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. An evolving k-threshold secret sharing scheme is recognized as a good scheme when it possesses the perfect security and a smaller share size. In this paper, we consider the evolving secret sharing scheme with the threshold \(k=3\) . First, we point out that the evolving 3-threshold secret sharing scheme proposed in D’Arco et al. (Theor Comput Sci 859:149–161, 2021) does not possess the perfect security. To solve this issue, we then present a revised version that attains perfect security. We follow the overall construction framework of D’Arco et al. (2021), which combines a normal evolving 3-threshold scheme and a 3-threshold scheme. Within this same framework, we adopt the scheme proposed in Cheng et al. (in: ASIACRYPT 2025, 2025) as the required normal evolving 3-threshold scheme and design a new 3-threshold secret sharing scheme with perfect security over an extension field as the required 3-threshold scheme, which serves as the core component of the entire evolving 3-threshold scheme. Finally, by analyzing, the t-th share size of the evolving 3-threshold scheme is \({\lg t+O( \lg \lceil \sqrt{\lg t} \rceil )+2\lceil \sqrt{\lg t} \rceil (\ell +1)-\ell +1}\) for an \(\ell \) -bit secret, where \(\lg \) denotes the binary logarithm. By comparing, for known evolving 3-threshold schemes’ share size, the most important term is \(2\lg t\) , the scheme reduces the most important term from \(2\lg t\) to \(\lg t\) . Thus, the proposed scheme can achieve perfect security and a smaller share size.
