Optimum general threshold secret sharing

Abstract

An important issue of threshold secret sharing (TSS) schemes is to minimize the size of shares. This issue is resolved for the simpler classes called (k,n)-TSS and (k,L,n)-threshold ramp secret sharing (TRSS). That is, for each of these two classes, an optimum construction which minimizes the share size was presented. The goal of this paper is to develop an optimum construction for a more general threshold class where the mutual information between the secret and a set of shares is defined by a discrete function which monotonically increases from zero to one with the number of shares. A tight lower bound of the entropy of shares is first derived and then an optimum construction is presented. The derived lower bound is larger than the previous one except for special functions such as convex and concave functions. The optimum construction encodes the secret by using one or more optimum TRSS schemes independently. The optimality is shown by devising a combination of TRSS schemes which achieves the new lower bound. © 2012 Springer-Verlag.