A New Metric and the Construction for Evolving 2-Threshold Secret Sharing Schemes Based on Prefix Coding of Integers

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.