Secret Sharing on Superconcentrator

Abstract: Using information inequalities, we prove any unrestricted arithmetic circuits computing the shares of any $(t, n)$-threshold secret sharing scheme must satisfy some superconcentrator-like connection properties. In the reverse direction, we prove, when the underlying field is large enough, any graph satisfying these connection properties can be turned into a linear arithmetic circuit computing the shares of a $(t, n)$-threshold secret sharing scheme. Specifically, $n$ shares can be computed by a linear arithmetic circuits with $O(n)$ wires in depth $O(\alpha(t, n))$, where $\alpha(t, n)$ is the two-parameter version of the inverse Ackermann function. For example, when $n \ge t^{2.5}$, depth $2$ would be enough; when $n \ge t \log^{2.5} t$, depth 3 would be enough.