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Non-Destructive Zero-Knowledge Proofs on Quantum States, and Multi-Party Generation of Authorized Hidden GHZ States

Léo Colisson, Frédéric Grosshans, Elham Kashefi

TL;DR

The paper tackles the challenge of verifying non-destructively whether quantum states possess certain properties and of generating distributed, authorized hidden GHZ states. It introduces Non-Interactive Zero-Knowledge Proofs on Quantum States (NIZKoQS), enabling a classical prover to certify quantum-state properties through a single message, leveraging post-quantum NIZK and LWE-based cryptography. It then provides an efficient, scalable method to generate large multi-qubit states in a single quantum superposition, and extends this to a multi-party setting where authorized applicants share a hidden GHZ state with blindness guarantees and secret-credential protections. The framework relies on MP11 trapdoors and LWE hardness, enabling a suite of protocols (BLIND, BLIND^sup, BLIND^sup_can, AUTH-BLIND^dist_can) and a general method to construct distributable GHZ-capable primitives, with potential applications to quantum secret sharing, anonymous transmission, and quantum-routing scenarios.

Abstract

We propose the first generalization of the famous Non-Interactive Zero-Knowledge (NIZK) proofs to quantum languages (NIZKoQS) and we provide a protocol to prove advanced properties on a received quantum state non-destructively and non-interactively (a single message being sent from the prover to the verifier). In our second orthogonal contribution, we improve the costly Remote State Preparation protocols [CCKW18,CCKW19,GV19] that can classically fake a quantum channel (this is at the heart of our NIZKoQS protocol) by showing how to create a multi-qubits state from a single superposition. Finally, we generalize these results to a multi-party setting and prove that multiple parties can anonymously distribute a GHZ state in such a way that only participants knowing a secret credential can share this state, which could have applications to quantum anonymous transmission, quantum secret sharing, quantum onion routing and more.

Non-Destructive Zero-Knowledge Proofs on Quantum States, and Multi-Party Generation of Authorized Hidden GHZ States

TL;DR

The paper tackles the challenge of verifying non-destructively whether quantum states possess certain properties and of generating distributed, authorized hidden GHZ states. It introduces Non-Interactive Zero-Knowledge Proofs on Quantum States (NIZKoQS), enabling a classical prover to certify quantum-state properties through a single message, leveraging post-quantum NIZK and LWE-based cryptography. It then provides an efficient, scalable method to generate large multi-qubit states in a single quantum superposition, and extends this to a multi-party setting where authorized applicants share a hidden GHZ state with blindness guarantees and secret-credential protections. The framework relies on MP11 trapdoors and LWE hardness, enabling a suite of protocols (BLIND, BLIND^sup, BLIND^sup_can, AUTH-BLIND^dist_can) and a general method to construct distributable GHZ-capable primitives, with potential applications to quantum secret sharing, anonymous transmission, and quantum-routing scenarios.

Abstract

We propose the first generalization of the famous Non-Interactive Zero-Knowledge (NIZK) proofs to quantum languages (NIZKoQS) and we provide a protocol to prove advanced properties on a received quantum state non-destructively and non-interactively (a single message being sent from the prover to the verifier). In our second orthogonal contribution, we improve the costly Remote State Preparation protocols [CCKW18,CCKW19,GV19] that can classically fake a quantum channel (this is at the heart of our NIZKoQS protocol) by showing how to create a multi-qubits state from a single superposition. Finally, we generalize these results to a multi-party setting and prove that multiple parties can anonymously distribute a GHZ state in such a way that only participants knowing a secret credential can share this state, which could have applications to quantum anonymous transmission, quantum secret sharing, quantum onion routing and more.

Paper Structure

This paper contains 36 sections, 1 theorem, 72 equations, 7 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Works and Further Applications
  4. Paper Organization
  5. Technical overview and presentation of the setup
  6. Initial setup
  7. List of the protocols proposed in this work
  8. Quick overview of our methods
  9. Open questions
  10. Preliminaries
  11. Notation
  12. Generalized hidden $$ GHZ state
  13. Classical Zero-Knowledge proofs and arguments for
  14. Classical Multi-Party Computations
  15. Introduction to the Learning With Errors (LWE) problem
  16. ...and 21 more sections

Key Result

theorem thmcountertheorem

Let $n \in \mathbb{N}$ be the size of the quantum state outputted by Bob in ${\tt BLIND-ZK}$ and $\delta=\negl[\lambda]$. The protocol ${\tt BLIND-ZK}$ (where Alice plays the role of the prover ${\sf P}$ and Bob the verifier ${\sf V}$) is a NIZKoQS for the quantum language, parameterized by the witn In particular, since by definition of $f_k$ we have $\mathbf{d} \xor \mathbf{d}' = h(x) \xor h(x')

Figures (7)

  • Figure 1: Construction from MP11
  • Figure 2: Circuit performed by the server Bob.
  • Figure 3: Game ${\tt IND- {\tt BLIND}^{\tt sup}_{\tt can} {}}$ required in \ref{['lem:securityBlindCanSup']}
  • Figure 4: The function to compute in the ${\tt AUTH-BLIND}^{\tt dist}_{\tt can}$ protocol in a MPC way. The first input is the input of the server, and the other inputs are from the applicants (the $y^{(i)}$ and $b^{(i)}$ are supposed to be equal to $y$ and $b$ and are just used to ensure that the server provided coherent inputs in the MPC, and $r^{(i)} \in \{0,1\}^n$ is a string supposed to be sampled uniformly at random).
  • Figure 6: Definition of the $\delta$-${\sf GHZ^H}$ capable family
  • ...and 2 more figures

Theorems & Definitions (27)

  • definition thmcounterdefinition: (Canonical) $$ GHZ state
  • definition thmcounterdefinition: ${\sf GHZ^G}$: Generalized $$ GHZ state
  • definition thmcounterdefinition: ${\sf GHZ^H}$: Hidden generalized $$ GHZ state
  • definition thmcounterdefinition: Computational indistinguishability BS_2019_ZK_Ct_rounds
  • definition thmcounterdefinition: Post-Quantum Zero-Knowledge Classical Protocol BS_2019_ZK_Ct_rounds
  • definition thmcounterdefinition: Post-Quantum Zero-Knowledge Proof of Knowledge Unruh_2012_Proof_of_Knowledge
  • definition thmcounterdefinition: $\mathsf{REAL}_{\Uppi,\pazocal{A}}(\lambda,\hbox{\boldmath$x$},\rho_\lambda)$
  • definition thmcounterdefinition: $\mathsf{IDEAL}_{f,\mathsf{Sim}}(\lambda,\hbox{\boldmath$x$},\rho_\lambda)$
  • definition thmcounterdefinition: Secure MPC ABG+21_PostQuantumMultiPartyComputation
  • definition thmcounterdefinition: Continuous and Discrete Gaussian
  • ...and 17 more