February 2018, 12(1): 199-214. doi: 10.3934/amc.2018014

Reduced access structures with four minimal qualified subsets on six participants

Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran

Received  March 2017 Revised  August 2017 Published  March 2018

In this paper, we discuss a point about applying known decomposition techniques in their most general form. Three versions of these methods, which are useful for obtaining upper bounds on the optimal information ratios of access structures, are known as: Stinson's $λ$-decomposition, $(λ, ω)$-decomposition and $λ$-weighted-decomposition, where the latter two are generalizations of the first one. We continue by considering the problem of determining the exact values of the optimal information ratios of the reduced access structures with exactly four minimal qualified subsets on six participants, which remained unsolved in Martí-Farré et al.'s paper [Des. Codes Cryptogr. 61 (2011), 167-186]. We improve the known upper bounds for all the access structures, except four cases, determining the exact values of the optimal information ratios. All three decomposition techniques are used while some cases are handled by taking full advantage of the generality of decompositions.

Citation: Motahhareh Gharahi, Shahram Khazaei. Reduced access structures with four minimal qualified subsets on six participants. Advances in Mathematics of Communications, 2018, 12 (1) : 199-214. doi: 10.3934/amc.2018014
References:
[1]

A. Beimel, Secret-sharing schemes: a survey, in Int. Conf. Coding Crypt., Springer, 2011, 11–46.

[2]

A. Beimel, A. Ben-Efraim, C. Padró and I. Tyomkin, Multi-linear secret-sharing schemes, in Theory of Cryptography Conference, Springer, Berlin, 2014,394–418.

[3]

G. R. Blakley, Safeguarding cryptographic keys, in Proceedings of the 1979 AFIPS National Computer Conference, Monval, NJ, USA, AFIPS Press, 1979, 313-317.

[4]

C. BlundoA. De SantisD. R. Stinson and U. Vaccaro, Graph decompositions and secret sharing schemes, J. Cryptology, 8 (1995), 39-64.

[5]

C. BlundoA. De SantisR. D. Simone and U. Vaccaro, Tight bounds on the information rate of secret sharing schemes, Des. Codes Crypt., 11 (1997), 107-122.

[6]

E. F. Brickell and D. R. Stinson, Some improved bounds on the information rate of perfect secret sharing schemes, J. Cryptology, 5 (1992), 153-166.

[7]

R. M. CapocelliA. D. SantisL. Gargano and U. Vaccaro, On the size of shares for secret sharing schemes, J. Cryptology, 6 (1993), 157-167.

[8]

L. Csirmaz, An impossibility result on graph secret sharing, Des. Codes Crypt., 53 (2009), 195-209.

[9]

L. Csirmaz, Secret sharing on the d-dimensional cube, Des. Codes Crypt., 74 (2015), 719-729.

[10]

L. Csirmaz and G. Tardos, Optimal information rate of secret sharing schemes on trees, IEEE Trans. Inf. Theory, 59 (2013), 2527-2530.

[11]

O. Farràs, T. B. Hansen, T. Kaced and C. Padró, Optimal non-perfect uniform secret sharing schemes, in Int. Crypt. Conf., Springer, Berlin, 2014,217–234.

[12]

O. Farràs, T. Kaced, S. Martin and C. Padro, Improving the linear programming technique in the search for lower bounds in secret sharing, Cryptology ePrint Archive, Report 2017/919,2017; available at https://eprint.iacr.org/2017/919

[13]

O. FarràsJ. R. Metcalf-BurtonC. Padró and L. Vázquez, On the optimization of bipartite secret sharing schemes, Des. Codes Crypt., 63 (2012), 255-271.

[14]

M. Gharahi and M. H. Dehkordi, Perfect secret sharing schemes for graph access structures on six participants, J. Math. Crypt., 7 (2013), 143-146.

[15]

M. Gharahi and M. H. Dehkordi, The complexity of the graph access structures on six participants, Des. Codes Crypt., 67 (2013), 169-173.

[16]

M. ItoA. Saito and T. Nishizeki, Secret sharing scheme realizing general access structure, Electr. Commun. Japan (Part Ⅲ: Fundam. Electr. Sci.), 72 (1989), 56-64.

