Fair quantum secret sharing based on symmetric bivariate polynomial

In this paper, we propose a new fair (t,n) threshold quantum secret sharing scheme based on the d-dimensional Bell state and symmetric bivariate polynomial. In the distribution phase, the dealer uses the symmetric bivariate polynomial to encode the secret and produces the corresponding share for each participant. To achieve fairness, we construct a secret sequence, which can guarantee that each participant can recover the correct secret if all participants are legal and honest. In reconstruction phase, the dealer prepares the d-dimensional Bell state, and all participants perform the unitary operations produced by the share of their polynomials on the transmitted particles to reconstruct the secret. Through the sequential communications, the proposed scheme has a good scalability. Furthermore, we consider the situation that these participants cooperate to recover the secret when the number of participants is more than t. At last, we analyze the correctness, security and fairness of the proposed protocol.