Access Structure in Graphs in High Dimension and Application to Secret Sharing

We give graphical characterisation of the access structure to both classical and quantum information encoded onto a multigraph defined for prime dimension q, as well as explicit decoding operations for quantum secret sharing based on graph state protocols. We give a lower bound on $k$ for the existence of a ((k,n))_q scheme and prove, using probabilistic methods, that there exists alpha such that a random multigraph has an accessing parameter k => alpha*n with high probability.

[1]  Sylvain Gravier,et al.  Optimal accessing and non-accessing structures for graph protocols , 2011, ArXiv.

[2]  Mehdi Mhalla,et al.  On the Minimum Degree Up to Local Complementation: Bounds and Complexity , 2012, WG.

[3]  Dirk Schlingemann,et al.  Quantum error-correcting codes associated with graphs , 2000, ArXiv.

[4]  R. Griffiths,et al.  Location of quantum information in additive graph codes , 2009, 0912.2017.

[5]  Santosh Kumar,et al.  Nonbinary Stabilizer Codes Over Finite Fields , 2005, IEEE Transactions on Information Theory.

[6]  Andreas Klappenecker,et al.  Graphs, quadratic forms, and quantum codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[7]  Valerio Scarani,et al.  Security proof for quantum key distribution using qudit systems , 2010, 1003.5464.

[8]  Matthias Christandl,et al.  Uncertainty, monogamy, and locking of quantum correlations , 2005, IEEE Transactions on Information Theory.

[9]  G. Aeppli,et al.  Proceedings of the International School of Physics Enrico Fermi , 1994 .

[10]  André Bouchet,et al.  Circle Graph Obstructions , 1994, J. Comb. Theory, Ser. B.

[11]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[12]  A. Broadbent,et al.  The GHZ State in Secret Sharing and Entanglement Simulation , 2008, 2009 Third International Conference on Quantum, Nano and Micro Technologies.

[13]  Damian Markham,et al.  Equivalence between sharing quantum and classical secrets and error correction , 2012, 1205.4182.

[14]  R. Cleve,et al.  HOW TO SHARE A QUANTUM SECRET , 1999, quant-ph/9901025.

[15]  J. Eisert,et al.  Entanglement in Graph States and its Applications , 2006, quant-ph/0602096.

[16]  Gilad Gour,et al.  Reducing the Quantum Communication Cost of Quantum Secret Sharing , 2011, IEEE Transactions on Information Theory.

[17]  Bart De Moor,et al.  Graphical description of the action of local Clifford transformations on graph states , 2003, quant-ph/0308151.

[18]  Avinatan Hassidim,et al.  Secure Multiparty Quantum Computation with (Only) a Strict Honest Majority , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[19]  Sylvain Gravier,et al.  On weak odd domination and graph-based quantum secret sharing , 2011, Theor. Comput. Sci..

[20]  Mehdi Mhalla,et al.  New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States , 2011, TQC.

[21]  M. Koashi,et al.  Quantum entanglement for secret sharing and secret splitting , 1999 .

[22]  Oscar H. Ibarra,et al.  A Generalization of the Fast LUP Matrix Decomposition Algorithm and Applications , 1982, J. Algorithms.

[23]  Damian Markham,et al.  Information Flow in Secret Sharing Protocols , 2009, DCM.

[24]  Sang-il Oum,et al.  Approximating rank-width and clique-width quickly , 2008, ACM Trans. Algorithms.

[25]  D. Markham,et al.  Graph states for quantum secret sharing , 2008, 0808.1532.

[26]  Vlad Gheorghiu.  Generalized semiquantum secret-sharing schemes , 2012 .

[27]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

[28]  Salman Beigi,et al.  Graph States Under the Action of Local Clifford Group in Non-Binary Case , 2006 .

[29]  J. Hopcroft,et al.  Triangular Factorization and Inversion by Fast Matrix Multiplication , 1974 .

[30]  D. Markham,et al.  Quantum secret sharing with qudit graph states , 2010, 1004.4619.

[31]  Salman Beigi,et al.  Graph concatenation for quantum codes , 2009, 0910.4129.

[32]  Mamadou Moustapha Kanté,et al.  The Rank-Width of Edge-Coloured Graphs , 2007, Theory of Computing Systems.

[33]  V. Buzek,et al.  Quantum secret sharing , 1998, quant-ph/9806063.

[34]  Simon Perdrix,et al.  Parameterized Complexity of Weak Odd Domination Problems , 2012, FCT.

[35]  Adi Shamir,et al.  How to share a secret , 1979, CACM.