Secret Sharing on Superconcentrator

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.