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Verifiable Quantum Advantage without Structure

Takashi Yamakawa, Mark Zhandry

TL;DR

By replacing the random oracle with a concrete cryptographic hash function such as SHA2, the authors obtain plausible Minicrypt instantiations of the above results, and do not appear to contradict the Aaronson-Ambanis conjecture.

Abstract

We show the following hold, unconditionally unless otherwise stated, relative to a random oracle: - There are NP search problems solvable by quantum polynomial-time machines but not classical probabilistic polynomial-time machines. - There exist functions that are one-way, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar separations hold for digital signatures and CPA-secure public key encryption (the latter requiring the assumption of a classically CPA-secure encryption scheme). Interestingly, the separation does not necessarily extend to the case of other cryptographic objects such as PRGs. - There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are non-interactive, whereas for non-uniform adversaries the proofs are two message public coin. - Our results do not appear to contradict the Aaronson-Ambanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction. By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond.

Verifiable Quantum Advantage without Structure

TL;DR

By replacing the random oracle with a concrete cryptographic hash function such as SHA2, the authors obtain plausible Minicrypt instantiations of the above results, and do not appear to contradict the Aaronson-Ambanis conjecture.

Abstract

We show the following hold, unconditionally unless otherwise stated, relative to a random oracle: - There are NP search problems solvable by quantum polynomial-time machines but not classical probabilistic polynomial-time machines. - There exist functions that are one-way, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar separations hold for digital signatures and CPA-secure public key encryption (the latter requiring the assumption of a classically CPA-secure encryption scheme). Interestingly, the separation does not necessarily extend to the case of other cryptographic objects such as PRGs. - There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are non-interactive, whereas for non-uniform adversaries the proofs are two message public coin. - Our results do not appear to contradict the Aaronson-Ambanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction. By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond.
Paper Structure (67 sections, 42 theorems, 142 equations, 1 figure)

This paper contains 67 sections, 42 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quantum speed-ups and structure.
  3. Our Results
  4. Random oracles.
  5. Instantiating the random oracle.
  6. Discussion
  7. Other sources of quantum advantage.
  8. Why $\NP$ search problems?
  9. The QROM
  10. Subsequent work.
  11. Overview
  12. Classical hardness.
  13. Quantum easiness.
  14. Certifiable randomness.
  15. Extension to non-uniform adversaries.
  16. ...and 52 more sections

Key Result

Theorem 1.2

Relative to a random oracle, there exists a non-interactive proof of quantumness, with unconditional security against any computationally-unbounded adversary making a polynomial number of classical queries.

Figures (1)

  • Figure 1: The algorithm $\mathsf{Prove}$ for computing $\pi$. Here, $n-k$ is the dimension of $C^\perp$, and $M_{C^\perp}$ is any invertible matrix whose first $n-k$ columns are a basis for $C^\perp$.

Theorems & Definitions (92)

  • Theorem 1.2: Informal
  • Remark 1
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2
  • Lemma 2.1: Parseval's equality
  • proof
  • Lemma 2.2
  • proof
  • ...and 82 more