Quantum-Computable One-Way Functions without One-Way Functions

TL;DR

This paper constructs a classical oracle where P equals NP but quantum-secure trapdoor one-way functions exist, demonstrating a separation between classical and quantum computational worlds with implications for cryptography.

Contribution

It introduces a new relativized model where classical and quantum worlds differ significantly, extending prior results to multi-copy pseudorandom states and classical cryptographic schemes.

Findings

Existence of quantum-secure trapdoor one-way functions in the relativized world

Classical P equals NP, but quantum cryptography remains secure in this model

New distributional block-insensitivity lemma for AC^0 circuits

Abstract

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.