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Alexander A. Razborov coauthored Natural Proofs by Alexander A. Razborov and Steven Rudich with Steven Rudich. They analyzed why many successful restricted-model circuit lower bounds do not straightforwardly establish strong lower bounds for general Boolean circuits.^[ref]

Their framework separates a constructive property of Boolean functions, a large property of Boolean functions and a useful property against a circuit class. Together these can provide a statistical test contradicting a quantitatively strong pseudorandom function family assumption. The test for a constructive property of Boolean functions is measured against truth table size, not merely the number of input bits; that distinction is essential when stating the cryptographic hypothesis precisely.

By Alexander A. Razborov, Steven Rudich.

Primary source: mit6875.github.io/PAPERS/natural_proofs.pdf

Section 4, quantitative pseudorandomness barrier.

Steven Rudich coauthored the natural proof framework with Alexander A. Razborov. Their 1997 paper identifies a barrier to a broad form of circuit lower bound argument, conditional on a sufficiently strong pseudorandom function family security assumption.^[ref]

A property that is efficiently recognizable from a truth table, holds for many random Boolean functions and excludes the target small-circuit class could distinguish pseudorandom function families from random ones. The required security must cover the time spent reading the entire table, which is exponential in the function's input length. The barrier restricts these proof methods under its assumption; it does not prove that circuit lower bounds are impossible.