Paper 2026/1939

Extendable Weighted Evolving Threshold Secret Sharing Schemes over Polynomial Quotient Rings

Abstract

In conventional $(k,n)$ secret sharing schemes, a dealer distributes shares of a secret among $n$ participants such that any subset of at least $k$ participants can reconstruct the secret. While such schemes often fail to accommodate practical scenarios in which participants have different management permissions over the secret. To address this problem, weighted $(t,n)$ secret sharing was introduced. In this case, each participant is assigned a specific weight, and the dealer distributes a share to each participant according to their weight, so that any subset of participants whose sum of weights is at least $t$ can recover the secret. However, in weighted $(t,n)$ secret sharing schemes, the number of participants $n$ is known in advance. When $n$ is uncertain and even grows over time, traditional weighted $(t,n)$ secret sharing schemes remain inadequate in such dynamic environments. To solve the problem, we propose the concept of a weighted evolving $t$-threshold secret sharing. Based on the prefix codes, we first propose a construction of a weighted evolving $3$-threshold scheme over the polynomial quotient ring for an $\ell$-bit secret. And then we extend the framework to a general weighted evolving $t$-threshold scheme for any $t\geq 3$. Moreover, we prove the correctness and security of the proposed scheme by leveraging the properties of the polynomial quotient ring and prefix codes. Finally, we analyze the corresponding share size. The result shows that for the $m$-th participant with the weight $w_m$ satisfying $1\leq w_m\leq t-1$, the size of the corresponding share is $w_m(t-1-\frac{w_m-1}{2})(\ell_m-1)+w_m\ell$ bits, where $\ell_m$ denotes the length of a binary prefix code of encoding integer $m$. In contrast, a baseline scheme that allocates $w_m$ independent full evolving shares to this participant would require $w_m(t-1)(\ell_m-1)+w_m\ell$ bits. Our proposed scheme saves $\frac{(w_m-1)w_m}{2}(\ell_m-1)$ bits, thereby achieving a smaller share size.