Secret Sharing Scheme: Vector Space Secret Sharing and F Function

Let P = {P₁, P₂, ..., P_n} be set of participants and Γ = {B_i|B_i ⊂ P, 1 ≤ i ≤ k} be access structure. Vector space secret sharing realizing access structure Γ requires existence of function φ : P → (Z_p)^d, where p is a prime number and d ≥ 2 is an integer, satisfying the following condition (1, 0, 0, ..., 0) =<; φ(P_i) : P_i ∈ B >;⇔ B ∈ Γ = {B₁, B₂,..., B_k}. There is no known algorithm to construct such a function φ in general. Constructions are mainly done by trial and error. In this paper, we developed a polynomial algorithm to construct a φ function for certain type of access structures. Some examples are given to illustrate the algorithms.