The threshold secret sharing scheme enables a 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 requires 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. Based on the prefix codes, 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 scheme is the first evolving k-threshold secret sharing scheme by generalizing Shamir’s scheme onto a polynomial ring. Besides, the proposed scheme also establishes the connection between prefix codes and the evolving schemes for \(k\ge 2\) . The analysis shows 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 is \((k-1)\lfloor \lg t\rfloor +2(k-1)\lfloor \lg ({\lfloor \lg t\rfloor +1}) \rfloor +\ell \) , 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. Specifically, when \(k=2\) , the proposal also provides a unified mathematical decryption for prior evolving 2-threshold secret sharing schemes and also achieves the minimal share size for a single-bit secret, which is the same as the best-known scheme.