Abstract In this paper, we consider to use the quantum stabilizer codes as secret sharing schemes for classical secrets. We give necessary and sufficient conditions for qualified and forbidden sets in terms of quantum stabilizers. Then, we give a Gilbert–Varshamov-type sufficient condition for existence of secret sharing schemes with given parameters, and by using that sufficient condition, we show that roughly 19% of participants can be made forbidden independently of the size of classical secret, in particular when an n -bit classical secret is shared among n participants having 1-qubit share each. We also consider how much information is obtained by an intermediate set and express that amount of information in terms of quantum stabilizers. All the results are stated in terms of linear spaces over finite fields associated with the quantum stabilizers.
Abstract We improve the flexibility in designing access structures of quantum stabilizer-based secret sharing schemes for classical secrets, by introducing message randomization in their encoding procedures. We generalize the Gilbert–Varshamov bound for deterministic encoding to randomized encoding of classical secrets. We also provide an explicit example of a ramp secret sharing scheme with which multiple symbols in its classical secret are revealed to an intermediate set, and justify the necessity of incorporating strong security criterion of conventional secret sharing. Finally, we propose an explicit construction of strongly secure ramp secret sharing scheme by quantum stabilizers, which can support twice as large classical secrets as the McEliece–Sarwate strongly secure ramp secret sharing scheme of the same share size and the access structure.
We investigate quantum secret-sharing schemes constructed from ${[[n,k,\ensuremath{\delta}]]}_{D}$ nonbinary stabilizer quantum error-correcting codes with carrier qudits of prime dimension
D. We provide a systematic way of determining the access structure, which completely determines the forbidden and intermediate structures. We then show that the information available to the intermediate structure can be fully described and quantified by what we call the information group, a subgroup of the Pauli group of
k qudits, and we employ this group structure to construct a method for hiding the information from the intermediate structure via twirling of the information group and sharing of classical bits between the dealer and the players. Our scheme allows for the transformation of a ramp (intermediate) quantum secret-sharing scheme into a semiquantum perfect secret-sharing scheme with the same access structure as the ramp one but without any intermediate subsets, and is optimal in the amount of classical bits the dealer has to distribute.
Quantum secret sharing is a scheme for encoding a quantum state (the secret) into multiple shares and distributing them among several participants. If a sufficient number of shares are put together, then the secret can be fully reconstructed. If an insufficient number of shares are put together however, no information about the secret can be revealed. In quantum ramp secret sharing, partial information about the secret is allowed to leak to a set of participants, called an unqualified set, that cannot fully reconstruct the secret. By allowing this, the size of a share can be drastically reduced. This paper introduces a quantum analog of classical strong security in ramp secret sharing schemes. While the ramp secret sharing scheme still leaks partial information about the secret to unqualified sets of participants, the strong security condition ensures that qudits with critical information can no longer be leaked.
In quantum secret sharing, a quantum secret state is mapped to multiple shares such that shares from qualified sets can recover the secret state and shares from other forbidden sets reveal nothing about the secret state; we study the setting where there are both classical shares and quantum shares. We show that the quantum secret sharing problem with both classical and quantum shares is feasible if and only if any two qualified sets have some quantum share in common. Next, for threshold quantum secret sharing where there are
N1 classical shares,
N2 quantum shares and qualified sets consist of any
K1 (or more) classical shares and any
K2>N2/2 (or more) quantum shares, we show that to share
1 qubit secret, each classical share needs to be at least
2 bits and each quantum share needs to be at least
1 qubit. Finally, we characterize the minimum share sizes for quantum secret sharing with at most
2 classical shares and at most
2 quantum shares. The converse proofs rely on quantum information inequalities and the achievable schemes use classical secret sharing, (encrypted) quantum secret sharing with only quantum shares, superdense coding, treating quantum digits as classical digits, and their various combinations.