Quantum Cryptography and Meta-Complexity

Abstract

In classical cryptography, one-way functions (OWFs) are the minimal assumption, while it is not the case in quantum cryptography. Several new primitives have been introduced such as pseudorandom state generators (PRSGs), one-way state generators (OWSGs), one-way puzzles (OWPuzzs), and EFI pairs. They seem to be weaker than OWFs, but still imply many useful applications. Now that the possibility of quantum cryptography without OWFs has opened up, the most important goal in the field is to build a foundation of it. In this paper, we, for the first time, characterize quantum cryptographic primitives with meta-complexity. We show that one-way puzzles (OWPuzzs) exist if and only if GapK is weakly-quantum-average-hard. GapK is a promise problem to decide whether a given bit string has a small Kolmogorov complexity or not. Weakly-quantum-average-hard means that an instance is sampled from a QPT samplable distribution, and for any QPT adversary the probability that it makes mistake is larger than ${\rm 1/poly}$. We also show that if quantum PRGs exist then GapK is strongly-quantum-average-hard. Here, strongly-quantum-average-hard is a stronger version of weakly-quantum-average-hard where the probability that the adversary makes mistake is larger than $1/2-1/{\rm poly}$. Finally, we show that if GapK is weakly-classical-average-hard, then inefficient-verifier proofs of quantumness (IV-PoQ) exist. Weakly-classical-average-hard is the same as weakly-quantum-average-hard except that the adversary is PPT. IV-PoQ are a generalization of proofs of quantumness (PoQ) that capture sampling-based and search-based quantum advantage, and an important application of OWpuzzs. This is the fist time that quantum advantage is based on meta-complexity. (Note: There are two concurrent works[Khurana-Tomer,arXiv:2409.15248; Cavalar-Goldin-Gray-Hall,arXiv:2410.04984].)