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Quantum multi-secret sharing via trap codes and discrete quantum walks

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Abstract

In this paper, we propose a fully quantum multi-secret sharing scheme. In contrast with regular secret sharing schemes, multi-secret sharing schemes share a set of unknown secrets, and during the reconstruction phase, all the secrets are reconstructed. The main technique is to suitably modify a quantum trap code to construct a scheme where increasing number of secret states are recovered as more and more participants combine their shares. It is desirable that the dimensions of the share states are within implementable limits. In view of this and due to the significantly large dimension of the share states produced by our first construction, we introduce a discrete-time quantum walk-based technique to reduce the dimension of the shares making the schemes more suitable for practical purposes. Our methods are unconditional and do not depend on any computational hardness assumptions like lattice-based problems. Our scheme is simple, secure against adversarial attacks and can be easily modified into several variants of multi-secret sharing schemes.

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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Abbreviations

\({\mathbb {N}}\), \({\mathbb {Z}}\), \({\mathbb {C}}\) :

Natural numbers, integers, complex numbers

\({\mathcal {P}}\), \(p_{i}\) :

Set of participants, i-th participant

\(\Gamma \), \(\Gamma _{YES}\), \(\Gamma _{NO}\) :

Access structure, qualified sets, forbidden sets

(n, k):

k-out-of-n threshold access structure

\(t_{i}\) :

i-th threshold

S, \({\mathcal {S}}\) :

Domain of secrets

(SHARE, RECON):

Secret sharing scheme

\(\Pi _{i}^{(s)}\) :

Distribution on share of i-th participant

\(I, X, \sigma _{x}, Y, Z, \sigma _{z}\) :

Pauli matrices

\({\mathbb {P}}_{n}\) :

Set of n-qubit Pauli matrices

[n]:

\(\{1,2, \ldots , n\}\)

\(\sigma \), \(\sigma _{n_{i}}\), \(\sigma _{k_{i}}\) :

Permutation map (not to be confused with the Pauli operators)

\(\left| \phi \right\rangle \) :

Qubit/quantum state/secret

\(Sh_{Th}(n,k), Rec_{Th}(n,k)\) :

Fully quantum threshold secret sharing scheme

\(Share_{k,n}, Rec_{k,n}\) :

Semi-quantum threshold secret sharing scheme

U, R, C, S, F, G :

Unitary operators

\(E_{ij}\) :

Elementary matrix

\({[}C_{1}, \ldots , C_{d}{]}\) :

Block diagonal matrix

SSS:

Secret sharing scheme

MSSS:

Multi-secret sharing scheme

QSSS:

Quantum secret sharing scheme

QMSSS:

Quantum multi-secret sharing scheme

QECC:

Quantum error-correcting code

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Acknowledgements

The authors thank the anonymous reviewers whose careful and insightful comments helped in correcting various errors and largely improved the quality of the manuscript. The first author thanks TCG Centres for Research and Education in Science and Technology for a post-doctoral fellowship which financially supported this work. Second author was financially supported by MITACS Accelerate fellowship, Canada vide Ref. No. IT25625, FR66861. The authors also thank Bimal Kr. Roy and Goutam Mukherjee for stimulating discussions.

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  1. Translational Research in Mathematics, TCG Centres for Research and Education in Science and Technology, Salt Lake Sector V, Kolkata, West Bengal, 700091, India

    Shion Samadder Chaudhury

  2. Department of Computer Science, University of Calgary, University Dr NW, Calgary, AB, T2N 1N4, Canada

    Sabyasachi Dutta

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Samadder Chaudhury, S., Dutta, S. Quantum multi-secret sharing via trap codes and discrete quantum walks. Quantum Inf Process 21, 380 (2022). https://doi.org/10.1007/s11128-022-03732-1

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