The advent of Cloud-Fog-Edge computing has transformed distributed data processing by performing computation closer to end devices. Due to resource constraints at edge nodes and the dynamic nature of fog-assisted communication, secure and efficient group key distribution and batch verification in such decentralized systems remain a major challenge. Many existing protocols based on Chinese remainder theorem (CRT) use straightforward scalar product with no non-linear transformation to mask the group key and hence fail to provide semantic security. Others suffer from architectural overhead since they require distinct and independent sets of moduli equations with multiple mathematical structures for different network layers, which increases computing overhead, limits scalability and synchronization delays during frequent node leave/join. To mitigate these challenges, this paper proposes a unified distributed CRT-based protocol for Cloud-Fog-Edge environments. Our protocol introduces a non-linear two-factor encryption logic, by incorporating a unique secret parameter for every edge node and breaks the algebraic linearity of the CRT. Additionally, our protocol uses a single set of moduli equations across Cloud-Fog-Edge networks which drastically reduces computation and storage costs at the fog layer. Our protocol achieves O(1) efficiency for rekeying. Formal security analysis using ProVerif and ROR model demonstrates that our protocol has considerable security advantages. To prove its practicality, an ESP32-based simulation on Wokwi is used to verifies the correctness of group key distribution, retrieval, and batch message verification. The performance analysis findings show that our protocol outperforms others in computation cost, communication cost, security and applicability for resource-constrained Cloud-Fog-Edge computing networks.

The advent of cloud–fog–edge computing has transformed distributed data processing by performing computation closer to end devices. Due to resource constraints at edge nodes and the dynamic nature of fog-assisted communication, secure and efficient group key distribution and batch verification in such decentralized systems remain a major challenge. Many existing protocols based on Chinese remainder theorem (CRT) use a straightforward scalar product to mask the group key and hence fail in multifactor security. Others suffer from architectural overhead since they require distinct and independent sets of moduli equations with multiple mathematical structures for different network layers, which increases computing overhead, limits scalability and delays synchronization during frequent node leave/join. To mitigate these challenges, this paper proposes a unified distributed CRT-based protocol for cloud–fog–edge environments. Our protocol introduces a two-factor modular key masking mechanism by incorporating a unique secret parameter for every edge node to strengthen group key protection and enhance the overall robustness of the key distribution mechanism. Additionally, our protocol uses a single set of moduli equations across cloud–fog–edge networks, which drastically reduces computation and storage costs at the fog layer. Our protocol achieves $\mathcal{O}$ (1) efficiency for rekeying. Formal security analysis using ProVerif and the ROR model demonstrates that our protocol has considerable security advantages. To prove its practicality, an ESP32-based simulation on Wokwi is used to verify the correctness of group key distribution, retrieval, and batch message verification. The performance analysis findings show that our protocol outperforms others in computation cost, communication cost, security and applicability for resource-constrained cloud–fog–edge computing networks.