Abstract
Various studies have focused on secret sharing schemes and in all of them, each shareholder is interested in a shorter share. The information rate of a secret sharing scheme shows the ratio between the size of the secret to the maximum number of shares given to each shareholder. In this regard, the researchers investigated the optimal information rate of the graph access structure. This paper aims to discover the exact values for the optimal information rates of the two graph access structures, which remained as open problems in Van Dijk’s paper. Furthermore, we introduced the developed multipartite graph and the developed cycle graph and calculated the exact value of their optimal information rate. Moreover, we presented a lower bound on the information rate for other developed graph access structures.
Highlights
The secret sharing scheme represents a method of distributing secrets between shareholders
The information rate is an important indicator to measure the performance of the secret sharing scheme
It is expressed as the ratio of the width of the secret to the maximum share width provided to shareholders
Summary
The secret sharing scheme represents a method of distributing secrets between shareholders. The access structure Γ represents the set of all approved subsets of shareholders that can resume the secret. The set of the minimal approved subsets of shareholders represents the basic access structure, which is called Γ0. The information rate is an important indicator to measure the performance of the secret sharing scheme. It is expressed as the ratio of the width of the secret to the maximum share width provided to shareholders. Song considered the case of a graph access structure with nine vertices [10]. Van Dijk considered all 112 graph access structures with six vertices, but only proved the exact value of the information rate in 94 cases [12]. There is insufficient exactness for nine of these access structures
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