AI 中文总结
针对单服务器矩阵乘法外包，研究秩至多r的加法掩蔽的统计隐私，证明秩球掩蔽和独立均匀因子乘积达到最大相关保密性q^{-r}，且渐近最优，并给出后验分布和差分隐私下界。
AI 中文摘要
我们研究了在有限域 ${\mathbb F_q}$ 上，使用秩至多为 $r$ 的加法掩蔽将矩阵乘法外包给单个服务器时的统计隐私。对于独立的均匀 $n\times n$ 输入，我们证明均匀的\emph{秩球掩蔽}和独立均匀因子的乘积在完整服务器视图下提供至多 $q^{-r}$ 的最大相关保密性，编码和解码的域运算复杂度为 $O(n^2r)$。这种保密性衡量了服务器在估计输入函数时被有效阻止的程度。我们证明了当 $r=o(n)$ 时，该保密性度量具有渐近匹配的下界，表明在秩至多为 $r$ 的输入无关加法掩蔽中，即使允许秘密可逆变换，这两种采样方法也是渐近最优的。我们还刻画了在任意联合输入分布下均匀秩球掩蔽的后验分布，并证明了在独立均匀输入下，行和列具有近似个体安全性。最后，我们证明对于固定的域大小 $q$ 和有界 $\varepsilon$，任何秩至多为 $r=o(n)$ 的输入无关加法掩蔽在条目级 $(\varepsilon,\delta)$-差分隐私中需要 $\delta\to1$。
英文摘要
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 $\varepsilon$.
