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EPJ Quantum Technology

High-dimensional GHZ-based multi-party quantum key agreement: rigorous security, fairness, and loss tolerance

Hamza Sohail, Burhan Ul Islam Khan, Nur Fatin Liyana Mohd Rosely, Khang Wen Goh, Dwi Sudarno Putra, Abdul Raouf Khan, Mesith Chaimanee

Published Apr 23, 2026• 0 citations
Publisher

Abstract

Abstract Multi-party quantum key agreement (MQKA) enables a group of mutually untrusted users to jointly establish a secret key with equal influence, a capability not provided by two-party quantum key distribution (QKD) and increasingly relevant for securing resilient infrastructure such as quantum-enabled smart grid networks and distributed cyber-physical systems. We present a high-dimensional Greenberger-Horne-Zeilinger (GHZ)-based MQKA that simultaneously targets rigorous security, provable fairness, and practical loss tolerance. The protocol distributes an N -partite d -level GHZ state in a star (with an optional star-of-stars hierarchy for large N ) and maps each user’s contribution to local $X_{d}^{a} Z_{d}^{b}$ X d a Z d b operations. A commit-then-reveal layer makes inputs binding and hiding, yielding equal influence fairness over $\mathbb{Z}_{d}$ Z d with a strict fair-abort guarantee. Composable secrecy is established via entropy bounds and a Devetak-Winter-style key rate analysis, strengthened by test-round statistics; we connect Mermin/Svetlichny-type correlators to min-entropy lower bounds and incorporate full leakage accounting and finite-key penalties. We model depolarizing, shift/phase (Weyl), amplitude-damping, erasures, detector efficiency, and dark counts, and derive closed-form mappings from device parameters to symbol/phase errors. Monte Carlo studies confirm that $d>2$ d > 2 expands the positive-rate region and reduces the global survival required for a given throughput compared with qubit GHZ MQKA under identical conditions. We discuss potential photonic realizations (time-bin or frequency-bin encoding, discrete Fourier transform networks, and superconducting nanowire detectors) and architectural pathways toward larger N via hierarchical clustering, while acknowledging the significant technological challenges that would need to be overcome to realize high-dimensional multipartite entanglement experimentally. Overall, the scheme delivers higher efficiency per entangled state, rigorous fairness, and composable security with realistic noise and loss, addressing key gaps in current MQKA designs for resilient infrastructure.

