Some Structural and Security Properties of the Modular Secret Sharing

We mainly study the modular secret sharing over the ring of univariate polynomials over the Galois field. The proposed modular schemes over these rings achieve improvements both in security and applicability compared to classical schemes over the ring of integers. Our threshold schemes are perfect i.e. secure in the sense of information theory and ideal i.e. efficient in the terms of memory storage (the size of the secret is equal to the size of the shares). Then we consider the modular secret sharing schemes with pairwise co-prime moduli over the ring of polynomials and the ring of integers. We give a complete description of the access structures that can be realized by such modular schemes.