# American Institute of Mathematical Sciences

February  2019, 13(1): 89-99. doi: 10.3934/amc.2019005

## Optimal information ratio of secret sharing schemes on Dutch windmill graphs

 Applied Mathematics and Cryptography Department, Malek Ashtar university of technology, Isfahan, Iran

* Corresponding author: Bagher Bagherpour

Received  April 2018 Revised  July 2018 Published  December 2018

One of the basic problems in secret sharing is to determine the exact values of the information ratio of the access structures. This task is important from the practical point of view, since the security of any system degrades as the amount of secret information increases.

A Dutch windmill graph consists of the edge-disjoint cycles such that all of them meet in one vertex. In this paper, we determine the exact information ratio of secret sharing schemes on the Dutch windmill graphs. Furthermore, we determine the exact ratio of some related graph families.

Citation: Bagher Bagherpour, Shahrooz Janbaz, Ali Zaghian. Optimal information ratio of secret sharing schemes on Dutch windmill graphs. Advances in Mathematics of Communications, 2019, 13 (1) : 89-99. doi: 10.3934/amc.2019005
##### References:
 [1] J. C. Benaloh and J. Leichter, Generalized secret sharing and monotone functions, Advances in Cryptology-Crpto 88 Proceedings, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 403 (1990), 27-35. doi: 10.1007/0-387-34799-2_3.  Google Scholar [2] G. R. Blakley, Safeguarding Cryptographic Keys, in AFIPS Conference Proceedings, 48 (1979), 313-317. Google Scholar [3] C. Blundo, A. De Santis, D. R. Stinson and U. Vaccaro, Graph decompositions and secret sharing schemes, Advances in Cryptology-Proceeding of Eurocrypt 92, Lecture Notes in Comput. Sci, 658 (1993), 1-24.  doi: 10.1007/3-540-47555-9_1.  Google Scholar [4] C. Blundo, A. De Santis and U. Vaccaro, Tight bounds on the information rate of secret sharing schemes, Designs Codes and Cryptography, 11 (1997), 107-122.  doi: 10.1023/A:1008216403325.  Google Scholar [5] C. Blundo, A. De Santis, L. Gargano and U. Vaccaro, On the information rate of secret sharing schemes, Theoretical Computer Science, 154 (1996), 283-306.  doi: 10.1016/0304-3975(95)00065-8.  Google Scholar [6] E. F. Brickell and D. M. Davenport, On the classification of ideal secret sharing schemes, Journal of Cryptology, 4 (1993), 157-167.   Google Scholar [7] R. M. Capocelli, A. De Santis, L. Gargano and U. Vaccaro, On the size of shares for secret sharing schemes, Journal of Cryptology, 6 (1993), 157-169.   Google Scholar [8] T. M. Cover and J.A. Thomas, Elements of Information Theory, 2$^{nd}$ edition, John Wiley and Sons, Inc., Hoboken, New Jersey, 2006. Google Scholar [9] L. Csirmas, The size of a share must be large, Journal of Cryptology, 10 (1997), 223-231.  doi: 10.1007/s001459900029.  Google Scholar [10] L. Csirmas, An impossibility result on graph secret sharing, Designs Codes and Cryptography, 53 (2009), 195-209.  doi: 10.1007/s10623-009-9304-0.  Google Scholar [11] L. Csirmaz and G. Tardos, Optimal information rate of secret sharing schemes on trees, IEEE Transaction on Information Theory, 59 (2013), 2527-2530.  doi: 10.1109/TIT.2012.2236958.  Google Scholar [12] L. Csirmaz and P. Ligeti, On an infinite family of graphs with information ratio $1-\frac{2}{k}$, Computing, 85 (2009), 127-136.  doi: 10.1007/s00607-009-0039-6.  Google Scholar [13] P. Erdős, A. Rényi and V. T. Sós, On a problem of graph theory, Studia Sci. Math. Hungar, 1 (1966), 215-235.   Google Scholar [14] R. G. Gallager, Information Theory and Reliable Communications, John Wiley, New York, 1986. Google Scholar [15] M. Ito, A. Saito and T. Nishizeki, Secret sharing scheme realizing general access structure, Proc. IEEE Globecorn, Tokyo, 87 (1987), 99-102.   Google Scholar [16] M. Ito, A. Saito and T. Nishizeki, Multiple assignment scheme for sharing secret, Journal of Cryptology, 6 (1993), 15-20.  doi: 10.1007/BF02620229.  Google Scholar [17] W. Jackson and Keith M. Martin, Perfect secret sharing schemes on five participants, Designs. Codes and Cryptography, 9 (1996), 267-286.  doi: 10.1007/BF00129769.  Google Scholar [18] C. Padró and G. Sáez, Secret sharing with bipartite access structure, IEEE Transaction on Information Theory, 46 (2000), 2596-2604.  doi: 10.1109/18.887867.  Google Scholar [19] C. Padró and L. Vazquez, Finding lower bounds on the complexity of secret sharing schemes by linear programming, Ninth Latin American Theoretical Informatics Symposium, LATIN 2010, Lecture Notes in Computer Science, 6034 (2010), 344-355.  doi: 10.1007/978-3-642-12200-2_31.  Google Scholar [20] A. Shamir, How to share a secret, Communication of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [21] D. R. Stinson, Decomposition constructions for secret sharing schemes, IEEE Transaction on Information Theory, 40 (1994), 118-125.  doi: 10.1109/18.272461.  Google Scholar [22] D. R. Stinson, An explication of secret sharing schemes, Designs Codes and Cryptography, 2 (1992), 357-390.  doi: 10.1007/BF00125203.  Google Scholar [23] H. M. Sun and B. L. Chen, Weighted decomposition construction for perfect secret sharing schemes, Compute Math. Appl., 43 (2002), 877-887.  doi: 10.1016/S0898-1221(01)00328-5.  Google Scholar [24] M. Van dijk, On the information rate of perfect secret sharing schemes, Designs, Codes and Cryptography, 6 (1995), 143-169.  doi: 10.1007/BF01398012.  Google Scholar

