A Generalization of Cover Free Families

TL;DR

This paper determines exact values and improves constructions for certain cover-free families, introduces a generalized concept motivated by key predistribution applications, and explores its properties and bounds.

Contribution

It provides exact values for N((r;w; d); t) in special cases, new constructions for specific cover-free families, and a novel generalization inspired by cryptographic applications.

Findings

Exact values of N((r;w; d); t) for some parameters

Improved constructions for (2; 1; d)-CFF and (2; 2; d)-CFF

Introduction and analysis of a generalized cover-free family

Abstract

An (r;w; d) -cover-free family (CFF) is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. The minimum number of elements for which there exists an (r;w; d)-CFF with t blocks is denoted by N((r;w; d); t). In this paper, we determine the exact value of N((r;w; d); t) for some special parameters. Also, we present two constructions for (2; 1; d)- CFF and (2; 2; d)-CFF which improve the existing constructions. Moreover, we introduce a generalization of cover-free families which is motivated by an application of CFF in the key predistribution schemes. Also, we investigate some properties and bounds on the parameters of this generalization.