Randomness Certification from Statistically Hiding Commitments

Abstract

Randomness certification aims to guarantee that the output of an untrusted quantum device contains genuine randomness that remains unpredictable even given quantum side information held by an adversary. For a classical verifier interacting with a single quantum device, such a guarantee cannot rely on the physical separation underlying standard Bell tests and must instead follow from computational restrictions on the prover. Building on the framework of Computational Bell Inequalities, we distinguish classical and quantum virtual-input hiding, showing that the former suffices for proofs of quantumness, whereas the latter enables randomness certification against quantum side information. Our main technical contribution is to establish quantum virtual-input hiding for a commitment-based protocol derived from the proof of quantumness from one-way-functions by Morimae and Yamakawa. We show that statistical hiding together with ordinary computational binding against quantum polynomial-time senders suffices, without requiring stronger quantum binding notions such as sum-binding or collapse-binding. This yields a randomness-certification protocol from post-quantum statistically hiding bit commitments. In particular, existing constructions of such commitments from post-quantum one-way permutations imply that post-quantum one-way permutations suffice for randomness certification. We further instantiate our construction using quantum-secure trapdoor claw-free functions, obtaining an efficient classical verifier, and in the quantum random oracle model, obtaining randomness certification without additional computational assumptions but with an inefficient verifier. These instantiations separate the cryptographic properties sufficient for randomness certification from the additional structure required for efficient verification within our commitment-based framework.