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##### References:
 [1] J. C. Benaloh and J. Leichter, Generalized secret sharing and monotone functions, Advances in Cryptology-Crpto 88 Proceedings, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 403 (1990), 27-35. doi: 10.1007/0-387-34799-2_3.  Google Scholar [2] G. R. Blakley, Safeguarding Cryptographic Keys, in AFIPS Conference Proceedings, 48 (1979), 313-317. Google Scholar [3] C. Blundo, A. De Santis, D. R. Stinson and U. Vaccaro, Graph decompositions and secret sharing schemes, Advances in Cryptology-Proceeding of Eurocrypt 92, Lecture Notes in Comput. Sci, 658 (1993), 1-24.  doi: 10.1007/3-540-47555-9_1.  Google Scholar [4] C. Blundo, A. De Santis and U. Vaccaro, Tight bounds on the information rate of secret sharing schemes, Designs Codes and Cryptography, 11 (1997), 107-122.  doi: 10.1023/A:1008216403325.  Google Scholar [5] C. Blundo, A. De Santis, L. Gargano and U. Vaccaro, On the information rate of secret sharing schemes, Theoretical Computer Science, 154 (1996), 283-306.  doi: 10.1016/0304-3975(95)00065-8.  Google Scholar [6] E. F. Brickell and D. M. Davenport, On the classification of ideal secret sharing schemes, Journal of Cryptology, 4 (1993), 157-167.   Google Scholar [7] R. M. Capocelli, A. De Santis, L. Gargano and U. Vaccaro, On the size of shares for secret sharing schemes, Journal of Cryptology, 6 (1993), 157-169.   Google Scholar [8] T. M. Cover and J.A. Thomas, Elements of Information Theory, 2$^{nd}$ edition, John Wiley and Sons, Inc., Hoboken, New Jersey, 2006. Google Scholar [9] L. Csirmas, The size of a share must be large, Journal of Cryptology, 10 (1997), 223-231.  doi: 10.1007/s001459900029.  Google Scholar [10] L. Csirmas, An impossibility result on graph secret sharing, Designs Codes and Cryptography, 53 (2009), 195-209.  doi: 10.1007/s10623-009-9304-0.  Google Scholar [11] L. Csirmaz and G. Tardos, Optimal information rate of secret sharing schemes on trees, IEEE Transaction on Information Theory, 59 (2013), 2527-2530.  doi: 10.1109/TIT.2012.2236958.  Google Scholar [12] L. Csirmaz and P. Ligeti, On an infinite family of graphs with information ratio $1-\frac{2}{k}$, Computing, 85 (2009), 127-136.  doi: 10.1007/s00607-009-0039-6.  Google Scholar [13] P. Erdős, A. Rényi and V. T. Sós, On a problem of graph theory, Studia Sci. Math. Hungar, 1 (1966), 215-235.   Google Scholar [14] R. G. Gallager, Information Theory and Reliable Communications, John Wiley, New York, 1986. Google Scholar [15] M. Ito, A. Saito and T. Nishizeki, Secret sharing scheme realizing general access structure, Proc. IEEE Globecorn, Tokyo, 87 (1987), 99-102.   