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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.

The Razborov–Rudich natural-proofs barrier connects lower bounds in circuit complexity with the security of pseudorandom function families. Suppose a property of Boolean functions is constructive, large and useful against a class containing the relevant pseudorandom function family.

Query all $2^n$ inputs of a candidate Boolean function, then run the property's membership test on its truth table. Constructivity makes this take $2^{O(n)}$ time. Largeness gives acceptance probability at least $2^{-O(n)}$ for a uniformly random Boolean function. Usefulness gives rejection for all functions in the chosen small-circuit family at sufficiently large lengths. The property therefore supplies a statistical distinguisher.

A security assumption must cover that running time and distinguishing advantage, at the chosen relation between input length and key length. Security only against polynomial-time algorithms in $n$ does not cover a full truth table test. Razborov and Rudich use a stronger quantitative hardness assumption to obtain the barrier against P/poly. The conclusion restricts a form of lower-bound proof under the assumption; it does not settle whether P equals NP.^[ref]

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.