Black-Box Separations for Non-interactive Classical Commitments in a Quantum World

Abstract

Commitments are fundamental in cryptography. In the classical world, commitments are equivalent to the existence of one-way functions. It is also known that the most desired form of commitments in terms of their round complexity, i.e., non-interactive commitments, cannot be built from one-way functions in a black-box way [Mahmoody-Pass, Crypto’12]. However, if one allows the parties to use quantum computation and communication, it is known that non-interactive commitments (to classical bits) are in fact possible [Koshiba-Odaira, Arxiv’11 and Bitansky-Brakerski, TCC’21]. We revisit the assumptions behind non-interactive commitments in a quantum world and study whether they can be achieved using quantum computation and classical communication based on a black-box use of one-way functions. We prove that doing so is impossible unless the Polynomial Compatibility Conjecture [Austrin et al. Crypto’22] is false. We further extend our impossibility to protocols with quantum decommitments. This complements the positive result of Bitansky and Brakerski [TCC’21], as they only required a classical decommitment message. Because non-interactive commitments can be based on injective one-way functions, assuming the Polynomial Compatibility Conjecture, we also obtain a black-box separation between one-way functions and injective one-way functions (e.g., one-way permutations) even when the construction and the security reductions are allowed to be quantum. This improves the separation of Cao and Xue [Theoretical Computer Science’21] in which they only allowed the security reduction to be quantum. At a technical level, we prove that sampling oracles at random from “sufficiently large” sets (of oracles) will make them one-way against polynomial quantum-query adversaries who also get arbitrary polynomial-size quantum advice about the oracle. This gives a natural generalization of the recent results of Hhan et al. [Asiacrypt’19] and Chung et al. [FOCS’20].