A new efficient hierarchical multi-secret sharing scheme based on linear homogeneous recurrence relations

Abstract

Hierarchical secret sharing is an important key management technique since it is specially customized for hierarchical organizations with different departments allocated with different privileges, such as the government agencies or companies. Hierarchical access structures have been widely adopted in secret sharing schemes, where efficiency is the primary consideration for various applications. How to design an efficient hierarchical secret sharing scheme is an important issue. A famous hierarchical secret sharing (HSS) scheme was proposed by Tassa based on Birkhoff interpolation. Later, based on the same method, many other HSS schemes were proposed. However, these schemes all depend on Polya’s condition, which is a necessary condition, not a sufficient condition. It cannot guarantee that Tassa’s HSS scheme always exists. Furthermore, this condition needs to check the non-singularity of many matrices.We propose a hierarchical multi-secret sharing scheme based on the linear homogeneous recurrence (LHR) relations and the one-way function. In our scheme, we select m linearly independent homogeneous recurrence relations. The participants in the highly-ranked subsets γ1,γ2,⋯,γj-1 join in the jth subset to construct the jth LHR relation. In addition, the proposed hierarchical multi-secret sharing scheme just requires one share for each participant. Besides, our scheme is both perfect and ideal. Furthermore, our scheme avoids many checks of the non-singularity of many matrices in the presented hierarchical secret sharing schemes. Although we need to publish more public values, our scheme reduces the computational complexity of the hierarchical secret sharing schemes from exponential time to polynomial time, i.e., O(nkm-1logn), which is relatively more efficient than schemes in the literature.