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International Doctoral Workshop on Mathematical and Engineering Methods in Computer Science

MEMICS 2012: Mathematical and Engineering Methods in Computer Science pp 15–31Cite as

Quantum Secret Sharing with Graph States

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Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 7721))

Abstract

We study the graph-state-based quantum secret sharing protocols [24,17] which are not only very promising in terms of physical implementation, but also resource efficient since every player’s share is composed of a single qubit. The threshold of a graph-state-based protocol admits a lower bound: for any graph of order n, the threshold of the corresponding n-player protocol is at least 0.506n. Regarding the upper bound, lexicographic product of the C 5 graph (cycle of size 5) are known to provide n-player protocols which threshold is n − n 0.68. Using Paley graphs we improve this bound to n − n 0.71. Moreover, using probabilistic methods, we prove the existence of graphs which associated threshold is at most 0.811n.Albeit non-constructive, probabilistic methods permit to prove that a random graph G of order n has a threshold at most 0.811n with high probability. However, verifying that the threshold of a given graph is acually smaller than 0.811n is hard since we prove that the corresponding decision problem is NP-Complete.These results are mainly based on the graphical characterization of the graph-state-based secret sharing properties, in particular we point out strong connections with domination with parity constraints.

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Author information

Authors and Affiliations

  1. CNRS, France

    Sylvain Gravier, Mehdi Mhalla & Simon Perdrix

  2. Institut Fourier, University of Grenoble, France

    Sylvain Gravier

  3. LIG, University of Grenoble, France

    Jérôme Javelle, Mehdi Mhalla & Simon Perdrix

Authors
  1. Sylvain Gravier
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  2. Jérôme Javelle
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  3. Mehdi Mhalla
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  4. Simon Perdrix
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Editor information

Editors and Affiliations

  1. Faculty of Informatics, Masaryk University, Botanická 68a, 602 00, Brno, Czech Republic

    Antonín Kučera

  2. Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Thomas A. Henzinger

  3. Faculty of Mathematics and Physics, Charles University in Prague, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jaroslav Nešetřil

  4. Faculty of Information Technology, Brno University of Technology, Božetěchova 2, 612 66, Brno, Czech Republic

    Tomáš Vojnar

  5. Institute of Computer Science, Masaryk University, Botanicka 68a, 602 00, Brno, Czec Republic

    David Antoš

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Gravier, S., Javelle, J., Mhalla, M., Perdrix, S. (2013). Quantum Secret Sharing with Graph States. In: Kučera, A., Henzinger, T.A., Nešetřil, J., Vojnar, T., Antoš, D. (eds) Mathematical and Engineering Methods in Computer Science. MEMICS 2012. Lecture Notes in Computer Science, vol 7721. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36046-6_3

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  • DOI: https://doi.org/10.1007/978-3-642-36046-6_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36044-2

  • Online ISBN: 978-3-642-36046-6

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