A first approach to the solution of that problem for the family of the $3$-homogeneous access structures is made in this paper. First, we present an ideal $3$-homogeneous access structure that is not vector space. Afterwards, we prove that the $3$-homogeneous access structures that can be realized by a ${\bf Z}_2$-vector space secret sharing scheme are {\em sparse\/}, that is, any subset of four participants contains at most two minimal qualified subsets. Finally, we solve the characterization problem for the family of the sparse $3$-homogeneous access structures. Specifically, we completely characterize the ideal access structures in this family, we prove that they coincide with the ${\bf Z}_2$-vector space ones and, besides, we demonstrate that there is no structure in this family having optimal information rate between $2/3$ and $1$. That is, we establish that the properties that were previously proved for several families also hold for the family of the sparse $3$-homogeneous access structures.
Category / Keywords: cryptographic protocols / Secret sharing, Information rate, Ideal secret sharing schemes. Publication Info: This is the preliminary version of the paper that appeared in ElectronicJournal of Combinatorics. A previous version appeared in the Proceedings of the International Workshop on Coding and Cryptography WCC 2003, Versailles, France. Date: received 30 Jul 2003, last revised 5 Jan 2007 Contact author: matcpl at mat upc es Available formats: PDF | BibTeX Citation Note: 5 Jan 2007: Publication info updated Version: 20070105:120513 (All versions of this report) Discussion forum: Show discussion | Start new discussion