[17]

W.-A. Jackson and K. M. Martin, Geometric secret sharing schemes and their duals, Des. Codes Crypt., 4 (1994), 83-95.

[18]

W.-A. Jackson and K. M. Martin, Perfect secret sharing schemes on five participants, Des. Codes Crypt., 9 (1996), 267-286.

[19]

E. D. KarninJ. W. Greene and M. E. Hellman, On secret sharing systems, IEEE Trans. Inf. Theory, 29 (1983), 35-41.

[20]

J. Martí-Farré and C. Padró, Secret sharing schemes with three or four minimal qualified subsets, Des. Codes Crypt., 34 (2005), 17-34.

[21]

J. Martí-Farré and C. Padró, Secret sharing schemes on access structures with intersection number equal to one, Discrete Appl. Math., 154 (2006), 552-563.

[22]

J. Martí-FarréC. Padró and L. Vázquez, Optimal complexity of secret sharing schemes with four minimal qualified subsets, Des. Codes Crypt., 61 (2011), 167-186.

[23]

K. M. Martin, New secret sharing schemes from old, J. Combin. Math. Combin. Comp., 14 (1993), 65-77.

[24]

C. Padró and G. Sáez, Secret sharing schemes with bipartite access structure, IEEE Trans. Inf. Theory, 46 (2000), 2596-2604.

[25]

C. Padró and G. Sáez, Lower bounds on the information rate of secret sharing schemes with homogeneous access structure, Inf. Process. Lett., 83 (2002), 345-351.

[26]

C. PadróL. Vázquez and A. Yang, Finding lower bounds on the complexity of secret sharing schemes by linear programming, Discrete Appl. Math., 161 (2013), 1072-1084.

[27]

A. Shamir, How to share a secret, Commun. ACM, 22 (1979), 612-613.

[28]

D. R. Stinson, An explication of secret sharing schemes, Des. Codes Crypt., 2 (1992), 357-390.

[29]

D. R. Stinson, Decomposition constructions for secret-sharing schemes, IEEE Trans. Inf. Theory, 40 (1994), 118-125.

[30]

H.-M. Sun and B.-L. Chen, Weighted decomposition construction for perfect secret sharing schemes, Comp. Math. Appl., 43 (2002), 877-887.

[31]

H.-M. SunH. WangB.-H. Ku and J. Pieprzyk, Decomposition construction for secret sharing schemes with graph access structures in polynomial time, SIAM J. Discrete Math., 24 (2010), 617-638.

[32]

M. Van Dijk, On the information rate of perfect secret sharing schemes, Des. Codes Crypt., 6 (1995), 143-169.

[33]

M. Van DijkW.-A. Jackson and K. M. Martin, A general decomposition construction for incomplete secret sharing schemes, Des. Codes Crypt., 15 (1998), 301-321.

[34]

M. Van DijkT. KevenaarG.-J. Schrijen and P. Tuyls, Improved constructions of secret sharing schemes by applying ($λ$, $ω$)-decompositions, Inf. Process. Lett., 99 (2006), 154-157.

show all references

References:
[1]

A. Beimel, Secret-sharing schemes: a survey, in Int. Conf. Coding Crypt., Springer, 2011, 11–46.

[2]

A. Beimel, A. Ben-Efraim, C. Padró and I. Tyomkin, Multi-linear secret-sharing schemes, in Theory of Cryptography Conference, Springer, Berlin, 2014,394–418.

[3]

G. R. Blakley, Safeguarding cryptographic keys, in Proceedings of the 1979 AFIPS National Computer Conference, Monval, NJ, USA, AFIPS Press, 1979, 313-317.

[4]

C. BlundoA. De SantisD. R. Stinson and U. Vaccaro, Graph decompositions and secret sharing schemes, J. Cryptology, 8 (1995), 39-64.

[5]

C. BlundoA. De SantisR. D. Simone and U. Vaccaro, Tight bounds on the information rate of secret sharing schemes, Des. Codes Crypt., 11 (1997), 107-122.

[6]

E. F. Brickell and D. R. Stinson, Some improved bounds on the information rate of perfect secret sharing schemes, J. Cryptology, 5 (1992), 153-166.

