Low-Rank Masking for Single-Server Matrix Multiplication

Abstract

We study the statistical privacy of outsourcing matrix multiplication over a finite field ${\mathbb F_q}$ to a single server using additive masks of rank at most $r$. For independent uniform $n\times n$ inputs, we show that uniform \emph{rank-ball masks} and products of independent uniform factors give maximal-correlation secrecy of at most $q^{-r}$ against the complete server view, with $O(n^2r)$ field operations for encoding and decoding. This secrecy captures how effectively the server is prevented from estimating functions of the inputs. We prove an asymptotically matching lower bound of this secrecy measure for $r=o(n)$, showing that both sampling methods are asymptotically optimal among input-independent additive masks of rank at most $r$, even when secret invertible transformations are allowed. We also characterize the posterior distribution for uniform rank-ball masks under arbitrary joint input distributions and prove approximate individual security for rows and columns under independent uniform inputs. Finally, we show that every input-independent additive mask of rank at most $r=o(n)$ requires $δ\to1$ in entry-level $(\varepsilon,δ)$-differential privacy for fixed field size $q$ and bounded $\varepsilon$.