Secure frameproof codes, key distribution patterns, group testing algorithms and related structures

Abstract

Frameproof codes were introduced by Boneh and Shaw as a method of “digital fingerprinting” which prevents a coalition of a specified size c from framing a user not in the coalition. Stinson and Wei then gave a combinatorial formulation of the problem in terms of certain types of extremal set systems. In this paper, we study frameproof codes that provide a certain (weak) form of traceability. We extend our combinatorial formulation to address this stronger requirement, and show that the problem is solved by using (i,j)-separating systems, as defined by Friedman, Graham and Ullman. Using constructions based on perfect hash families, we give the first efficient explicit constructions for these objects for general values of i and j. We also review nonconstructive existence results that are based on probabilistic arguments. Then we look at two other, related concepts, namely key distribution patterns and nonadaptive group testing algorithms. We again approach these problems from the point of view of extremal set systems, and we describe a natural common setting in which these two problems are complementary special cases. This approach also demonstrates a close relationship between these two problems and frameproof codes. Explicit constructions are given, and some nonconstructive existence results are reviewed. In the case of key distribution patterns, our explicit constructions are the most efficient ones known.