Separating Pseudorandom Generators from Logarithmic Pseudorandom States
Mohammed Barhoush
TL;DR
This work resolves a central question in quantum cryptography by establishing two quantum black-box separations: (i) no fully black-box construction can transform a pseudorandom generator ($\mathsf{PRG}$) into a pseudorandom quantum state primitive of logarithmic or linear size ($\ell$-$\mathsf{PRS}$), and (ii) no BB construction can derive a PRG from a $\bot$-pseudodeterministic PRG, both relative to a unitary oracle with inverse access. The authors achieve this via two unitary oracle frameworks: a PSPACE oracle to defeat any BB PRG construction and a Common Haar Function-Like State (CHFS) oracle that yields Haar-like random states of size $\ell(|x|)$, enabling a careful stability/continuity (geodesic) argument across oracle hybrids. The results imply that several cryptographic primitives that are currently realized from SPRSs (such as digital signatures and quantum public-key encryption with tamper-resilient keys) cannot be achieved via reductions from PRGs in a black-box manner. Moreover, a second separation demonstrates that even translating from pseudodeterministic PRGs to standard PRGs via BB reductions fails, strengthening the separation from SP RS-based cryptographic applications. Overall, the paper delineates a hierarchy between quantum pseudorandom primitives and clarifies the limitations of black-box reductions in quantum cryptography.
Abstract
Pseudorandom generators (PRGs) are a foundational primitive in classical cryptography, underpinning a wide range of constructions. In the quantum setting, pseudorandom quantum states (PRSs) were proposed as a potentially weaker assumption that might serve as a substitute for PRGs in cryptographic applications. Two primary size regimes of PRSs have been studied: logarithmic-size and linear-size. Interestingly, logarithmic PRSs have led to powerful cryptographic applications, such as digital signatures and quantum public-key encryption with tamper-resilient keys, that have not been realized from their linear counterparts. However, PRGs have only been black-box separated from linear PRSs, leaving open the fundamental question of whether PRGs are also separated from logarithmic PRSs. In this work, we resolve this open problem. We establish a quantum black-box separation between (quantum-evaluable) PRGs and PRSs of either size regime. Specifically, we construct a unitary quantum oracle with inverse access relative to which no black-box construction of PRG from (logarithmic or linear) PRS exists. This does not directly separate PRG from some of the applications of SPRS since these applications involve, as a first step, a non-black-box construction of a notion termed bot-PRGs. To address this, we present another unitary separation showing that PRG are also separated from bot-PRGs. Thus, we obtain separation from digital signatures and quantum public-key encryption.