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

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