Efficient pseudorandom unitaries with general (adaptive) query security

Establish the existence of efficient pseudorandom unitary ensembles that are computationally indistinguishable from Haar measure against any polynomial-time quantum adversary making general (adaptive) oracle queries, under standard cryptographic assumptions such as the existence of quantum-secure one-way functions.

Background

Pseudorandom unitaries (PRUs) are the unitary analog of pseudorandom functions, but constructing them under standard assumptions has been challenging. While several pseudorandom state constructions exist and intermediate objects (e.g., pseudorandom state scramblers and pseudorandom isometries) have been realized, full PRUs with general query security have remained elusive.

This paper achieves a construction of PRUs with nonadaptive (parallel) security from quantum-secure one-way functions, narrowing the gap. However, full adaptive-query security—where the adversary can choose queries based on prior outcomes—remains unresolved and is explicitly identified as open.

References

However, the existence of efficient pseudorandom unitaries with general query security remains open.

— Efficient unitary designs and pseudorandom unitaries from permutations  (2404.16751 - Chen et al., 2024) in Introduction (Section 1), Computational pseudorandomness discussion

Finally, the optimal depth of strong pseudorandom unitaries remains open. The present theorem gives a logarithmic-depth information-theoretic strong-design construction, but it does not provide a keyed family or computational security against efficient distinguishers. A logarithmic-depth, system-only construction of strong pseudorandom unitaries would require a suitable keyed local primitive together with a security-composition theorem that remains valid under inverse, transpose, and complex-conjugate queries.

— Strong unitary designs in optimal depth and space  (2608.13491 - Parella-Dilmé et al., 13 Aug 2026) in Section Discussion