Google Scholar [16] M. Ito, A. Saito and T. Nishizeki, Multiple assignment scheme for sharing secret, Journal of Cryptology, 6 (1993), 15-20.  doi: 10.1007/BF02620229.  Google Scholar [17] W. Jackson and Keith M. Martin, Perfect secret sharing schemes on five participants, Designs. Codes and Cryptography, 9 (1996), 267-286.  doi: 10.1007/BF00129769.  Google Scholar [18] C. Padró and G. Sáez, Secret sharing with bipartite access structure, IEEE Transaction on Information Theory, 46 (2000), 2596-2604.  doi: 10.1109/18.887867.  Google Scholar [19] C. Padró and L. Vazquez, Finding lower bounds on the complexity of secret sharing schemes by linear programming, Ninth Latin American Theoretical Informatics Symposium, LATIN 2010, Lecture Notes in Computer Science, 6034 (2010), 344-355.  doi: 10.1007/978-3-642-12200-2_31.  Google Scholar [20] A. Shamir, How to share a secret, Communication of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [21] D. R. Stinson, Decomposition constructions for secret sharing schemes, IEEE Transaction on Information Theory, 40 (1994), 118-125.  doi: 10.1109/18.272461.  Google Scholar [22] D. R. Stinson, An explication of secret sharing schemes, Designs Codes and Cryptography, 2 (1992), 357-390.  doi: 10.1007/BF00125203.  Google Scholar [23] H. M. Sun and B. L. Chen, Weighted decomposition construction for perfect secret sharing schemes, Compute Math. Appl., 43 (2002), 877-887.  doi: 10.1016/S0898-1221(01)00328-5.  Google Scholar [24] M. Van dijk, On the information rate of perfect secret sharing schemes, Designs, Codes and Cryptography, 6 (1995), 143-169.  doi: 10.1007/BF01398012.  Google Scholar
The Dutch windmill graph $D_4^{(k)}$ with predefined labelling
The friendship graph $F_k$ with predefined labelling
Subgraphs of the graph $C(\mathcal{F}_k)$
 $G$ $V$ $S_1(V)$ $\{v_c, v^{1}_2, v^{1}_{n_1}, \ldots, v^{k}_{2}, v^{k}_{n_k}\}$ $\Pi_1 =\{(2k-1) \times S_{1}(V)\}$ $S^{i}_{1}(V)$$i\in \{1, \ldots, k\} \{v_c, v^{i}_2, v^{i}_{3}\} \Pi_2 =\{1 \times S^{i}_1(V): i\in \{1, \ldots, k\}\} S^{i}_{2}(V)$$i\in \{1, \ldots, k\}$ $\{v_c, v^{i}_{n_i}, v^{i}_{n_i-1}\}$ $\Pi_3 =\{1 \times S^{i}_{2}(V): i\in \{1, \ldots, k\}\}$ $P^{i}_1(V)$$i\in \{1, \ldots, k\} \{v^{i}_2, v^{i}_3, \ldots, v^{i}_{n_i}\} \Pi_4 =\{(2k-1) \times P^{i}_1(V): i \in \{1, \ldots, k\}\} P^{i}_2(V)$$i\in \{1, \ldots, k\}$ $\{v^{i}_3, v^{i}_{4}, \ldots, v^{i}_{n_i-1}\}$ $\Pi_5 =\{1 \times P^{i}_2(V): i \in \{1, \ldots, k\}\}$
