Access structures determined by uniform polymatroids

Abstract

Abstract In this article, all multipartite access structures obtained from uniform integer polymatroids were investigated using the method developed by Farràs, Martí-Farré, and Padró. They are matroid ports, i.e., they satisfy the necessary condition to be ideal. Moreover, each uniform integer polymatroid defines some ideal access structures. Some objects in this family can be useful for the applications of secret sharing. The method presented in this article is universal and can be continued with other classes of polymatroids in further similar studies. Here, we are especially interested in hierarchy of participants determined by the access structure, and we distinguish two main classes: they are compartmented and hierarchical access structures. The main results obtained for access structures determined by uniform integer polymatroids and a monotone increasing family Δ \Delta can be summarized as follows. If the increment sequence of the polymatroid is non-constant, then the access structure is connected. If Δ \Delta does not contain any singletons or the height of the polymatroid is maximal and its increment sequence is not constant starting from the second element, then the access structure is compartmented. If Δ \Delta is generated by a singleton or the increment sequence of the polymatroid is constant starting from the second element, then the obtained access structures are hierarchical. They are proven to be ideal, and their hierarchical orders are completely determined. Moreover, if the increment sequence of the polymatroid is constant and ∣ Δ ∣ > 1 | \Delta | \gt 1 , then the hierarchical order is not antisymmetric, i.e., some different blocks are equivalent. The hierarchical order of access structures obtained from uniform integer polymatroids is always flat, that is, every hierarchy chain has at most two elements.