MicroCrypt Assumptions with Quantum Input Sampling and Pseudodeterminism: Constructions and Separations
Mohammed Barhoush, Ryo Nishimaki, Takashi Yamakawa
TL;DR
This work investigates two natural relaxations in MicroCrypt: quantum input sampling and bot-pseudodeterminism. It defines PRG^{qs} and PRS^{qs}, showing that under quantum-input sampling, PRG^{qs}, BC-SPRS^{qs}, and SPRS^{qs} become interconvertible in certain parameter regimes, a phenomenon not known in the uniform-input setting. The authors establish striking black-box separations, including a separation between PRG and PRF^{qs} with inverse access and CPTP-based separations that separate OWSG and bot-PRG, and bot-PRG from PRF^{qs}, thereby illuminating a hierarchy among uniform-input, pseudodeterministic, and quantum-input primitives in MicroCrypt. The results demonstrate that quantum-input sampling can yield inherently weaker primitives and reveal a nuanced structure of relationships and boundaries within MicroCrypt primitives, with implications for constructing cryptographic primitives from weaker assumptions. Overall, the paper advances a clearer map of which primitive relationships and separations persist when quantum input and pseudodeterminism are allowed, shaping future designs and impossibility results in quantum cryptography.
Abstract
We investigate two natural relaxations of quantum cryptographic primitives. The first involves quantum input sampling, where inputs are generated by a quantum algorithm rather than sampled uniformly at random. Applying this to pseudorandom generators ($\textsf{PRG}$s) and pseudorandom states ($\textsf{PRS}$s), leads to the notions denoted as $\textsf{PRG}^{qs}$ and $\textsf{PRS}^{qs}$, respectively. The second relaxation, $\bot$-pseudodeterminism, relaxes the determinism requirement by allowing the output to be a special symbol $\bot$ on an inverse-polynomial fraction of inputs. We demonstrate an equivalence between bounded-query logarithmic-size $\textsf{PRS}^{qs}$, logarithmic-size $\textsf{PRS}^{qs}$, and $\textsf{PRG}^{qs}$. Moreover, we establish that $\textsf{PRG}^{qs}$ can be constructed from $\bot$-$\textsf{PRG}$s, which in turn were built from logarithmic-size $\textsf{PRS}$. Interestingly, these relations remain unknown in the uniform key setting. To further justify these relaxed models, we present black-box separations. Our results suggest that $\bot$-pseudodeterministic primitives may be weaker than their deterministic counterparts, and that primitives based on quantum input sampling may be inherently weaker than those using uniform sampling. Together, these results provide numerous new insights into the structure and hierarchy of primitives within MicroCrypt.