MicroCrypt Assumptions with Quantum Input Sampling and Pseudodeterminism: Constructions and Separations

TL;DR

This paper explores relaxed quantum cryptographic primitives involving quantum input sampling and pseudodeterminism, establishing equivalences, constructions, and separations that deepen understanding of their hierarchy and limitations.

Contribution

It introduces and analyzes quantum input sampling and $ot$-pseudodeterminism, establishing new equivalences, constructions, and black-box separations in quantum cryptography.

Findings

Equivalence between bounded-query and logarithmic-size $ extsf{PRS}^{qs}$ and $ extsf{PRG}^{qs}$.

Construction of $ extsf{PRG}^{qs}$ from $ot$-$ extsf{PRG}$s.

Separation results indicating weaker nature of $ot$-pseudodeterministic primitives.

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.