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On the Relativistic Zero Knowledge Quantum Proofs of Knowledge

Kaiyan Shi, Kaushik Chakraborty, Wen Yu Kon, Omar Amer, Marco Pistoia, Charles Lim

TL;DR

It is shown that there exists quantum proofs of knowledge with knowledge error 1/2 + negl({\eta}) for all relations in NP via a construction of such a system for the Hamiltonian cycle relation using a general relativistic commitment scheme exhibiting the fairly-binding property due to Fehr and Fillinger.

Abstract

We initiate the study of relativistic zero-knowledge quantum proof of knowledge systems with classical communication, formally defining a number of useful concepts and constructing appropriate knowledge extractors for all the existing protocols in the relativistic setting which satisfy a weaker variant of the special soundness property due to Unruh (EUROCRYPT 2012). We show that there exists quantum proofs of knowledge with knowledge error 1/2 + negl(η) for all relations in NP via a construction of such a system for the Hamiltonian cycle relation using a general relativistic commitment scheme exhibiting the fairly-binding property due to Fehr and Fillinger (EUROCRYPT 2016). We further show that one can construct quantum proof of knowledge extractors for proof systems which do not exhibit special soundness, and therefore require an extractor to rewind multiple times. We develop a new multi-prover quantum rewinding technique by combining ideas from monogamy of entanglement and gentle measurement lemmas that can break the quantum rewinding barrier. Finally, we prove a new bound on the impact of consecutive measurements and use it to significantly improve the soundness bound of some existing relativistic zero knowledge proof systems, such as the one due to Chailloux and Leverrier (EUROCRYPT 2017).

On the Relativistic Zero Knowledge Quantum Proofs of Knowledge

TL;DR

It is shown that there exists quantum proofs of knowledge with knowledge error 1/2 + negl({\eta}) for all relations in NP via a construction of such a system for the Hamiltonian cycle relation using a general relativistic commitment scheme exhibiting the fairly-binding property due to Fehr and Fillinger.

Abstract

We initiate the study of relativistic zero-knowledge quantum proof of knowledge systems with classical communication, formally defining a number of useful concepts and constructing appropriate knowledge extractors for all the existing protocols in the relativistic setting which satisfy a weaker variant of the special soundness property due to Unruh (EUROCRYPT 2012). We show that there exists quantum proofs of knowledge with knowledge error 1/2 + negl(η) for all relations in NP via a construction of such a system for the Hamiltonian cycle relation using a general relativistic commitment scheme exhibiting the fairly-binding property due to Fehr and Fillinger (EUROCRYPT 2016). We further show that one can construct quantum proof of knowledge extractors for proof systems which do not exhibit special soundness, and therefore require an extractor to rewind multiple times. We develop a new multi-prover quantum rewinding technique by combining ideas from monogamy of entanglement and gentle measurement lemmas that can break the quantum rewinding barrier. Finally, we prove a new bound on the impact of consecutive measurements and use it to significantly improve the soundness bound of some existing relativistic zero knowledge proof systems, such as the one due to Chailloux and Leverrier (EUROCRYPT 2017).
Paper Structure (33 sections, 29 theorems, 125 equations, 1 table)

This paper contains 33 sections, 29 theorems, 125 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our Results
  3. Rewinding Oracles in the Relativistic Setting:
  4. Relativistic $\Sigma$-Protocol:
  5. Quantum Proofs of Knowledge for Relativistic $\Sigma$-Protocols with $\delta_{SS}$-Special Soundness
  6. Quantum Proofs of Knowledge for Symmetric Relativistic Zero-Knowledge Protocols:
  7. Improved Soundness Bounds from a New Consecutive Measurement Lemma:
  8. Preliminaries
  9. Quantum Proofs of Knowledge in the Relativistic Setting
  10. Interactive Machines and Rewinding Oracles with Relativistic Constraints
  11. Execution of three interactive machines in relativistic setting.
  12. Oracle algorithms with rewinding
  13. QPoKs for Relativistic $\Sigma$-Protocols with $\delta_{SS}$-Special Soundness
  14. Canonical extractor for Relativistic $\Sigma$-Protocols:
  15. Lower Bound on the Knowledge Error
  16. ...and 18 more sections

Key Result

theorem 1

If a relativistic $\Sigma$-protocol has the $\delta_{SS}$-special soundness property then, it has a quantum proof of knowledge with knowledge error $\frac{1}{2} + \textsf{negl}(\eta)$.

Theorems & Definitions (54)

  • definition 1: Relativistic $\Sigma$-protocol
  • definition 2: $\delta_{SS}$-Special soundness unruh2012quantum
  • theorem 1: Informal
  • corollary 1: Relativistic QPoK for all languages in NP (Informal)
  • theorem 2: Informal
  • theorem 3: Informal
  • lemma 1: Informal
  • lemma 2: Informal
  • definition 3: Relativistic commitment scheme
  • definition 4: $\epsilon_{FB}$-fairly-binding
  • ...and 44 more