On the role of mutual infomation between the shares in a robust (k, n)-threshold scheme

In this paper we unveil basic properties of a (k, n)-threshold scheme secure against an extended impersonation attack by an opponent. We consider a scenario in which an opponent tries to impersonate one of the participants under knowledge of other shares. We first establish lower bounds on not only the sizes of the n shares and the uniform random number needed for generation of the shares but also the probability of successful impersonation. Next, we construct a (k, n)-threshold scheme with detectability of the extended impersonation attack, which turns out to be optimal in a certain case. Mutual information between the shares plays key roles in the analysis.