[7]

R. M. CapocelliA. D. SantisL. Gargano and U. Vaccaro, On the size of shares for secret sharing schemes, J. Cryptology, 6 (1993), 157-167.

[8]

L. Csirmaz, An impossibility result on graph secret sharing, Des. Codes Crypt., 53 (2009), 195-209.

[9]

L. Csirmaz, Secret sharing on the d-dimensional cube, Des. Codes Crypt., 74 (2015), 719-729.

[10]

L. Csirmaz and G. Tardos, Optimal information rate of secret sharing schemes on trees, IEEE Trans. Inf. Theory, 59 (2013), 2527-2530.

[11]

O. Farràs, T. B. Hansen, T. Kaced and C. Padró, Optimal non-perfect uniform secret sharing schemes, in Int. Crypt. Conf., Springer, Berlin, 2014,217–234.

[12]

O. Farràs, T. Kaced, S. Martin and C. Padro, Improving the linear programming technique in the search for lower bounds in secret sharing, Cryptology ePrint Archive, Report 2017/919,2017; available at https://eprint.iacr.org/2017/919

[13]

O. FarràsJ. R. Metcalf-BurtonC. Padró and L. Vázquez, On the optimization of bipartite secret sharing schemes, Des. Codes Crypt., 63 (2012), 255-271.

[14]

M. Gharahi and M. H. Dehkordi, Perfect secret sharing schemes for graph access structures on six participants, J. Math. Crypt., 7 (2013), 143-146.

[15]

M. Gharahi and M. H. Dehkordi, The complexity of the graph access structures on six participants, Des. Codes Crypt., 67 (2013), 169-173.

[16]

M. ItoA. Saito and T. Nishizeki, Secret sharing scheme realizing general access structure, Electr. Commun. Japan (Part Ⅲ: Fundam. Electr. Sci.), 72 (1989), 56-64.

[17]

W.-A. Jackson and K. M. Martin, Geometric secret sharing schemes and their duals, Des. Codes Crypt., 4 (1994), 83-95.

[18]

W.-A. Jackson and K. M. Martin, Perfect secret sharing schemes on five participants, Des. Codes Crypt., 9 (1996), 267-286.

[19]

E. D. KarninJ. W. Greene and M. E. Hellman, On secret sharing systems, IEEE Trans. Inf. Theory, 29 (1983), 35-41.

[20]

J. Martí-Farré and C. Padró, Secret sharing schemes with three or four minimal qualified subsets, Des. Codes Crypt., 34 (2005), 17-34.

[21]

J. Martí-Farré and C. Padró, Secret sharing schemes on access structures with intersection number equal to one, Discrete Appl. Math., 154 (2006), 552-563.

[22]

J. Martí-FarréC. Padró and L. Vázquez, Optimal complexity of secret sharing schemes with four minimal qualified subsets, Des. Codes Crypt., 61 (2011), 167-186.

[23]

K. M. Martin, New secret sharing schemes from old, J. Combin. Math. Combin. Comp., 14 (1993), 65-77.

[24]

C. Padró and G. Sáez, Secret sharing schemes with bipartite access structure, IEEE Trans. Inf. Theory, 46 (2000), 2596-2604.

[25]

C. Padró and G. Sáez, Lower bounds on the information rate of secret sharing schemes with homogeneous access structure, Inf. Process. Lett., 83 (2002), 345-351.

[26]

C. PadróL. Vázquez and A. Yang, Finding lower bounds on the complexity of secret sharing schemes by linear programming, Discrete Appl. Math., 161 (2013), 1072-1084.

[27]

A. Shamir, How to share a secret, Commun. ACM, 22 (1979), 612-613.

[28]

D. R. Stinson, An explication of secret sharing schemes, Des. Codes Crypt., 2 (1992), 357-390.

[29]

D. R. Stinson, Decomposition constructions for secret-sharing schemes, IEEE Trans. Inf. Theory, 40 (1994), 118-125.

[30]

H.-M. Sun and B.-L. Chen, Weighted decomposition construction for perfect secret sharing schemes, Comp. Math. Appl., 43 (2002), 877-887.

