A Meta-Complexity Characterization of Quantum Cryptography

TL;DR

This paper establishes a fundamental link between quantum cryptographic primitives called one-way puzzles and the average-case hardness of Kolmogorov complexity, extending classical cryptography concepts into the quantum realm.

Contribution

It provides the first meta-complexity characterization of quantum one-way puzzles, connecting their existence to the hardness of approximating Kolmogorov complexity in quantum settings.

Findings

One-way puzzles exist iff there is a quantum samplable distribution with hard-to-approximate Kolmogorov complexity.

The existence of one-way puzzles is equivalent to the hardness of probability estimation over certain distributions.

Relativized worlds suggest that characterizing one-way puzzles via NP or QMA problems may be infeasible.

Abstract

We prove the first meta-complexity characterization of a quantum cryptographic primitive. We show that one-way puzzles exist if and only if there is some quantum samplable distribution of binary strings over which it is hard to approximate Kolmogorov complexity. Therefore, we characterize one-way puzzles by the average-case hardness of a uncomputable problem. This brings to the quantum setting a recent line of work that characterizes classical cryptography with the average-case hardness of a meta-complexity problem, initiated by Liu and Pass. Moreover, since the average-case hardness of Kolmogorov complexity over classically polynomial-time samplable distributions characterizes one-way functions, this result poses one-way puzzles as a natural generalization of one-way functions to the quantum setting. Furthermore, our equivalence goes through probability estimation, giving us the additional equivalence that one-way puzzles exist if and only if there is a quantum samplable distribution over which probability estimation is hard. We also observe that the oracle worlds of defined by Kretschmer et. al. rule out any relativizing characterization of one-way puzzles by the hardness of a problem in NP or QMA, which means that it may not be possible with current techniques to characterize one-way puzzles with another meta-complexity problem.