Union–intersection-bounded families and their applications

Abstract Cover-free families have been widely studied over recent decades due to their applications in numerous subjects. In this paper, we introduce the concept of ( s , t ; d ) -union–intersection-bounded families, which is a generalization of t -cover-free families. We provide a general upper bound on the maximum size of an ( s , t ; d ) -union–intersection-bounded family, and show a probabilistic lower bound for the case that the ground set is sufficiently large. They have the same order of magnitude for certain cases. We also discuss the applications of ( s , t ; d ) -union–intersection-bounded families in broadcast encryption, and derive a better upper bound for ( 1 , t ; d ) -union–intersection-bounded families (also known as superimposed distance codes).