[31]

H.-M. SunH. WangB.-H. Ku and J. Pieprzyk, Decomposition construction for secret sharing schemes with graph access structures in polynomial time, SIAM J. Discrete Math., 24 (2010), 617-638.

[32]

M. Van Dijk, On the information rate of perfect secret sharing schemes, Des. Codes Crypt., 6 (1995), 143-169.

[33]

M. Van DijkW.-A. Jackson and K. M. Martin, A general decomposition construction for incomplete secret sharing schemes, Des. Codes Crypt., 15 (1998), 301-321.

[34]

M. Van DijkT. KevenaarG.-J. Schrijen and P. Tuyls, Improved constructions of secret sharing schemes by applying ($λ$, $ω$)-decompositions, Inf. Process. Lett., 99 (2006), 154-157.

Table 1.  An ideal $2$-decomposition for $\Gamma = \Gamma_{4}(\{1, 2, 3, 5, 9, C\})$
$[\Gamma^-]=23+5C+9C+1359$
$[{\Gamma ^{j}}^-]$ ${{\mathbf{\sigma }}^{j}}=\text{(}\sigma _{p}^{j}\text{)}_{p\in\mathcal{P}}$
$23$ $(0, 1, 1, 0, 0, 0)$
$5C+9C$ $(0, 0, 0, 1, 1, 1)$
$5C+1359 $ $(1, 0, 1, 1, 1, 1)$
$23+9C+1359+125C$ $(1, 1, 1, 1, 1, 1)$
$[\Gamma^-]=23+5C+9C+1359$
$[{\Gamma ^{j}}^-]$ ${{\mathbf{\sigma }}^{j}}=\text{(}\sigma _{p}^{j}\text{)}_{p\in\mathcal{P}}$
$23$ $(0, 1, 1, 0, 0, 0)$
$5C+9C$ $(0, 0, 0, 1, 1, 1)$
$5C+1359 $ $(1, 0, 1, 1, 1, 1)$
$23+9C+1359+125C$ $(1, 1, 1, 1, 1, 1)$
Table 2.  An ideal $(3, 1)$-decomposition for $\Gamma = \Gamma_{4}(\{1, 3, 5, A, B, C\})$
$[\Gamma^-]=5C + 3AB + ABC +135B$ $\Gamma^+=\{13BC, 13AC, 15AB, 135A, 35B\}$
$[{\Gamma ^{j}}^-]$$a_1\dots a_4$$b_1\dots b_5$${{\mathbf{\sigma }}^{j}}=\text{(}\sigma _{p}^{j}\text{)}_{p\in\mathcal{P}}$
$5C$$1000$$00000$$(0, 0, 1, 0, 0, 1)$
$ 5+AB$$1111$$00111$$(0, 0, 1, 1, 1, 0)$
$C+3AB+13B$$1111$$11000$$(1, 1, 0, 1, 1, 1)$
$3AB+ABC+135B+15BC$$0111$$00000$$(1, 1, 1, 1, 1, 1)$
Note. Consider an access structure $\Gamma$ with $\Gamma^-=\{A_1, \dots, A_m\}$ and $\Gamma^+=\{B_1, \dots, B_{M}\}$. Each bit $a_i$ of binary string $a_1\dots a_m$ in the second column indicates if $A_i$ is a qualified subset of $\Gamma^j$; that is, $a_i=1$ iff $A_i\in{\Gamma^j} $. Similarly, each bit $b_i$ of binary string $b_1\dots b_M$ in third column indicates if $B_i$ is a qualified subset of ${\Gamma^j}$; that is, $b_i=1$ iff $B_i\in{\Gamma^j} $.
$[\Gamma^-]=5C + 3AB + ABC +135B$ $\Gamma^+=\{13BC, 13AC, 15AB, 135A, 35B\}$
$[{\Gamma ^{j}}^-]$$a_1\dots a_4$$b_1\dots b_5$${{\mathbf{\sigma }}^{j}}=\text{(}\sigma _{p}^{j}\text{)}_{p\in\mathcal{P}}$
