Constructing explicit RIP matrices is an open problem in compressed sensing theory. In particular, it is quite challenging to construct explicit RIP matrices that break the square-root bottleneck. On the other hand, providing explicit $2$-source extractors is a fundamental problem in theoretical computer science, cryptography and combinatorics. Nowadays, there are only a few known constructions for explicit $2$-source extractors (with negligible errors) that break the half barrier for min-entropy. In this paper, we establish a new connection between RIP matrices breaking the square-root bottleneck and $2$-source extractors breaking the half barrier for min-entropy. Here we focus on an RIP matrix (called the Paley ETF) and a $2$-source extractor (called the Paley graph extractor), where both are defined from quadratic residues over the finite field of odd prime order $p\equiv 1 \pmod{4}$. As a main result, we prove that if the Paley ETF breaks the square-root bottleneck, then the Paley graph extractor breaks the half barrier for min-entropy as well. Since it is widely believed that the Paley ETF breaks the square-root bottleneck, our result accordingly provides a new affirmative intuition on the conjecture for the Paley graph extractor by Benny Chor and Oded Goldreich.

翻译：构建显式RIP矩阵是压缩感知理论中的一个开放性问题。特别地，构造打破平方根瓶颈的显式RIP矩阵极具挑战性。另一方面，提供显式2-源提取器是理论计算机科学、密码学和组合数学中的基本问题。目前，仅有少数已知构造能够实现打破最小熵半障碍的显式2-源提取器（误差可忽略）。本文建立了打破平方根瓶颈的RIP矩阵与打破最小熵半障碍的2-源提取器之间的新联系。我们聚焦于一种RIP矩阵（称为Paley ETF）和一种2-源提取器（称为Paley图提取器），两者均基于奇素数阶有限域$\mathbb{F}_p$（$p\equiv 1 \pmod{4}$）上的二次剩余定义。主要结果表明：若Paley ETF打破平方根瓶颈，则Paley图提取器同样打破最小熵半障碍。由于学术界普遍认为Paley ETF能打破平方根瓶颈，本研究因此为Benny Chor与Oded Goldreich关于Paley图提取器的猜想提供了新的肯定性直觉依据。