Secret Sharing Schemes on Sparse Homogeneous Access Structures with Rank Three

One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most $2/3$. An access structure is said to be $r$-homogeneous if there are exactly $r$ participants in every minimal qualified subset. A first approach to the characterization of the ideal $3$-homogeneous access structures is made in this paper. We show that the results in the previously studied families can not be directly generalized to this one. Nevertheless, we prove that the equivalences above apply to the family of the sparse $3$-homogeneous access structures, that is, those in which any subset of four participants contains at most two minimal qualified subsets. Besides, we give a complete description of the ideal sparse $3$-homogeneous access structures.

[1]  Ernest F. Brickell,et al.  Some Ideal Secret Sharing Schemes , 1990, EUROCRYPT.

[2]  Alfredo De Santis,et al.  On the Information Rate of Secret Sharing Schemes (Extended Abstract) , 1992, CRYPTO.

[3]  Douglas R Stinson,et al.  Some improved bounds on the information rate of perfect secret sharing schemes , 1990, Journal of Cryptology.

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

[5]  K. Martin,et al.  Perfect secret sharing schemes on five participants , 1996 .

[6]  Avi Wigderson,et al.  On span programs , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[7]  G. R. Blakley,et al.  Safeguarding cryptographic keys , 1899, 1979 International Workshop on Managing Requirements Knowledge (MARK).

[8]  Ernest F. Brickell,et al.  On the classification of ideal secret sharing schemes , 1989, Journal of Cryptology.

[9]  Alfredo De Santis,et al.  On the Information Rate of Secret Sharing Schemes , 1996, Theor. Comput. Sci..

[10]  Alexei E. Ashikhmin,et al.  Almost Affine Codes , 1998, Des. Codes Cryptogr..

[11]  Paul D. Seymour.  On secret-sharing matroids , 1992, J. Comb. Theory, Ser. B.

[12]  Alfredo De Santis,et al.  Graph decompositions and secret sharing schemes , 2004, Journal of Cryptology.

[13]  Douglas R. Stinson,et al.  Decomposition constructions for secret-sharing schemes , 1994, IEEE Trans. Inf. Theory.

[14]  Carles Padró,et al.  Secret sharing schemes with bipartite access structure , 2000, IEEE Trans. Inf. Theory.

[15]  Carles Padró,et al.  Secret Sharing Schemes with Three or Four Minimal Qualified Subsets , 2005, Des. Codes Cryptogr..

[16]  Alfredo De Santis,et al.  On the size of shares for secret sharing schemes , 1991, Journal of Cryptology.

[17]  Alfredo De Santis,et al.  Tight Bounds on the Information Rate of Secret Sharing Schemes , 1997, Des. Codes Cryptogr..

[18]  Carles Padró,et al.  Lower bounds on the information rate of secret sharing schemes with homogeneous access structure , 2002, Inf. Process. Lett..

[19]  Douglas R. Stinson,et al.  An explication of secret sharing schemes , 1992, Des. Codes Cryptogr..

[20]  van Me Marten Dijk.  Secret key sharing and secret key generation , 1997 .

[21]  Carles Padró,et al.  Secret Sharing Schemes on Access Structures with Intersection Number Equal to One , 2002, SCN.

[22]  James L. Massey,et al.  Minimal Codewords and Secret Sharing , 1999 .