A New Multi-stage Secret Sharing Scheme for Hierarchical Access Structure with Existential Quantifier

Abstract

Multi-stage secret sharing scheme is practical in the case that there is a security system with m ordered checkpoints.
It is natural to divide the m checkpoints into m different levels. There are m different secrets, and each
of them with a different importance corresponds to a checkpoint/level. The participants are also divided into
m disjoint levels as they do in the hierarchical threshold access structure. Hierarchical threshold access structure
with the existential quantifier ( HTAS∃ ) does not cover the common practice that at least a few numbers
of high-ranking participants are required to be involved in any recovery of the secret. The popular schemes
with hierarchical access structure were needed to check many matrices for non-singularity. We propose a
multi-stage secret sharing scheme for HTAS∃ , and the tools are based on the linear homogeneous recurrence
relations (LHRRs) and one-way functions. We give the HTAS∃ a modification, so that this hierarchical access
structure can satisfy the common practice. In our scheme, if the participants are divided into m levels, there
usually has m secrets. But before the (j − 1)-th secret is recovered, the j-th secret cannot be recovered. Our
scheme is a computational secure. The proposed scheme requires a share for each participant and the share
is as long as each secret. Our scheme has high efficiency by comparing with the state-of-the-art hierarchical
secret sharing schemes.