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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.

A natural proof lower bound is based on a property of Boolean functions that is simultaneously constructive, large, and useful against the circuit class being bounded.

A property $\mathcal{P}=({\mathcal{P}}_n)$ of $n$-variable Boolean functions is:

A property satisfying all three conditions is a natural proof. The Razborov–Rudich natural-proofs barrier states that if exponentially secure pseudorandom function families exist in $\mathrm{P}/\mathrm{poly}$, then no property that is constructive and large is useful against polynomial-size circuits.

Suppose such a property $\mathcal{P}$ existed. Given oracle access to an unknown $n$-variable function, query its full truth table and run the constructive membership test. This takes $2^{O(n)}$ time. For a truly random function, largeness makes the test accept with probability at least $2^{-O(n)}$. Repetition amplifies this to a constant acceptance probability within $2^{O(n)}$ time.

For a function drawn from the assumed pseudorandom family, each keyed function has polynomial-size circuits, so usefulness makes the test reject for the relevant lengths. The amplified membership test therefore distinguishes the pseudorandom family from a truly random function with constant advantage in exponential time, contradicting exponential pseudorandomness. Hence the three conditions cannot coexist.

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]