Secret Sharing on Superconcentrator

TL;DR

This paper characterizes the graph-theoretic properties of arithmetic circuits for threshold secret sharing, establishing that such circuits must be superconcentrator-like, and derives bounds on their complexity.

Contribution

It introduces a graph-theoretic characterization of secret sharing schemes via superconcentrator properties and connects these to circuit complexity bounds.

Findings

Circuits must satisfy superconcentrator connectivity properties.

Any such graph can be transformed into a linear arithmetic circuit for secret sharing.

Derived bounds on the arithmetic circuit complexity of secret sharing schemes.

Abstract

We study the arithmetic circuit complexity of threshold secret sharing schemes by characterizing the graph-theoretic properties of arithmetic circuits that compute the shares. Using information inequalities, we prove that any unrestricted arithmetic circuit (with arbitrary gates and unbounded fan-in) computing the shares must satisfy superconcentrator-like connectivity properties. Specifically, when the inputs consist of the secret and $t-1$ random elements, and the outputs are the $n$ shares of a $(t, n)$-threshold secret sharing scheme, the circuit graph must be a $(t, n)$-concentrator; moreover, after removing the secret input, the remaining graph is a $(t-1, n)$-concentrator. Conversely, we show that any graph satisfying these properties can be transformed into a linear arithmetic circuit computing the shares of a threshold secret sharing scheme, assuming a sufficiently large field. As a consequence, we derive upper and lower bounds on the arithmetic circuit complexity of computing the shares in threshold secret sharing schemes.