Differentially Private Distributed Matrix Multiplication: Fundamental Accuracy-Privacy Trade-Off Limits

The classic BGW algorithm of Ben Or, Goldwasser and Wigderson for secure multiparty computing demonstrates that secure distributed matrix multiplication over finite fields is possible over 2t+1 computation nodes, while keeping the input matrices private from every t colluding computation nodes. In this paper, we develop and study a novel coding formulation to explore the trade-offs between computation accuracy and privacy in secure multiparty computing for real-valued data, even with fewer than 2t+1 nodes, through a differential privacy perspective. For the case of t = 1, we develop achievable schemes and converse arguments that bound ϵ - the differential privacy parameter that measures the privacy loss - for a given accuracy level. Our achievable coding schemes are specializations of Shamir secret sharing applied to real-valued data, coupled with appropriate choice of evaluation points. We develop converse arguments that apply for general additive noise based schemes.