Abstract: Aiming at the problems of the insufficient security and the limited application areas in the threshold scheme based on univariate Lagrange interpolation polynomial, by studying existing threshold schemes and the bivariate Lagrange interpolation theory in the real field, on the basis of finite fields, a threshold scheme based on bivariate Lagrange interpolation polynomial is proposed. The structure and the numerical example of the scheme are given, which proves the rationality and feasibility of the scheme. Finally, the scheme is compared with the threshold scheme based on univariate Lagrange interpolation polynomial. The analysis shows that the loss caused by the loss of sub-secrets is lower, the difficulty of collusion is bigger in the new scheme, so the security of the scheme is higher. At the same time, this scheme can expand the application area of threshold schemes.

Key words: finite fields, bivariate Lagrange interpolation polynomial, secret sharing, threshold scheme, rectangular points mesh

摘要： 针对基于一元Lagrange插值多项式的门限方案中存在的安全性不足及应用领域受限问题，通过研究现有的门限方案和实数域上的二元Lagrange插值理论，在有限域的基础上，提出一种基于二元Lagrange插值多项式的门限方案。给出了方案的构造及其数值算例，证明了方案的合理性和可行性。将该方案与基于一元Lagrange插值多项式的门限方案进行对比分析，表明新的方案中子秘密丢失所造成的损失更低、合谋难度更大，方案的安全性更高。同时，该方案可以拓宽门限方案的应用领域。

关键词: 有限域, 二元Lagrange插值多项式, 秘密共享, 门限方案, 矩形网点