This paper presents a perfectly secure matrix multiplication (PSMM) protocol for multiparty computation (MPC) of $\mathrm{A}^{\top}\mathrm{B}$ over finite fields. The proposed scheme guarantees correctness and information-theoretic privacy against threshold-bounded, semi-honest colluding agents, under explicit local storage constraints. Our scheme encodes submatrices as evaluations of sparse masking polynomials and combines coefficient alignment with Beaver-style randomness to ensure perfect secrecy. We demonstrate that any colluding set of parties below the security threshold observes uniformly random shares, and that the recovery threshold is optimal, matching existing information-theoretic limits. Building on this framework, we introduce a learning-augmented extension that integrates tensor-decomposition-based local block multiplication, capturing both classical and learned low-rank methods. We demonstrate that the proposed learning-based PSMM preserves privacy and recovery guarantees for MPC, while providing scalable computational efficiency gains (up to $80\%$) as the matrix dimensions grow.

翻译：本文提出了一种用于有限域上$\mathrm{A}^{\top}\mathrm{B}$多方计算（MPC）的完美安全矩阵乘法（PSMM）协议。该方案在明确的本地存储约束下，针对阈值受限的半诚实合谋参与者，保证了正确性和信息论隐私性。我们的方案将子矩阵编码为稀疏掩码多项式的求值，并结合系数对齐与Beaver式随机性来确保完美保密性。我们证明了任何低于安全阈值的合谋参与方集合仅能观察到均匀随机份额，且恢复阈值是最优的，与现有信息论极限相匹配。基于此框架，我们进一步提出了一种学习增强扩展，该扩展集成了基于张量分解的本地块乘法，兼容经典方法与学习型低秩方法。我们证明了所提出的基于学习的PSMM在保持MPC隐私性与恢复保证的同时，能随着矩阵维度增长提供可扩展的计算效率提升（最高达$80\%$）。