24 M.-M. Wang et al.
highly sensitive information. In their secret sharing scheme, a dealer can split a
secret into several pieces and send these pieces to different participants in such
a way that any qualified participants set can reconstruct the secret coopera-
tively, but any unqualified participants set cannot get any information about
the secret. As an important area of quantum cryptograph, quantum secret shar-
ing (QSS) [17–19] has also been developed to achieve high-level security than
their classical counterparts [15,16].
The first QSS scheme was proposed by Hillary et al. in 1999 based on entan-
gled Greenberger-Horne-Zeilinger (GHZ) state [17], which will be called as the
HBB-QSS protocol hereafter. In the well-known HBB-QSS scheme, the dealer
Alice can create a secret for two players Bob and Charlie in such a way that the
secret can only be recovered if Bob and Charlie work together, but one of them
cannot get anything useful about the secret. A GHZ triplet is shared among
three participants as quantum resource. Each of them measures their own qubit
either in the Pauli X or Y basis, randomly. In half of the chance, their measure-
ment results will be correlated, which means their measurement results could
be used for secret generating or eavesdropping detection. The quantum resource
efficiency of the HBB-QSS scheme is only 50% and half of the entangled GHZ
states have to be discarded.
Besides of entanglement-based QSS protocols, QSS protocols without entan-
glement have also been developed [20,21]. According to the secret information
shared among players, QSS can be divided into two types, i.e., QSS proto-
cols for sharing classical information and QSS protocols for sharing quantum
states [22,23], which is also known as quantum state sharing (QSTS) [24]. It
should be mentioned that a QSTS protocol has also been proposed in Ref. [17]
Since the appearance of the HBB-QSS scheme, multiparty QSS has been
developed based on entangled GHZ states. In Ref. [25], Xiao et al. general-
ized the HBB-QSS scheme into arbitrary multiparties. To improve the efficiency
of the HBB-QSS scheme, they also proposed two methods, i.e., the favored-
measuring-basis and the measuring-basis-encrypted, respectively. However, some
per-shared keys [25] or different probability parameters are required in these
methods. Furthermore, Yu et al. further extended [26] the HBB-QSS scheme
in high-dimensional system. By using mutually unbiased or biased bases, their
scheme is secure against the intercept-resend attack. The efficiency of the scheme
in high-dimensional system is
1
d
, where d is the dimension of the quantum system,
which is similar to HBB-QSS scheme.
In this paper, we study the problem of how to improve the efficiency of HBB-
QSS scheme for sharing classical information. By using a simple measurement-
delay strategy, we present two improved versions of QSS schemes based GHZ
states, i.e. multiparty QSS and high-dimensional QSS. The quantum resource
efficiency of these two proposed schemes can be increased to 100% rather than
50% in [17], asymptotically 100% in [25], or
1
d
in [26]. The organization of the
paper is as follows. In Sect. 2, we propose the improved multiparty QSS scheme.
Then we extend the idea to high-dimension situation in Sect. 3. The paper is
discussed and concluded in Sect. 4.