high computational overhead cost, requiring 2t + 1 compute nodes to ensure privacy against any t colluding nodes. By allowing for approximate computation and operations over the real numbers, we demonstrate that noise can be added to data shared with computing nodes in order to ensure differential privacy instead of perfect privacy. Specifically, the signal-to-noise ratio of the data received by colluding nodes can be mapped to differential privacy guarantees. We precisely characterize the trade-off between differential privacy and accuracy in this setting, and prove that a degree of differential privacy against t colluding nodes can always be ensured whenever there are more than t+1 computing node - a reduction of t nodes compared to perfect privacy. A particularly novel technical aspect is an achievable scheme that carefully encodes the data and noise at different magnitude levels. This coding scheme ensures that the adversary's input appears to be layers of noise, whereas the legitimate decoder is able to uncover the desired computation by "peeling"off the noise layers.