 $G$ $V$ $S_1(V)$ $\{v_c, v^{1}_2, v^{1}_{n_1}, \ldots, v^{k}_{2}, v^{k}_{n_k}\}$ $\Pi_1 =\{(2k-1) \times S_{1}(V)\}$ $S^{i}_{1}(V)$$i\in \{1, \ldots, k\} \{v_c, v^{i}_2, v^{i}_{3}\} \Pi_2 =\{1 \times S^{i}_1(V): i\in \{1, \ldots, k\}\} S^{i}_{2}(V)$$i\in \{1, \ldots, k\}$ $\{v_c, v^{i}_{n_i}, v^{i}_{n_i-1}\}$ $\Pi_3 =\{1 \times S^{i}_{2}(V): i\in \{1, \ldots, k\}\}$ $P^{i}_1(V)$$i\in \{1, \ldots, k\} \{v^{i}_2, v^{i}_3, \ldots, v^{i}_{n_i}\} \Pi_4 =\{(2k-1) \times P^{i}_1(V): i \in \{1, \ldots, k\}\} P^{i}_2(V)$$i\in \{1, \ldots, k\}$ $\{v^{i}_3, v^{i}_{4}, \ldots, v^{i}_{n_i-1}\}$ $\Pi_5 =\{1 \times P^{i}_2(V): i \in \{1, \ldots, k\}\}$
Subgraphs of the graph $C'(\mathcal{F}_k)$
 $G$ $V$ $S_1(V)$ $\{v_c, v', v^{1}_2, v^{1}_{n_1}, \ldots, v^{k}_{2}, v^{k}_{n_k}\}$ $\Pi_1 =\{(2k) \times S_{1}(V)\}$ $S^{i}_{1}(V)$$i\in \{1, \ldots, k\} \{v_c, v^{i}_2, v^{i}_{3}\} \Pi_2 =\{1 \times S^{i}_1(V): i\in \{1, \ldots, k\}\} S^{i}_{2}(V)$$i\in \{1, \ldots, k\}$ $\{v_c, v^{i}_{n_i}, v^{i}_{n_i-1}\}$ $\Pi_3 =\{1 \times S^{i}_{2}(V): i\in \{1, \ldots, k\}\}$ $S'(V)$ $\{v_c, v' \}$ $\Pi_4 =\{1 \times S'(V) \}$ $P^{i}_1(V)$$i\in \{1, \ldots, k\} \{v^{i}_2, v^{i}_3, \ldots, v^{i}_{n_i}\} \Pi_5 =\{2k \times P^{i}_1(V): i \in \{1, \ldots, k\}\} P^{i}_2(V)$$i\in \{1, \ldots, k\}$ $\{v^{i}_3, v^{i}_{4}, \ldots, v^{i}_{n_i-1}\}$ $\Pi_6 =\{1 \times P^{i}_2(V): i \in \{1, \ldots, k\}\}$
 $G$ $V$ $S_1(V)$ $\{v_c, v', v^{1}_2, v^{1}_{n_1}, \ldots, v^{k}_{2}, v^{k}_{n_k}\}$ $\Pi_1 =\{(2k) \times S_{1}(V)\}$ $S^{i}_{1}(V)$$i\in \{1, \ldots, k\} \{v_c, v^{i}_2, v^{i}_{3}\} \Pi_2 =\{1 \times S^{i}_1(V): i\in \{1, \ldots, k\}\} S^{i}_{2}(V)$$i\in \{1, \ldots, k\}$ $\{v_c, v^{i}_{n_i}, v^{i}_{n_i-1}\}$ $\Pi_3 =\{1 \times S^{i}_{2}(V): i\in \{1, \ldots, k\}\}$ $S'(V)$ $\{v_c, v' \}$ $\Pi_4 =\{1 \times S'(V) \}$ $P^{i}_1(V)$$i\in \{1, \ldots, k\} \{v^{i}_2, v^{i}_3, \ldots, v^{i}_{n_i}\} \Pi_5 =\{2k \times P^{i}_1(V): i \in \{1, \ldots, k\}\} P^{i}_2(V)$$i\in \{1, \ldots, k\}$ $\{v^{i}_3, v^{i}_{4}, \ldots, v^{i}_{n_i-1}\}$ $\Pi_6 =\{1 \times P^{i}_2(V): i \in \{1, \ldots, k\}\}$
Subgraphs of the graph $D^{(k)}_{4}$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v_c, v_1, v_3, \ldots, v_{3k-2}, v_{3k}\}$ $\{v_c v_j , v_c v_{j+2}:$ $j \in \{1, 4, \ldots, 3k-2 \}\}$ $G_{1+(i+2)/3}$$i\in \{1, 4, \ldots 3k-2 \} \{v_c, v_i, v_{i+1}, v_{i+2}\} \{ v_c v_i, v_c v_{i+2}, v_i v_{i+1}, v_{i+1} v_{i+2}\} G_{k+1+(i+2)/3}$$i\in \{1, 4, \ldots, 3k-2 \}$ $\{v_i, v_{i+1}, v_{i+2}\}$ $\{ v_i