General Results on Secret Sharing Based on General Access Structure

For secret sharing, current researches mainly focus on perfect access structures with a very limited number of access subsets, where each subset is either a qualified set or a forbidden set and no semi-access subset exists, as well as on the shares bounds under a uniform distribution, where the number of the bits required by a share is used as the measurement of the bounds. Therefore, the research results are inevitably limited to some extent. Based on general access structures, some generalized information-theoretic results that are suitable for both perfect and non-perfect access structures with an unlimited number of access subsets identified by qualified, forbidden or semi-access are presented in this paper. These results are the general conclusions of many current related works and can be used as the basis for further researches. Meantime, using the information entropy of a share as the measurement of the bounds, some generalized bounds that are suitable for all shares and bounds that are suitable only for particular shares are given too. The bounds are also the generalization of many current related results under arbitrary probability distributions. Some of the bounds are tighter than those well-known ones. Additionally, with the help of the above new generalized results, some potential results can be easily deduced and the proof for many well-known results can be easier and more concise.