A Construction of Evolving $k$-threshold Secret Sharing Scheme over A Polynomial Ring

The threshold secret sharing scheme allows the dealer to distribute the share to every participant such that the secret is correctly recovered from a certain amount of shares. The traditional $(k, n)$-threshold secret sharing scheme requests that the number of participants $n$ is known in advance. In contrast, the evolving secret sharing scheme allows that $n$ can be uncertain and even ever-growing. In this paper, we consider the evolving secret sharing scenario. Using the prefix codes and the properties of the polynomial ring, we propose a brand-new construction of evolving $k$-threshold secret sharing scheme for an $\ell$-bit secret over a polynomial ring, with correctness and perfect security. The proposed schemes establish the connection between prefix codes and the evolving schemes for $k\geq2$, and are also first evolving $k$-threshold secret sharing schemes by generalizing Shamir's scheme onto a polynomial ring. Specifically, the proposal also provides an unified mathematical decryption for prior evolving $2$-threshold secret sharing schemes. Besides, the analysis of the proposed schemes show that the size of the $t$-th share is $(k-1)(\ell_t-1)+\ell$ bits, where $\ell_t$ denotes the length of a binary prefix code of encoding integer $t$. In particular, when $\delta$ code is chosen as the prefix code, the share size achieves $(k-1)\lfloor\lg t\rfloor+2(k-1)\lfloor\lg ({\lfloor\lg t\rfloor+1}) \rfloor+\ell$, which improves the prior best result $(k-1)\lg t+6k^4\ell\lg{\lg t}\cdot\lg{\lg {\lg t}}+ 7k^4\ell\lg k$, where $\lg$ denotes the binary logarithm. When $k=2$, the proposed scheme also achieves the minimal share size for single-bit secret, which is the same as the best known scheme.