Fiat-Shamir for Proofs Lacks a Proof Even in the Presence of Shared Entanglement

We explore the cryptographic power of arbitrary shared physical resources. The most general such resource is access to a fresh entangled quantum state at the outset of each protocol execution. We call this the Common Reference Quantum State (CRQS) model, in analogy to the well-known Common Reference String (CRS). The CRQS model is a natural generalization of the CRS model but appears to be more powerful: in the two-party setting, a CRQS can sometimes exhibit properties associated with a Random Oracle queried once by measuring a maximally entangled state in one of many mutually unbiased bases. We formalize this notion as a Weak One-Time Random Oracle (WOTRO), where we only ask of the m-bit output to have some randomness when conditioned on the n-bit input. We show that WOTRO with $n - m \in {\omega}(\lg n)$ is black-box impossible in the CRQS model, meaning that no protocol can have its security black-box reduced to a cryptographic game. We define a (inefficient) quantum adversary against any WOTRO protocol that can be efficiently simulated in polynomial time, ruling out any reduction to a secure game that only makes black-box queries to the adversary. On the other hand, we introduce a non-game quantum assumption for hash functions that implies WOTRO in the CRQ\$model (where the CRQS consists only of EPR pairs). We first build a statistically secure WOTRO protocol where$m = n$, then hash the output. The impossibility of WOTRO has the following consequences. First, we show the black-box impossibility of a quantum Fiat-Shamir transform, extending the impossibility result of Bitansky et al. (TCC '13) to the CRQS model. Second, we show a black-box impossibility result for a strenghtened version of quantum lightning (Zhandry, Eurocrypt '19) where quantum bolts have an additional parameter that cannot be changed without generating new bolts.$