We construct a classical oracle relative to which $\mathsf{ P} = \mathsf{ NP}$ but quantum-computable quantum-secure trapdoor one-way functions exist. This is a substantial strengthening of the result of Kretschmer, Qian, Sinha, and Tal (STOC 2023), which only achieved single-copy pseudorandom quantum states relative to an oracle that collapses $\mathsf{ NP}$ to $\mathsf{ P}$. For example, our result implies multi-copy pseudorandom states and pseudorandom unitaries, but also classical-communication public-key encryption, signatures, and oblivious transfer schemes relative to an oracle on which $\mathsf{ P}=\mathsf{ NP}$. Hence, in our new relativized world, classical computers live in "Algorithmica" whereas quantum computers live in "Cryptomania," using the language of Impagliazzo's worlds. Our proof relies on a new distributional block-insensitivity lemma for $\mathsf{ AC^0}$ circuits, wherein a single block is resampled from an arbitrary distribution.
Categories: quant-ph cs.CC cs.CR