Multi-secret sharing schemes based on finite geometry over rings and modules

Secret sharing schemes have been proven to be useful in various fields of interest such as cryptologic key management and multi-party secure protocols. Recent developments show the importance of what is called multiple-secret sharing schemes. This paper gives a new and order theoretic approach to this topic. Its advantage is the capability of modelling implications in a set of involved secrets. We introduce a concept of homogeneity and adapt classical notions like that of information rate and perfectness to the generalized context. Our main results are conceptual and furthermore yield constructions of large classes of homogeneous and perfect control schemes using unital modules and their automorphism groups.