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.

翻译：利用信息不等式，我们证明任何计算$(t, n)$-门限秘密共享方案份额的不受限算术电路必须满足某种类似超集中器的连接性质。反之，我们证明，当底层域足够大时，任何满足这些连接性质的图均可转化为计算$(t, n)$-门限秘密共享方案份额的线性算术电路。具体而言，$n$个份额可通过一个具有$O(n)$条线、深度为$O(\alpha(t, n))$的线性算术电路计算，其中$\alpha(t, n)$是逆Ackermann函数的双参数版本。例如，当$n \ge t^{2.5}$时，深度为2即可满足要求；当$n \ge t \log^{2.5} t$时，深度为3即可满足要求。