Nonperfect secret sharing schemes

A nonperfect secret sharing scheme (NSS) consists of a family of access subsets Γ₁, a family of semi-access subsets Γ₂ and a family of non-access subsets Γ₃. In an NSS, it is possible that ¦V_i¦<¦S¦, where ¦V_i¦ is the size of the share and ¦S¦ is the size of the secret. This paper characterizes nonperfect secret sharing schemes. First, we show that (Γ₁, Γ₂, Γ₃) is realizable if and only if Γ₁ is monotone and Γ₁ ∪ Γ₂ is monotone. Then, we derive a lower bound of ¦V_i¦ in terms of a distance between Γ₁ and Γ₃. Finally, we show a condition for (Γ₁, Γ₂, Γ₃) to achieve ¦V _i ¦=¦S¦/2 for all i.