v_{i+1}, v_{i+1} v_{i+2}\}$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v_c, v_1, v_3, \ldots, v_{3k-2}, v_{3k}\}$ $\{v_c v_j , v_c v_{j+2}:$ $j \in \{1, 4, \ldots, 3k-2 \}\}$ $G_{1+(i+2)/3}$$i\in \{1, 4, \ldots 3k-2 \} \{v_c, v_i, v_{i+1}, v_{i+2}\} \{ v_c v_i, v_c v_{i+2}, v_i v_{i+1}, v_{i+1} v_{i+2}\} G_{k+1+(i+2)/3}$$i\in \{1, 4, \ldots, 3k-2 \}$ $\{v_i, v_{i+1}, v_{i+2}\}$ $\{ v_i v_{i+1}, v_{i+1} v_{i+2}\}$
Subgraphs of the graph $D'^{(k)}_{4}$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v_c, v_1, v_3, \ldots, v_{3k-2}, v_{3k},$ $v'\}$ $\{v_c v_j , v_c v_{j+2}, v_c v': j\in\{1, 4,$ $\ldots, 3k-2\}\}$ $G_{1+(i+2)/3}$$i\in \{1, 4, \ldots 3k-2 \} \{v_c, v_i, v_{i+1}, v_{i+2}\} \{ v_c v_i, v_c v_{i+2}, v_i v_{i+1}, v_{i+1} v_{i+2}\} G_{k+2} \{v_c, v' \} \{v_c v' \} G_{k+2+(i+2)/3}$$i\in \{1, 4, \ldots, 3k-2 \}$ $\{v_i, v_{i+1}, v_{i+2}\}$ $\{ v_i v_{i+1}, v_{i+1} v_{i+2}\}$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v_c, v_1, v_3, \ldots, v_{3k-2}, v_{3k},$ $v'\}$ $\{v_c v_j , v_c v_{j+2}, v_c v': j\in\{1, 4,$ $\ldots, 3k-2\}\}$ $G_{1+(i+2)/3}$$i\in \{1, 4, \ldots 3k-2 \} \{v_c, v_i, v_{i+1}, v_{i+2}\} \{ v_c v_i, v_c v_{i+2}, v_i v_{i+1}, v_{i+1} v_{i+2}\} G_{k+2} \{v_c, v' \} \{v_c v' \} G_{k+2+(i+2)/3}$$i\in \{1, 4, \ldots, 3k-2 \}$ $\{v_i, v_{i+1}, v_{i+2}\}$ $\{ v_i v_{i+1}, v_{i+1} v_{i+2}\}$
Subgraphs of the graph $v\nabla \mathcal{F}_k$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v, V(H_1), V(H_2),$ $\ldots, V(H_k)\}$ $\{v w: w \in \{ V(H_1), \ldots, V(H_k)\}\}$ $G_{1+i}$$i\in \{1, \ldots, k\} \{v, V(H_i)\} \{ E(H_i), v w:w \in V(H_i) \} G_{k+1+i}$$i\in \{1, \cdots, k \}$ $V(H_i)$ $E(H_i)$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v, V(H_1), V(H_2),$ $\ldots, V(H_k)\}$ $\{v w: w \in \{ V(H_1), \ldots, V(H_k)\}\}$ $G_{1+i}$$i\in \{1, \ldots, k\} \{v, V(H_i)\} \{ E(H_i), v w:w \in V(H_i) \} G_{k+1+i}$$i\in \{1, \cdots, k \}$ $V(H_i)$ $E(H_i)$
Subgraphs of the graph $v\nabla \mathcal{F'}_k$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v, V(H_1), \ldots, V(H_k), v'\}$ $\{v v', v w: w \in \{ V(H_1), \ldots, V(H_k) \}$ $G_{1+i}$$i\in \{1, \ldots, k\} \{v, V(H_i)\} \{ E(H_i), v w:w \in V(H_i) \} G_{k+2} \{v, v'\} \{v v' \} G_{k+2+i}$$i\in \{1, \cdots, k \}$ $V(H_i)$ $E(H_i)$
 $G$ $V(G)$ $E(G)$ $G_1$ $\{v, V(H_1), \ldots, V(H_k), v'\}$ $\{v v', v w: w \in \{ V(H_1), \ldots, V(H_k) \}$ $G_{1+i}$$i\in \{1, \ldots, k\} \{v, V(H_i)\} \{ E(H_i), v w:w \in V(H_i) \} G_{k+2} \{v, v'\} \{v v' \} G_{k+2+i}$$i\in \{1, \cdots, k \}$ $V(H_i)$ $E(H_i)$
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