Differentially Private Secure Multiplication with Erasures and Adversaries

We consider a private distributed multiplication problem involving $N$ computation nodes and $T$ colluding nodes. Shamir's secret sharing algorithm provides perfect informationtheoretic privacy, while requiring an honest majority, i.e., $\mathrm{N} \geq 2 \mathrm{T}+1$. Recent work has investigated approximate computation and characterized privacy-accuracy trade-offs for the honest minority setting ($N \leq 2 \mathrm{T}$) for real-valued data, quantifying privacy leakage via the differential privacy (DP) framework and accuracy via the mean squared error. However, it does not incorporate the error correction capabilities of Shamir's secret-sharing algorithm. This paper develops a new polynomialbased coding scheme for secure multiplication with an honest minority, and characterizes its achievable privacy-utility tradeoff, showing that the tradeoff can approach the converse bound as closely as desired. Unlike previous schemes, the proposed scheme inherits the capability of the Reed-Solomon (RS) code to tolerate erasures and adversaries. We utilize a modified Berlekamp-Welch algorithm over the real number field to detect adversarial nodes.