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Perfect Secret Sharing Schemes

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Presentation on theme: "Perfect Secret Sharing Schemes"— Presentation transcript:

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2 Perfect Secret Sharing Schemes
On the Complexity of Perfect Secret Sharing Schemes Motahhareh Gharahi

3 How to Share a Secret S Shamir (1979) Blakley (1979)
Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

4 Secret Sharing scheme A secret sharing scheme is a method which allows a secret to be shared among a set of participants, in such a way that only qualified subsets of participants can reconstruct the secret from their shares. The secret sharing scheme is called perfect if unqualified subsets of participants cannot obtain any information about the secret. Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

5 Access Structure The access structure is the collection of all subsets of participants that are qualified to reconstruct the secret.   Monotone Access Structures  The basis of an access structure is the collection of all minimal qualified subsets of participants. Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

6 Efficiency of a Secret Sharing Scheme
The efficiency of a secret sharing scheme is measured by The ratio between the maximum size of the shares given to any participant and the size of the secret Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

7 Complexity of Secret Sharing Scheme
The complexity of secret sharing scheme is defined by The complexity of an access structure is defined by The access structure with complexity equal to one is called ideal Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

8 Toy Example Graph Access Structures
The access structure is based on a (connected) graph G if the participants and the minimal qualified subsets are corresponding to the vertices and the edges of the graph G, respectively r+s r r r r r+s Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

9 Upper Bounds on the Complexity
D. R. Stinson (1994) Sun and Chen (2002) M. van Dijk et al. (2006) Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

10 Theorem (D. R. Stinson, 1994) Motahhareh Gharahi Mar 2016
‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

11 Example 2 2 2 2 Motahhareh Gharahi Mar 2016
‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

12 Constructing Perfect Secret Sharing Scheme
Theorem (M. Gharahi and M. Hadian, 2013) Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

13 Entropy and its Properties
Shannon Entropy Properties Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

14 Perfect Secret Sharing Scheme
Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

15 Lower Bounds on the Complexity
Entropy Method Theorem (Csirmaz, 1997) Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

16 Lower Bounds on the Complexity
Theorem (M. Gharahi and M. Hadian, 2013) Proof Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

17 Lower Bounds on the Complexity
Corollary (M. Gharahi and M. Hadian, 2013) Example Corollary Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

18 Miscellany Recent results
Csirmaz, Secret sharing on infinite graphs (2008) Tatra Mountains Mathematical Publications Csirmaz , Ligeti, On an infinite families of graphs with information ratio 2 −1/k (2009) Computing Csirmaz, Secret sharing on the d-dimensional cube (2013) Designs, Codes and Cryptography Gharahi, Hadian Dehkordi, The complexity of the graph access structures on six participants (2013) Designs, Codes and Cryptography Gharahi, Hadian Dehkordi, Perfect secret sharing schemes for graph access structures on six participants (2013) Journal of Mathematical Cryptology Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

19 Miscellany Martí-Farré, Padró, Secret sharing schemes on access structures with intersection number equal to one (2006) Discrete Applied Mathematics Martí-Farré, Padró, On Secret Sharing Schemes, Matroids and Polymatroids (2010) Journal of Mathematical Cryptology Martí-Farré, Padró, Vázquez, Optimal complexity of secret sharing schemes with four minimal qualified subsets (2011) Designs, Codes and Cryptography Padró, Vázquez, Finding lower bounds on the complexity of secret sharing schemes by linear programming (2013) Discrete Applied Mathematics  Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

20 Access Structure Induced by a Matroid
 Induced access structure by a matroid Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

21 Ideal access structures
Theorem (Brickell-Davenport, 1991) Question Does there exist an access structure induced by a matroid that does not have an ideal scheme? Yes Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

22 Non-Shannon Information Inequalities
Theorem (Z. Zhang and R. W. Yeung, 1998) Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

23 Vámos Matroid Motahhareh Gharahi Mar 2016
‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

24 Complexity of Vámos Access Structure
Theorem (C. Padro et al ) Theorem (J. Mart-Farre and C. Padro ) Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

25 Improved Lower Bounds Theorem (J. R. Metcalf-Burton, 2010)
Theorem (M. Gharahi and M. Hadian, 2013) Motahhareh Gharahi ‎‎Complexity ‎of ‎Perfect Secret Sharing Scheme Mar 2016

26 Thank You for Your Attention


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