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arXiv:2404.15646

Advance Sharing with Ogawa et al.'s Ramp Quantum Secret Sharing Scheme

Satoshi Masumori, Ryutaroh Matsumoto

Published Apr 24, 2024• 0 citations
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Abstract

The ramp quantum secret sharing proposed by Ogawa et al. has the highest possible coding rate given a threshold type access structure. On the other hand, in some quantum secret sharing schemes, it is known that some shares can be distributed to participants before a secret is given to the dealer. However, it is unclear whether some shares can be distributed before a secret is given in Ogawa et al.'s scheme. In this paper, we propose a method to distribute some shares before a secret is given in Ogawa et al.'s scheme, then determine a necessary and sufficient condition on sets of shares that can be distributed before a given secret.

DOI
10.48550/arxiv.2404.15646

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The ramp quantum secret sharing proposed by Ogawa et al. has the highest possible coding rate given a threshold type access structure. On the other hand, in some quantum secret sharing schemes, it is known that some shares can be distributed to participants before a secret is given to the dealer. However, it is unclear whether some shares can be distributed before a secret is given in Ogawa et al.'s scheme. In this paper, we propose a method to distribute some shares before a secret is given in Ogawa et al.'s scheme, then determine a necessary and sufficient condition on sets of shares that can be distributed before a given secret.
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In some quantum secret sharing schemes, it is known that some shares can be distributed to participants before a secret is given to the dealer. However, it is unclear whether some shares can be distributed before a secret is given in the ramp quantum secret sharing schemes with the highest coding rate. This article proposes procedures to distribute some shares before a secret is given in those schemes. The new procedures enhance the applicability of the secret sharing schemes to wider scenarios as some participants can be unavailable when the dealer obtains the quantum secret. Then, it is proved that our new encoding procedures retain the correspondences between quantum secrets and quantum shares in the original schemes, which ensures that the highest coding rates of the original schemes are also retained.
Secret sharing is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified sets of participants can restore the original secret from their shares. When we encode a secret by a secret sharing scheme and distribute shares, sometimes not all participants are accessible, and it is desirable to distribute shares to those participants before a secret information is determined. Secret sharing schemes for classical secrets have been known to be able to distribute some shares before a given secret. Lie et al. found a ((2, 3))-threshold secret sharing for quantum secrets can distribute some shares before a given secret. However, it is unknown whether distributing some shares before a given secret is possible with other access structures of secret sharing for quantum secrets. We propose a quantum secret sharing scheme for quantum secrets that can distribute some shares before a given secret with other access structures.
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.
A quantum secret sharing scheme is a method to share a quantum secret among participants in such a way that only certain specified subsets of the set of participants can combine to recover the secret. A quantum evolving secret sharing scheme is a recently introduced variant of secret sharing schemes where the sets of participants is not fixed and an unbounded and unspecified number of participants arrive one by one with time. The number of participants can be potentially infinite and and a quantum secret is shared and protected among the participants. The existing construction has some major drawbacks such as 1) the exponential quantum memory requirement and 2) the very high dimensions of the shares which makes the scheme difficult to implement. In this paper we overcome these drawbacks by constructing a scheme which uses quantum memory linear in the number of participants and significantly improves on the dimensions of the shares of the participants. This construction uses quantum secret redistribution and trap codes. The construction is flexible and can be modified to different types of access structures(subsets of participants which can recover the secret). Certain ramp properties can also be incorporated in the scheme.