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A Meta-Complexity Characterization of Minimal Quantum Cryptography

Bruno Cavalar, Boyang Chen, Andrea Coladangelo, Matthew Gray, Zihan Hu, Zhengfeng Ji, Xingjian Li

TL;DR

The paper presents a meta-complexity framework for minimal quantum cryptography centered on EFI pairs, showing that EFI existence is equivalent to hardness of estimating a Kolmogorov-like complexity on non-uniform single-copy state families. It connects EFI to non-uniform 1PRS and to hard-on-average instances of GapH and GapU, providing both forward and reverse reductions. A universal EFI construction is given, together with a unifying view based on the span of easy states and robust entropy relations. The work deepens the link between quantum cryptographic primitives and algorithmic information, yielding new characterizations and potential avenues for uniformity and broader cryptographic implications in the quantum regime.

Abstract

We give a meta-complexity characterization of EFI pairs, which are considered the "minimal" primitive in quantum cryptography (and are equivalent to quantum commitments). More precisely, we show that the existence of EFI pairs is equivalent to the following: there exists a non-uniformly samplable distribution over pure states such that the problem of estimating a certain Kolmogorov-like complexity measure is hard given a single copy. A key technical step in our proof, which may be of independent interest, is to show that the existence of EFI pairs is equivalent to the existence of non-uniform single-copy secure pseudorandom state generators (nu 1-PRS). As a corollary, we get an alternative, arguably simpler, construction of a universal EFI pair.

A Meta-Complexity Characterization of Minimal Quantum Cryptography

TL;DR

The paper presents a meta-complexity framework for minimal quantum cryptography centered on EFI pairs, showing that EFI existence is equivalent to hardness of estimating a Kolmogorov-like complexity on non-uniform single-copy state families. It connects EFI to non-uniform 1PRS and to hard-on-average instances of GapH and GapU, providing both forward and reverse reductions. A universal EFI construction is given, together with a unifying view based on the span of easy states and robust entropy relations. The work deepens the link between quantum cryptographic primitives and algorithmic information, yielding new characterizations and potential avenues for uniformity and broader cryptographic implications in the quantum regime.

Abstract

We give a meta-complexity characterization of EFI pairs, which are considered the "minimal" primitive in quantum cryptography (and are equivalent to quantum commitments). More precisely, we show that the existence of EFI pairs is equivalent to the following: there exists a non-uniformly samplable distribution over pure states such that the problem of estimating a certain Kolmogorov-like complexity measure is hard given a single copy. A key technical step in our proof, which may be of independent interest, is to show that the existence of EFI pairs is equivalent to the existence of non-uniform single-copy secure pseudorandom state generators (nu 1-PRS). As a corollary, we get an alternative, arguably simpler, construction of a universal EFI pair.

Paper Structure

This paper contains 42 sections, 51 theorems, 127 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Meta-complexity.
  3. Quantum Cryptography
  4. Our contributions
  5. Connections
  6. $\mathsf{OWPuzz} \Rightarrow \mathsf{EFI}$.
  7. Unitary Synthesis and Quantum Complexity.
  8. Open problems
  9. Technical overview
  10. Equivalence of EFI pairs and (non-uniform) 1PRS
  11. From a (non-uniform) 1PRS to a "universal" EFI pair.
  12. A meta-complexity characterization of EFI pairs
  13. A unifying viewpoint on various notions of Kolmogorov complexity of states
  14. Preliminaries
  15. Quantum information
  16. ...and 27 more sections

Key Result

Theorem 1.1

$\mathsf{EFI}$ exist if and only if non-uniform $\mathsf{1PRS}$ with advice size $O(\log \lambda)$ exist.

Figures (1)

  • Figure 1: Outline of reductions. "HoA" stands for "hard on average". The six problems at the bottom are all equivalent, as the reductions form cycles. The top three are also equivalent and imply the bottom six.

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2: Informal
  • Corollary 1.1
  • Theorem 1.3: Informal
  • Lemma 3.1: Fuchs-van de Graaf inequalities, cf., for example, Theorem 3.33 of Wat18
  • Lemma 3.2
  • proof
  • Lemma 3.3: Result 5 and Corollary 6.5 of Tom12
  • Corollary 3.1
  • proof
  • ...and 114 more