Optimal Uniform Secret Sharing

An important problem in secret sharing schemes is minimizing the share size. For ( $k$ , $n$ )-threshold schemes and ( $k$ , $L$ , $n$ )-ramp schemes, constructions that minimize the share size are known. This paper presents optimal constructions for a more general class of access structures in which subsets with the same cardinality have the same amount of information about the secret. We refer to schemes with such uniform access structures as uniform secret sharing. We first derive a tight lower bound for share entropy and then present an optimal construction. Our lower bound exceeds that previously reported. The optimal construction encodes the secret value using one or more ramp schemes.