Abstract: Based on symmetric bivariate polynomial, this paper proposes a new fair [(t,n)] threshold secret sharing scheme, which can guarantee that each participant can recover the correct secret if all participants are legal and honest; when there is a cheater, all participants are unable to recover the correct secret. In the proposed scheme, a symmetric bivariate polynomial is used to generate session key for any two participants; moreover, combined with discrete logarithm, the symmetric bivariate polynomial enables the scheme to choose a sufficiently long sequence of secrets to guarantee the fairness while each participant holds a small number of shares. Additionally, the scheme also can achieve fair secret recovery in asynchronous environment. Compared with Harn’s scheme, the proposed scheme is fairer and more flexible.

Key words: secret sharing, cheater, symmetric bivariate polynomial, fairness

摘要： 基于二元对称多项式，提出一种新的公平[(t,n)]门限秘密共享方案，能够确保：所有参与者都合法且诚实时，均能恢复正确的秘密；存在欺骗者时，所有参与者都无法恢复正确的秘密。该方案利用二元对称多项式不仅为任意两个参与者提供会话密钥;结合离散对数，在确保每个share持有者拥有较少share的情况下，使得Dealer可以选取足够长的秘密序列，从而确保方案的公平性。此外，方案在异步环境下也能实现公平秘密恢复。与Harn方案相比，该方案更加公平和灵活。

关键词: 秘密共享, 欺骗者, 二元对称多项式, 公平性