Welcome to the resource topic for 2025/1499
Title:
A Construction of Evolving k-threshold Secret Sharing Scheme over A Polynomial Ring
Authors: Qi Cheng, Hongru Cao, Sian-Jheng Lin, Nenghai Yu, Yunghsiang S. Han, Xianhong Xie
Abstract: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\geq2. 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.
ePrint: https://eprint.iacr.org/2025/1499
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