Complexity Lower Bounds of Small Matrix Multiplication over Finite Fields via Backtracking and Substitution

2026-03-07

AI Summary

This work addresses the problem of establishing lower bounds on the bilinear complexity of small-scale matrix multiplication over finite fields. By integrating substitution, backtracking search, symmetry reduction, and dynamic programming, the authors systematically enumerate classes of linear constraints on input matrices and derive rank-based lower bounds within each class. This approach yields the first automated proof that the bilinear complexity of $3 \times 3$ matrix multiplication over $\mathbb{F}_2$ is at least 20, surpassing the longstanding record of 19 that had remained unimproved for over two decades. The complete proof can be automatically generated on a standard laptop in under 1.5 hours and verified in seconds, significantly advancing the automation and efficiency of lower-bound certification in algebraic complexity theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We introduce a new method for proving bilinear complexity lower bounds for matrix multiplication over finite fields. The approach combines the substitution method with a systematic backtracking search over linear restrictions on the first matrix $A$ in the product $AB = C^T$. We enumerate restriction classes up to symmetry; for each class we either obtain a rank lower bound by classical arguments or branch further via the substitution method. The search is organized by dynamic programming on the restricted matrix $A$. As an application we prove that the bilinear complexity of multiplying two $3 \times 3$ matrices over $\mathbb{F}_2$ is at least $20$, improving the longstanding lower bound of $19$ (Bl\"aser 2003). The proof is found automatically within 1.5 hours on a laptop and verified in seconds.

Problem

• matrix multiplication
• bilinear complexity
• finite fields
• lower bounds
• 3x3 matrices

Innovation

• bilinear complexity
• matrix multiplication
• finite fields
• backtracking search
• substitution method