A Construction of Evolving $3$-threshold Secret Sharing Scheme with Perfect Security and Smaller Share Size

The evolving $k$-threshold secret sharing scheme allows the dealer to distribute the secret to many participants such that only no less than $k$ shares together can restore the secret. In contrast to the conventional secret sharing scheme, the evolving scheme allows the number of participants to be uncertain and even ever-growing. In this paper, we consider the evolving secret sharing scheme with $k=3$. First, we point out that the prior approach has risks in the security. To solve this issue, we then propose a new evolving $3$-threshold scheme with perfect security. Given a $\ell$-bit secret, the $t$-th share of the proposed scheme has $\lceil\log_2 t\rceil +O({\lceil \log_4 \log_2 t\rceil}^2)+\log_2 p(2\lceil \log_4 \log_2 t\rceil-1)$ bits, where $p$ is a prime. Compared with the prior result $2 \lfloor\log_2 t\rfloor+O(\lfloor\log_2 t\rfloor)+\ell$, the proposed scheme reduces the leading constant from $2$ to $1$. Finally, we propose a conventional $3$-threshold secret sharing scheme over a finite field. Based on this model of the revised scheme and the proposed conventional $3$-threshold scheme, we present a brand-new and more concise evolving $3$-threshold secret sharing scheme.