基于有限域上多项式的存在特权集的门限签名研究

Research on Threshold Signature Scheme with Privilege Set Based on Polynomials over Finite Fields

摘要: 针对存在特权集的门限签名中传统双重秘密共享机制存在的密钥分发复杂及安全短板问题，本文提出了一种基于有限域上相似多项式的改进方案。该方案利用Schnorr签名体制，构建了一个仅在常数项上存在差异的主多项式与辅助多项式。通过这种构造，将特权成员与普通成员绑定在同一数学框架下，既实现了特权集成员对签名过程的强制参与，又消除了两组防护水平不一致带来的安全隐患。安全性分析表明，该方案满足信息论安全，优化了密钥分发与管理的复杂度，且能有效抵抗合谋攻击。

Abstract: To address the issues of complex key distribution and the "security short-board effect" inherent in the traditional dual secret sharing mechanism within privilege set threshold signatures, this paper proposes an improved scheme based on similar polynomials over a finite field. Leveraging the Schnorr signature scheme, the proposed method constructs a primary polynomial and an auxiliary polynomial that differ solely in their constant terms. By binding privileged members and ordinary members within a unified mathematical framework, this construction not only enforces the mandatory participation of the privileged set in the signing process but also eliminates the security vulnerabilities arising from inconsistent protection levels between the two groups. Security analysis demonstrates that the proposed scheme achieves information-theoretic security, optimizes the complexity of key distribution and management, and effectively resists collusion attacks.