A New Metric and the Construction for Evolving 2-Threshold Secret Sharing Schemes Based on Prefix Coding of Integers
Wei Yan, Sian-Jheng Lin, Yunghsiang Sam Han · IEEE Transactions on Communications · 2023
Evolving secret sharing schemes do not require prior knowledge of the number of parties$n$, which 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 unknown what prefix coding of integers should be used to construct a better secret sharing scheme. In this paper, we introduce a metric$K_{\Sigma }$to evaluate evolving 2-threshold secret sharing schemes$\Sigma $such that a smaller$K_{\Sigma }$of a scheme is better. The metric$K_{\Sigma }$is related to the ratio of the sum of the share sizes for the first$n$parties in scheme$\Sigma $and the sum of the share sizes for the optimal$(2,n)$-threshold secret sharing scheme. Then we prove that the metric$K_{\Sigma }\geq 1.5$and construct a new prefix coding of integers, termed$\lambda $code, to achieve the metric$K_{\Lambda }=1.59375$. Thus, this shows that the range of the metric$K_{\Sigma }$for the optimal$(2,\infty)$-threshold secret sharing scheme is$1.5\leq K_{\Sigma }\leq 1.59375$. In addition, an achievable lower bound on the sum of share sizes for$(2,n)$-threshold secret sharing schemes is also provided.