Quantum Cryptography in Algorithmica

Abstract: We construct a classical oracle relative to which $\mathsf{P} = \mathsf{NP}$ yet single-copy secure pseudorandom quantum states exist. In the language of Impagliazzo's five worlds, this is a construction of pseudorandom states in "Algorithmica," and hence shows that in a black-box setting, quantum cryptography based on pseudorandom states is possible even if one-way functions do not exist. As a consequence, we demonstrate that there exists a property of a cryptographic hash function that simultaneously (1) suffices to construct pseudorandom states, (2) holds for a random oracle, and (3) is independent of $\mathsf{P}$ vs. $\mathsf{NP}$ in the black-box setting. We also introduce a conjecture that would generalize our results to multi-copy secure pseudorandom states. We build on the recent construction by Aaronson, Ingram, and Kretschmer (CCC 2022) of an oracle relative to which $\mathsf{P} = \mathsf{NP}$ but $\mathsf{BQP} \neq \mathsf{QCMA}$, based on hardness of the OR $\circ$ Forrelation problem. Our proof also introduces a new discretely-defined variant of the Forrelation distribution, for which we prove pseudorandomness against $\mathsf{AC^0}$ circuits. This variant may be of independent interest.