  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.
