A New Metric and Its Scheme Construction for Evolving $2$-Threshold Secret Sharing Schemes

Evolving secret sharing schemes do not require prior knowledge of the number of parties $n$ and $n$ may be infinitely countable. It is known that the evolving $2$-threshold secret sharing scheme and prefix coding of integers have a one-to-one correspondence. However, it is not known what prefix coding of integers to use to construct the scheme better. In this paper, we propose a new metric $K_{\Sigma}$ for evolving $2$-threshold secret sharing schemes $\Sigma$. We prove that the metric $K_{\Sigma}\geq 1.5$ and construct a new prefix coding of integers, termed $\lambda$ code, to achieve the metric $K_{\Lambda}=1.59375$. Thus, it is proved that the range of the metric $K_{\Sigma}$ for the optimal $(2,\infty)$-threshold secret sharing scheme is $1.5\leq K_{\Sigma}\leq1.59375$. In addition, the reachable lower bound of the sum of share sizes for $(2,n)$-threshold secret sharing schemes is proved.