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

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.