Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Secret sharing schemes on sparse homogeneous access structures with rank three. | Secret sharing schemes on sparse homogeneous access structures with rank three Jaume Marti-Farre Carles Padró Dept. Matematica Aplicada IV Universitat Politecnica de Catalunya C. Jordi Girona 1-3 Modul C3 Campus Nord 08034 Barcelona Spain jaumem@ matcpl@ Submitted May 17 2004 Accepted Sep 22 2004 Published Oct 7 2004 Mathematics Subject Classifications 94A62 94A60 Abstract 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. Keywords. Cryptography Secret sharing schemes Information rate Ideal secret sharing schemes. 1 Introduction A secret sharing scheme E is a method to distribute a secret value k E K among a set of participants P. Every participant p E P receives a share sp E Sp in such a way that This work was partially supported by the Spanish Ministerio de Ciencia y Tecnologia under projects TIC 2000-1044 and TIC 2003-00866. The material in this paper was presented in part at the International Workshop on Coding and Cryptography WCC 2003 Versailles France. An earlier version of this paper