$5C$$1000$$00000$$(0, 0, 1, 0, 0, 1)$
$ 5+AB$$1111$$00111$$(0, 0, 1, 1, 1, 0)$
$C+3AB+13B$$1111$$11000$$(1, 1, 0, 1, 1, 1)$
$3AB+ABC+135B+15BC$$0111$$00000$$(1, 1, 1, 1, 1, 1)$
Note. Consider an access structure $\Gamma$ with $\Gamma^-=\{A_1, \dots, A_m\}$ and $\Gamma^+=\{B_1, \dots, B_{M}\}$. Each bit $a_i$ of binary string $a_1\dots a_m$ in the second column indicates if $A_i$ is a qualified subset of $\Gamma^j$; that is, $a_i=1$ iff $A_i\in{\Gamma^j} $. Similarly, each bit $b_i$ of binary string $b_1\dots b_M$ in third column indicates if $B_i$ is a qualified subset of ${\Gamma^j}$; that is, $b_i=1$ iff $B_i\in{\Gamma^j} $.
Table 3.  A $2$-weighted-decomposition for $\Gamma = \Gamma_{4}(\{3, 5, 6, 9, A, D\})$
$[\Gamma^-]=359D+36A+56D+9AD$
$[W_{j}^{-}]$ $\Sigma^j$ ${{\mathbf{\sigma }}^{j}}=\text{(}\sigma _{p}^{j}\text{)}_{p\in\mathcal{P}}$
$ 1\times(359D+36AD+56D+9AD) $ $(1, 1, 1, 1, 1, 1)$
$ 1 \times(359D+56D+9AD)+2\times(36A)$$\Sigma^2$ $(2, 1, 2, 1, 2, 2)$
Note. In $\Sigma^2$, the shares of participants are assigned as follows: $\mathbf{s}_3= (r_2+r_4-s_1, r_5)$, $\mathbf{s}_5 =r_3+r_4$, $\mathbf{s}_6 = (r_4, r_6), \mathbf{s}_9 =r_1+r_2, \mathbf{s}_A= (r_2, r_5+r_6+s_2)$, $\mathbf{s}_ D=(r_3+s_1, r_1+s_1)$.
$[\Gamma^-]=359D+36A+56D+9AD$
$[W_{j}^{-}]$ $\Sigma^j$ ${{\mathbf{\sigma }}^{j}}=\text{(}\sigma _{p}^{j}\text{)}_{p\in\mathcal{P}}$
$ 1\times(359D+36AD+56D+9AD) $ $(1, 1, 1, 1, 1, 1)$
$ 1 \times(359D+56D+9AD)+2\times(36A)$$\Sigma^2$ $(2, 1, 2, 1, 2, 2)$
Note. In $\Sigma^2$, the shares of participants are assigned as follows: $\mathbf{s}_3= (r_2+r_4-s_1, r_5)$, $\mathbf{s}_5 =r_3+r_4$, $\mathbf{s}_6 = (r_4, r_6), \mathbf{s}_9 =r_1+r_2, \mathbf{s}_A= (r_2, r_5+r_6+s_2)$, $\mathbf{s}_ D=(r_3+s_1, r_1+s_1)$.
Table 4.  Results obtained by ideal $\lambda$-decomposition
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{1}$$12359C$ $ 23 + 5C + 9C + 1359 $$[3/2, 5/3]$$3/2$
$\mathcal{A}_{2}$$ 12569C $ $ 26 + 9C + 159 + 56C $
$\mathcal{A}_{3}$$ 13569A $$ 56 + 9A + 36A + 1359 $
$\mathcal{A}_{4}$$ 1356AC $$ AC + 135 + 56C + 36A $
$\mathcal{A}_{5}$$35679A$$ 9A+567+367A+3579$
$\mathcal{A}_{6}$$ 127BCD $ $17BD+27B+7CD+BCD$ $[5/3, 11/6]$ $5/3$
Note. Details of decompositions can be found in Appendix A.1.
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{1}$$12359C$ $ 23 + 5C + 9C + 1359 $$[3/2, 5/3]$$3/2$
$\mathcal{A}_{2}$$ 12569C $ $ 26 + 9C + 159 + 56C $
