https://doi.org/10.1140/epjqt/s40507-026-00510-1
Research
High-dimensional GHZ-based multi-party quantum key agreement: rigorous security, fairness, and loss tolerance
1
Pak-Austria Fachhochschule Institute of Applied Sciences and Technology, 22620, Haripur, Pakistan
2
Faculty of Computer Science and Information Technology, Universiti Malaya, 50603, Kuala Lumpur, Malaysia
3
School of Computing Science, Taylor’s University, 47500, Selangor, Malaysia
4
Faculty of Data Science and Information Technology, INTI International University, 71800, Nilai, Malaysia
5
Fakultas Teknik, Universitas Negeri Padang, 25132, Padang, Indonesia
6
Department of Computer Sciences, King Faisal University, 31982, Al-Ahsa, Saudi Arabia
7
Faculty of Engineering and Technology, Shinawatra University, 12160, Pathum Thani, Thailand
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
b
This email address is being protected from spambots. You need JavaScript enabled to view it.
c
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
4
January
2026
Accepted:
10
April
2026
Published online:
23
April
2026
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 
operations. A commit-then-reveal layer makes inputs binding and hiding, yielding equal influence fairness over 
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 
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.
Key words: Multi-party quantum key agreement / GHZ states / High-dimensional qudits / Fairness / Composable security / Resilient infrastructure / Loss tolerance
© The Author(s) 2026
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
