Secure construction for nonlinear function threshold ramp secret sharing

There are two types of threshold ramp secret sharing (TRSS) schemes: Linear function ramp and nonlinear function ramp. A linear (resp. nonlinear) function TRSS scheme reveals information of the secret linearly (resp. nonlinearly). There are many studies on the linear ones and various secure and efficient constructions have been proposed. In contrast, the notion of the nonlinear function scheme was recently introduced, and any previous construction is either insecure or inefficient. This paper first points out defects of the previous insecure construction, and then presents the first secure and efficient construction. The proposed construction can achieves H(V-i) < H(S) while in the previous secure construction H(V-i) = H(S) where H(V-i) and H(S) are the entropies of each share and the secret, respectively.