Learning-Augmented Perfectly Secure Collaborative Matrix Multiplication

TL;DR

This work introduces Learning-Augmented Perfectly Secure Matrix Multiplication (LA-PSMM), a secure multiparty protocol for computing $A^{\top}B$ over finite fields with information-theoretic privacy against up to $t-1$ semi-honest colluders and explicit local storage constraints. It marries a PSMM core—built on sparse masking polynomials and Beaver-style randomness to achieve perfect privacy and optimal recovery thresholds—with a learning-driven local multiplication layer that uses tensor-decomposition-based low-rank forms (inspired by AlphaTensor) to dramatically reduce per-agent computation without degrading security or recoverability. The main results include a tight recovery bound $N \le \min\{2k^2+2t-3,\; k^2+kt+t-2\}$, a DOF-based improvement for structured target products reducing required agents to $N_{\text{struct}}$, and a rigorous demonstration that LA-PSMM preserves privacy and the recovery guarantees while delivering substantial computational efficiency gains (up to ~80% locally). The numerical experiments compare against BGW-style job-splitting, showing PSMM's advantages in agent count and communication, and demonstrate how learning-based decompositions scale favorably with matrix size while maintaining security, making the approach viable for large-scale MPC-enabled matrix computations in control and learning pipelines.

Abstract

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.

Figures (5)

• Figure 1: An NCS architecture incorporating a multiparty computation setup.
• Figure 2: Agents required $N$ vs. storage split $k$ for multiple $t$.
• Figure 3: Per-agent communication vs. number of agents $N$ for $(m,k,t)=(1024,8,8)$. Our scheme requires fewer bytes per agent than BGW and also operates at a smaller $N$.
• Figure 4: The agent computation complexity comparison of PSMM vs. LA-PSMM with different learned rank $T_l$.
• Figure 5: The agent computation gain of LA-PSMM with different learned rank $T_l$.

Theorems & Definitions (14)

• Lemma 1: Masking lemma
• proof
• Proposition 1: Correctness of triple-based multiplication
• proof
• Theorem 1: Agent bound for secure MM
• proof
• Remark 1: Positioning relative to polynomial sharing
• Remark 2: Improvement via polynomial sharing
• Proposition 2: Advantage under reduced degrees-of-freedom
• proof