$\mathcal{A}_{3}$$ 13569A $$ 56 + 9A + 36A + 1359 $
$\mathcal{A}_{4}$$ 1356AC $$ AC + 135 + 56C + 36A $
$\mathcal{A}_{5}$$35679A$$ 9A+567+367A+3579$
$\mathcal{A}_{6}$$ 127BCD $ $17BD+27B+7CD+BCD$ $[5/3, 11/6]$ $5/3$
Note. Details of decompositions can be found in Appendix A.1.
Table 5.  Result obtained by non-ideal $\lambda$-decomposition
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{7}$$167ABD$$17BD+67AB+67D+ABD $$[3/2, 5/3]$$3/2$
Note. Details of decomposition can be found in Appendix A.2.
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{7}$$167ABD$$17BD+67AB+67D+ABD $$[3/2, 5/3]$$3/2$
Note. Details of decomposition can be found in Appendix A.2.
Table 6.  Results obtained by ideal $(\lambda, \omega)$-decomposition
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{8}$$ 135ABC $$ 5C + 3AB + ABC + 135B $$[3/2, 5/3]$$3/2$
$\mathcal{A}_{9}$$ 125ACD $$ 2A+ 15D+ 5CD + ACD $
$\mathcal{A}_{10}$$ 136ACE $$ 13 + ACE + 6CE + 36AE $
$\mathcal{A}_{11}$$ 167ABC $$ 17B + 67C + ABC + 67AB $
Note. Details of decompositions can be found in Appendix A.3.
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{8}$$ 135ABC $$ 5C + 3AB + ABC + 135B $$[3/2, 5/3]$$3/2$
$\mathcal{A}_{9}$$ 125ACD $$ 2A+ 15D+ 5CD + ACD $
$\mathcal{A}_{10}$$ 136ACE $$ 13 + ACE + 6CE + 36AE $
$\mathcal{A}_{11}$$ 167ABC $$ 17B + 67C + ABC + 67AB $
Note. Details of decompositions can be found in Appendix A.3.
Table 7.  Results obtained by $\lambda$-weighted decomposition
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{12}$$ 3569AD $$359D+36A+56D+9AD$$[3/2, 5/3]$$3/2$
$\mathcal{A}_{13}$$ 1249AC $ $ 19 + 2A + 4C + 9AC $
$\mathcal{A}_{14}$$ 35679E $$ 3579+367E+567E+9E $
$\mathcal{A}_{15}$$ 3569BE $$359B+36BE+56E+9BE$$[3/2, 7/4]$
Note. Details of decompositions can be found in Appendix A.4.
$\mathcal{P}$Access Structure $\sigma$ from [22] $\sigma$
$\mathcal{A}_{12}$$ 3569AD $$359D+36A+56D+9AD$$[3/2, 5/3]$$3/2$
$\mathcal{A}_{13}$$ 1249AC $ $ 19 + 2A + 4C + 9AC $
$\mathcal{A}_{14}$$ 35679E $$ 3579+367E+567E+9E $
$\mathcal{A}_{15}$$ 3569BE $$359B+36BE+56E+9BE$$[3/2, 7/4]$
Note. Details of decompositions can be found in Appendix A.4.
Table 8.  Results obtained from the corresponding dual graph access structures
$\mathcal{P}$Access Structure $(\cong \Gamma^*)$ $\sigma$ from [22] $\sigma$
$167BDE $$ 17BD+67BE+67DE+BDE $$(\cong \Gamma^*_{62})$$[3/2, 5/3]$$3/2$[14,32]
$356BDE$$35BD+36BE+56DE+BDE$$ (\cong\Gamma^*_{68} )$
