Secret Sharing Scheme: Vector Space Secret Sharing and F Function

• Authors: Mustafa Atici
• Venue: ASONAM '12 Proceedings of the 2012 International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2012)
• Year: 2012

Let ${\cal P}=\{P_{1}, P_{2}, ..., P_{n}\}$ be set of participants and $\Gamma=\{B_{i}| B_{i}\subset {\cal P}, 1\leq i \leq k \}$ be access structure. Vector space secret sharing realizing access structure $\Gamma$ requires existence of function $\phi : {\cal P}\longrightarrow ({\cal Z}_{p})^{d}$, where $p$ is a prime number and $d\geq 2$ is an integer, satisfying the following condition $(1, 0, 0, ..., 0)= \Leftrightarrow B \in \Gamma=\{ B_{1}, B_{2}, ..., B_{k}\}$. There is no known algorithm to construct such a function $\phi$ in general. Constructions are mainly done by trial and error. In this paper, we developed a polynomial algorithm to construct a $\phi$ function for certain type of access structures. Some examples are given to illustrate the algorithms.