Strongly Refuting Semirandom Linear Systems in Subexponential Time

2026-09-24

AI Summary

This study addresses the challenge of efficiently certifying the unsatisfiability of semi-random linear equation systems over $\mathbb{F}_2$ subject to majority constraints, particularly in the regime of extremely few samples. To this end, it refines the BKW and Lyubashevsky search algorithms to propose a randomized refutation strategy with sub-exponential time complexity, thereby overcoming the limitations inherent in traditional Sum-of-Squares (SOS) hierarchies. The primary contribution lies in achieving rapid refutation within $2^{O(n/\log n)}$ time using merely $n^{1+\gamma}$ samples. This result exposes a significant performance gap between the proposed algorithmic framework and SOS-based methods. Furthermore, it establishes a new benchmark for natural noise-tolerant signal recovery problems, offering deeper insights into the computational boundaries of learning from sparse, noisy observations.

Abstract

In this paper, we consider the problem of refuting $\mathbb{F}_2$-linear equations with random right-hand sides. Formally, we give a sub-exponential $2^{O(n/\log n)}$-time randomized algorithm that takes as input an arbitrary $m \times n$ matrix $A$ and a uniformly random vector $b \in \mathbb{F}_2^m$, and outputs a witness showing that no assignment satisfies more than a $\frac{1}{2}+ε$ fraction of the equations provided that $m \geq 2^{O(n/\log n)}$. The setting above is the semirandom refutation variant of the famous work [BKW03] that gives a $2^{O(n/\log n)}$-time search algorithm for the learning parity with noise (LPN) problem with $m \geq 2^{O(n/\log n)}$ equations. Building on the search algorithm of [Lyub05], we also give a $2^{O(n/\log \log n)}$-time refutation algorithm that succeeds with only $m \geq n^{1 + γ}$ equations, for a small constant $γ$. Finally, we prove that our algorithm is not captured by the sum-of-squares hierarchy by proving a degree-$Ω(n)$ sum-of-squares lower bound, showing that ''[BKW03]-style'' algorithms achieve better runtime than can be done under sum-of-squares. We thus obtain a natural example of a noise-tolerant signal recovery problem that exhibits a nontrivial gap between the performance of efficient algorithms and that of those based on the sum-of-squares hierarchy.

Problem

• semirandom refutation
• linear equations over F2
• learning parity with noise
• subexponential time
• sum-of-squares lower bound

Innovation

• Semirandom Refutation
• Subexponential Time
• Learning Parity with Noise
• Sum-of-Squares Lower Bound
• Signal Recovery