$357ABC$$357B+37AB+57C+ABC$$( \cong \Gamma^*_{33}) $$[3/2, 7/4]$
$357ACE$ $ 357+37AE+57CE+ACE $$ (\cong\Gamma^*_{36}) $
$37BCDE $$37BD+37BE+7CDE+BCDE $$(\cong \Gamma^*_{102}) $$[3/2, 11/6]$
$125ADE$$15D+2AE+5DE+ADE $$(\cong \Gamma^*_{14}) $$[5/3, 7/4]$$5/3$[32]
$135ADE$$135D+3AE+5DE+ADE $ $(\cong \Gamma^*_{29} )$
$137BCE$$ 137B+37BE+7CE+BCE $ $(\cong \Gamma^*_{48} ) $$[5/3, 11/6]$
$124BDE$$1BD+2BE+4DE+BDE $$ (\cong \Gamma^*_{9} ) $$[7/4, 11/6]$$ 7/4$ [30,32,15]
$ 125BDE $$ 15BD+2BE+5DE+BDE$$ (\cong \Gamma^*_{22}) $
$ 127BDE $$17BD+27BE+7DE+BDE $$ (\cong \Gamma^*_{40}) $
$ 135BDE $$135BD+3BE+5DE+BDE $$ (\cong \Gamma^*_{42}) $
$ 136BDE $$13BD+36BE+6DE+BDE $$(\cong \Gamma^*_{43}) $
$ 137BDE $$ 137BD+37BE+7DE+BDE $$ (\cong\Gamma^*_{61}) $
$\mathcal{P}$Access Structure $(\cong \Gamma^*)$ $\sigma$ from [22] $\sigma$
$167BDE $$ 17BD+67BE+67DE+BDE $$(\cong \Gamma^*_{62})$$[3/2, 5/3]$$3/2$[14,32]
$356BDE$$35BD+36BE+56DE+BDE$$ (\cong\Gamma^*_{68} )$
$357ABC$$357B+37AB+57C+ABC$$( \cong \Gamma^*_{33}) $$[3/2, 7/4]$
$357ACE$ $ 357+37AE+57CE+ACE $$ (\cong\Gamma^*_{36}) $
$37BCDE $$37BD+37BE+7CDE+BCDE $$(\cong \Gamma^*_{102}) $$[3/2, 11/6]$
$125ADE$$15D+2AE+5DE+ADE $$(\cong \Gamma^*_{14}) $$[5/3, 7/4]$$5/3$[32]
$135ADE$$135D+3AE+5DE+ADE $ $(\cong \Gamma^*_{29} )$
$137BCE$$ 137B+37BE+7CE+BCE $ $(\cong \Gamma^*_{48} ) $$[5/3, 11/6]$
$124BDE$$1BD+2BE+4DE+BDE $$ (\cong \Gamma^*_{9} ) $$[7/4, 11/6]$$ 7/4$ [30,32,15]
$ 125BDE $$ 15BD+2BE+5DE+BDE$$ (\cong \Gamma^*_{22}) $
$ 127BDE $$17BD+27BE+7DE+BDE $$ (\cong \Gamma^*_{40}) $
$ 135BDE $$135BD+3BE+5DE+BDE $$ (\cong \Gamma^*_{42}) $
$ 136BDE $$13BD+36BE+6DE+BDE $$(\cong \Gamma^*_{43}) $
$ 137BDE $$ 137BD+37BE+7DE+BDE $$ (\cong\Gamma^*_{61}) $
Table 9.  Open access structures
$\mathcal{P}$Access structure $\sigma$ [32,22]
$\{3, 5, 7, A, D, E\}$ $ 357D+37AE+57DE+ADE$ $(\cong \Gamma^*_{75})$$[3/2, 5/3]$
$\{3, 5, 7, B, D, E\}$ $ 357BD+37BE+57DE+BDE$ $(\cong \Gamma_{84}^*) $
$\{1, 6, 7, A, D, E\}$$17D+67AE+67DE+ADE$
$ \{3, 5, 7, 9, B, E\} $$3579B+37BE+57E+9BE$$[3/2, 11/6]$
$\mathcal{P}$Access structure $\sigma$ [32,22]
$\{3, 5, 7, A, D, E\}$ $ 357D+37AE+57DE+ADE$ $(\cong \Gamma^*_{75})$$[3/2, 5/3]$
$\{3, 5, 7, B, D, E\}$ $ 357BD+37BE+57DE+BDE$ $(\cong \Gamma_{84}^*) $
$\{1, 6, 7, A, D, E\}$$17D+67AE+67DE+ADE$
$ \{3, 5, 7, 9, B, E\} $$3579B+37BE+57E+9BE$$[3/2, 11/6]$
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