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Optimal Shadow Allocations of Secret Sharing Schemes Arisen from the Dynamic Coloring of Extended Neighborhood Coronas

Author

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  • Raúl M. Falcón

    (Department Applied Mathematics I, School of Architecture, Universidad de Sevilla, 41012 Sevilla, Spain)

  • Nagaraj Mohanapriya

    (PG and Research Department of Mathematics, Kongunadu Arts and Science College, Coimbatore 641029, Tamil Nadu, India)

  • Venkitachalam Aparna

    (PG and Research Department of Mathematics, Kongunadu Arts and Science College, Coimbatore 641029, Tamil Nadu, India)

Abstract

Every t -dynamic proper n -coloring of a graph G describes a shadow allocation of any ( n , t + 1 ) -threshold secret sharing scheme based on G , so that, after just one round of communication, each participant can either reconstruct the secret, or obtain a different shadow from each one of his/her neighbors. Thus, for just one round of communication, this scheme is fair if and only if the threshold is either less than or equal to the minimum degree of G , or greater than or equal to its maximum degree. Despite that the dynamic coloring problem has widely been dealt with in the literature, a comprehensive study concerning this implementation in cryptography is still required. This paper delves into this topic by focusing on the use of extended neighborhood coronas for modeling communication networks whose average path lengths are small even after an asymptotic growth of their center and/or outer graphs. Particularly, the dynamic coloring problem is solved for any extended neighborhood corona with center path or star, for which we establish optimal shadow allocations of any (fair) threshold secret sharing scheme based on them. Some bounds are also established for the dynamic chromatic number of any extended neighborhood corona.

Suggested Citation

  • Raúl M. Falcón & Nagaraj Mohanapriya & Venkitachalam Aparna, 2022. "Optimal Shadow Allocations of Secret Sharing Schemes Arisen from the Dynamic Coloring of Extended Neighborhood Coronas," Mathematics, MDPI, vol. 10(12), pages 1-14, June.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2018-:d:836545
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    References listed on IDEAS

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    1. R. Rajkumar & S. Muthuraman, 2021. "Structural and spectral properties of the generalized corona networks," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-30, April.
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    3. Ramon Ferrer i Cancho & Christiaan Janssen & Ricard V. Solé, 2001. "The Topology of Technology Graphs: Small World Patterns in Electronic Circuits," Working Papers 01-05-029, Santa Fe Institute.
    4. Zhang, Xiaoyu & Xu, Maochao & Da, Gaofeng & Zhao, Peng, 2021. "Ensuring confidentiality and availability of sensitive data over a network system under cyber threats," Reliability Engineering and System Safety, Elsevier, vol. 214(C).
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