  In addition to secret splitting, secret reconstruction is another important
component of secret sharing. In this paper, the first quantum secret
reconstruction protocol based on cluster states is proposed. Before the
protocol, a classical secret is divided into multiple shares, which are
distributed among shareholders via secret splitting. In the protocol, the
dealer utilizes her secret to encrypt a private quantum state, and sends the
encrypted state to a combiner chosen by her from the shareholders. With the
help of other shareholders, the combiner utilizes the properties of cluster
states to recover the privacy quantum state. It is shown that the proposed
protocol is secure against several common attacks, including external and
internal attacks. Compared with classical secret reconstruction protocols, this
protocol not only achieves theoretical security of all shares, but also is more
efficient due to reducing the distribution cost and computation cost. To
demonstrate the feasibility of the protocol, a corresponding simulation quantum
experiment is conducted on the IBM Q platform. Furthermore, in conjunction with
quantum fingerprinting, it can be directly applied to achieve the task of
multiple secrets sharing, because the classical shares can be reused in the
proposed protocol.
