Simple Multi-Secret Sharing Schemes to Achieve the Optimal Maximum Improvement Ratio

Secret sharing scheme is a way to distribute a secret among several participants in such a way that only qualified subsets of the participants can reconstruct the secret and unqualified subsets have no information about the secret. A multi-secret sharing scheme is the extension of a secret sharing scheme to the case in which there are many secrets need to be shared, and each secret may with different qualified subsets of participants that can reconstruct the secret. The maximum improvement ratio is used to indicate the efficiency of a multi-secret sharing scheme. In 2001, Crescenzo proved a lower bound of the maximum improvement ratio and propose a multi-secret sharing scheme that can achieve this bound. But this scheme must use several complexity theorems that held in some literatures, and in their scheme, the number of participants must decided by the number of secrets. In this paper, we will propose two simple schemes that not only get the same conclusion directly, but also extend this result to all of the case that for any number of participants and secrets such that the number of participants only need to greater than the number of secrets