Paper 2018/143

Conjecturally Superpolynomial Lower Bound for Share Size

Shahram Khazaei

Abstract

Information ratio, which measures the maximum/average share size per shared bit, is a criterion of efficiency of a secret sharing scheme. It is generally believed that there exists a family of access structures such that the information ratio of any secret sharing scheme realizing it is $2^{\Omega(n)}$, where the parameter $n$ stands for the number of participants. The best known lower bound, due to Csirmaz (1994), is $\Omega(n/\log n)$. Closing this gap is a long-standing open problem in cryptology. In this paper, using a technique called \emph{substitution}, we recursively construct a family of access structures having information ratio $n^{\frac{\log n}{\log \log n}}$, assuming a well-stated information-theoretic conjecture is true. Our conjecture emerges after introducing the notion of \emph{convec set} for an access structure, a subset of $n$-dimensional real space. We prove some topological properties about convec sets and raise several open problems.