  Quantum secret sharing allows each player to have classical information for
secret sharing in quantum mechanical ways. In this work, we construct a class
of quantum states on which players can quantumly perform secret sharing secure
against dishonest players as well as eavesdropper. We here call them the
genuine secret-sharing states. In addition, we show that if $N$ players share
an $N$-party genuine secret-sharing state, then arbitrary $M$ players out of
the total players can share an $M$-party genuine secret-sharing state by means
of local operations and classical communication on the state. We also define
the distillable rate with respect to the genuine secret-sharing state, and
explain the connection between the distillable rate and the relative entropy of
entanglement.
