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A DC programming approach for solving a centralized group key management problem

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Abstract

A single trusted entity known as a Key Server is in charge of key generation, distribution, and management in centralized key management schemes. To prevent eavesdropping and protect the exchange content, a group key is used to encrypt the group communication. This management mechanism is typically based on a binary tree structure. When the membership of a group changes dynamically, the group key must be changed, triggering a certain updated cost. This paper addresses an important problem in centralized dynamic group key management. It consists in finding a set of leaf nodes in a binary key tree to insert new members with minimal insertion cost and keeping the tree as balanced as possible. The two mentioned important objectives are combined into a unified (deterministic) optimization model whose objective function contains discontinuous step functions with binary variables, which is known to be NP-hard. We then reformulate the problem as a combinatorial optimization program with continuous objective by introducing new binary variables. Applying penalty techniques, it results in a standard DC (Difference of Convex functions) program that can be solved efficiently by DCA (DC algorithm). Besides, the insertion nodes must be the leaf nodes, we introduce a two-step algorithm to reduce the model complexity: the first is to find the set of leaf nodes, while the second is to solve the simplified optimization problem. Numerical experiments have been studied intensively to justify the merit of our proposed model as well as the corresponding DCA.

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Data availability

The datasets generated during and/or analysed during the current study are not publicly available but are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and constructive suggestions, which have greatly helped to improve the presentation of this paper.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. LGIPM, Département IA, Université de Lorraine, F-57000, Metz, France

    Hoai An Le Thi, Thi Tuyet Trinh Nguyen & Hoang Phuc Hau Luu

  2. Institut Universitaire de France (IUF), Paris, France

    Hoai An Le Thi

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  1. Hoai An Le Thi
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  2. Thi Tuyet Trinh Nguyen
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Contributions

HALT: Conceptualization, Methodology, Validation, Review & Editing, Supervision, Visualization. TTTN: Conceptualization, Data preparation, Methodology, Writing—Original Draft & Editing, Investigation, Numerical Experiments. HPHL: Methodology, Writing—Original Draft & Editing, Validation.

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Correspondence to Hoang Phuc Hau Luu.

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Le Thi, H.A., Nguyen, T.T.T. & Luu, H.P.H. A DC programming approach for solving a centralized group key management problem. J Comb Optim 44, 3165–3193 (2022). https://doi.org/10.1007/s10878-022-00862-1

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  • DOI: https://doi.org/10.1007/s10878-022-00862-1

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