Security of the center in key distribution schemes

Abstract

This paper studies the security of the trusted center in a b-secure t-group key distribution scheme in which the center issues a private information v _i , ∈ V to each user i based on the center secret s ∈ S. User i can obtain the common key k ∈ K with t−1 other users from v _i so that any dishonest b users cannot learn any key they should not know. First, we show how to distribute the power of the center to m centers so that, even if l−1 centers and b users collaborate, they cannot learn any key that b users should not know. Next, we prove that

$$\log _2 |S| \geqslant \left( {\begin{array}{*{20}c}{b + t} \\t \\\end{array} } \right)\log _2 |K|$$

.