Classical Gowers conjecture for random reversible circuits

Determine whether sufficiently deep random reversible circuits form ensembles of cryptographic pseudorandom permutations, so that every efficient observer can distinguish them from uniformly random permutations only with negligible advantage.

Background

The paper describes Gowers’s conjecture as the computational strengthening of statistical mixing results for random reversible circuits. Although repeated local reversible operations achieve strong approximate-wise-independence properties, it remains unresolved whether polynomial-depth circuits are computationally indistinguishable from uniformly random permutations for all efficient distinguishers.

The paper emphasizes that a positive resolution would have major complexity-theoretic consequences, including the existence of one-way functions and therefore the separation of P from NP. The authors distinguish this unresolved conjecture from later results refuting the separate HMMR round-local formulation.

References

Despite substantial recent progress on the statistical mixing of random circuits, the computational conjecture still remains wide open as a positive resolution would imply the existence of one-way functions, and hence $\mathsf{P}\neq \mathsf{NP}$.

— On the pseudorandomness of simple quantum processes  (2610.02100 - Dujmovic et al., 1 Oct 2026) in Introduction, Section 1