DOI
10.1140/epjqt/s40507-026-00510-1

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EPJ Quantum Technology•Apr 22, 2025•12 citations
Abstract Multi-party quantum key agreement (MQKA) enables a group of mutually untrusted users to jointly establish a secret key with equal influence, a capability not provided by two-party quantum key distribution (QKD) and increasingly relevant for securing resilient infrastructure such as quantum-enabled smart grid networks and distributed cyber-physical systems. We present a high-dimensional Greenberger-Horne-Zeilinger (GHZ)-based MQKA that simultaneously targets rigorous security, provable fairness, and practical loss tolerance. The protocol distributes an N -partite d -level GHZ state in a star (with an optional star-of-stars hierarchy for large N ) and maps each user’s contribution to local XdaZdbX_{d}^{a} Z_{d}^{b}Xda​Zdb​ X d a Z d b operations. A commit-then-reveal layer makes inputs binding and hiding, yielding equal influence fairness over Zd\mathbb{Z}_{d}Zd​ Z d with a strict fair-abort guarantee. Composable secrecy is established via entropy bounds and a Devetak-Winter-style key rate analysis, strengthened by test-round statistics; we connect Mermin/Svetlichny-type correlators to min-entropy lower bounds and incorporate full leakage accounting and finite-key penalties. We model depolarizing, shift/phase (Weyl), amplitude-damping, erasures, detector efficiency, and dark counts, and derive closed-form mappings from device parameters to symbol/phase errors. Monte Carlo studies confirm that d>2d>2d>2 d > 2 expands the positive-rate region and reduces the global survival required for a given throughput compared with qubit GHZ MQKA under identical conditions. We discuss potential photonic realizations (time-bin or frequency-bin encoding, discrete Fourier transform networks, and superconducting nanowire detectors) and architectural pathways toward larger N via hierarchical clustering, while acknowledging the significant technological challenges that would need to be overcome to realize high-dimensional multipartite entanglement experimentally. Overall, the scheme delivers higher efficiency per entangled state, rigorous fairness, and composable security with realistic noise and loss, addressing key gaps in current MQKA designs for resilient infrastructure.
Multiparty quantum key agreement (MQKA) enables n≥3n \geq 3n≥3 mutually distrustful users to establish a shared secret key through collaborative quantum protocols. In this paper, we provide a comprehensive review where we argue that MQKA is best understood as a design space organized along three orthogonal but tightly coupled axes: (1) network architecture, which determines how quantum states flow between participants; (2) quantum resources, which encode the physical degrees of freedom used for implementation; and (3) security model, which defines trust assumptions about devices and infrastructure. Rather than treating MQKA as a linear sequence of isolated protocols, we develop this three-axis perspective to reveal recurrent patterns, sharp trade-offs, and unexplored design spaces. We classify MQKA protocols into structural families, map them to underlying quantum resources, and analyze how different security models shape fairness and collusion resistance. We further identify open challenges in composable security frameworks, network native integration, device-independent implementations, and propose a research roadmap toward hybrid-resource, bosonic-code-encoded, and fairness-aware MQKA suitable for the future quantum internet deployments in the post-NISQ era.
Quantum conference key agreement (QCKA) enables multiple spatially separated users to establish a common key with composable information-theoretic security by exploiting multipartite quantum correlations. Greenberger--Horne--Zeilinger (GHZ) states are a natural resource for this task, but their distribution over optical fiber is degraded by transmission loss, polarization decoherence, and detector imperfections. We investigate the finite-key performance of an NNN-party GHZ-based QCKA system using the NNN-BB84 protocol, where each optical link is modeled by extending the bipartite erasure--Pauli channel to incorporate detector dark counts. We characterize the impact of this quantum channel on the shared GHZ state and derive closed-form expressions for the error rates. The analytical results reveal that the local bit-error rate depends only on transmissivity and dark counts, while the global phase-error rate compounds over all N−1N-1N−1 links and therefore degrades faster with both distance and the number of parties. These error rates are integrated into a finite-key secret-key-rate (SKR) framework, in which we optimize the parameter-estimation probability via Brent's derivative-free method. Numerical results show that increasing fiber distance and network size reduce the achievable SKR. Furthermore, the optimized parameter-estimation probability consistently increases the SKR and extends the maximum secure distance relative to a fixed-allocation baseline, with comparison to the asymptotic limit quantifying the residual finite-key penalty.
Future quantum communication networks will conceivably support cryptographic tasks that require entanglement among more than two users. Quantum secret sharing is a prime example where entanglement provides a direct means to coordinate untrusted parties with security from eavesdropping in a multi-party setting. However, the canonical GHZ-based protocols can be vulnerable to participant attacks, in which untrusted parties try to learn the secret without collaborating. Here, we experimentally evaluate a discrete-variable (n,n)(n,n)(n,n)-threshold quantum secret-sharing protocol whose finite-key analysis provides composable security against general attacks, including participant attacks. Using two domain-engineered entangled photon pair sources, we generate 4-qubit GHZ states at rates above 5×1035\times10^35×103 fourfold events per second and a maximum asymptotic secret key rate of 750±10750 \pm 10750±10 bits per second. We then distribute the state through a 4-arm star network comprising 20km of fibre in total. From the measured event rates and error statistics, we infer that a randomised 24-hour execution with the optimised basis probability would yield a composable finite-key lower bound of 8.7 Mbits, under the assumption that the measured source and device statistics remain stationary.
As quantum communication evolves from point-to-point to multi-user networks, existing scalable schemes for multi-party quantum key distribution increasingly reveal their limitations in complex scenarios. These limitations are primarily manifested as a reliance on a central node, which results in insufficient hierarchical control capabilities of the system. Furthermore, existing schemes cannot enforce communication controllability at the quantum physical layer, making authorized communication difficult. To address these challenges, we propose a GHZ-state-based quantum-layer authorized hierarchical multi-user quantum key distribution scheme. By exploiting the global correlation of multipartite entanglement, authorization is mapped onto constraints imposed by managers’ measurement outcomes on receivers’ correlation structures, enabling hierarchically controlled key establishment. Specifically, any two users dynamically act as managers, whose measurements act as entanglement constraints dictating whether remaining users successfully establish a shared secret key. Theoretical analysis demonstrates the scheme guarantees correctness and consistency of key generation. Simulation results further indicate good robustness against typical types of channel noise, enabling stable controlled quantum key distribution within low-noise regimes. This work shifts communication authorization from classical mechanisms at the application layer to the quantum physical layer, providing a promising approach for secure quantum communication in complex networks.
As a burgeoning branch of quantum cryptography, quantum key agreement is a kind of key establishing processes where the security and fairness of the established common key should be guaranteed simultaneously. However, the difficulty on designing a qualified quantum key agreement protocol increases significantly with the increase of the number of the involved participants. Thus far, only few of the existing multiparty quantum key agreement (MQKA) protocols can really achieve security and fairness. Nevertheless, these qualified MQKA protocols are either too inefficient or too impractical. In this paper, an MQKA protocol is proposed with single photons in travelling mode. Since only one eavesdropping detection is needed in the proposed protocol, the qubit efficiency and measurement efficiency of it are higher than those of the existing ones in theory. Compared with the protocols which make use of the entangled states or multi-particle measurements, the proposed protocol is more feasible with the current technologies. Security and fairness analysis shows that the proposed protocol is not only immune to the attacks from external eavesdroppers, but also free from the attacks from internal betrayers.
This paper introduces the design process of a multiparty quantum key agreement protocol based on the Greenberger-Horne-Zeilinger state in detail. Building on the traditional circle-type quantum key agreement protocol, we introduce a star structure, which significantly improves the speed and efficiency of key agreement. To facilitate the reader’s understanding, we provide an example of a four participants quantum key agreement protocol. In the process of quantum state transmission, we perform operations using the Pauli matrix and the Hadamard matrix to ensure that the quantum state remains in one of the four basis states. This significantly enhances the security of the protocol. After rigorous security analysis, we find that the protocol can effectively resist intercept-resend attack, entangle-measure attack, collective attack, and dishonest participant attack. Under a collective attack, if the first particle is subjected to bit-flipping noise, then p<0.2430p<0.2430p<0.2430 only guarantees r>0.2r>0.2r>0.2 when a=1a=1a=1 . Additionally, we conduct a fairness analysis and evaluate the practical performance of the proposed protocol. In an ideal depolarization noise-free environment, the protocol can achieve a positive key rate only when the global detection efficiency exceeds 0.9636. Finally, we conduct a comprehensive comparative analysis of the protocols. The results show that our proposed protocol is superior to other existing schemes in terms of efficiency and running time.