Secure Frameproof Code Through Biclique Cover

TL;DR

This paper establishes a connection between secure frameproof codes and biclique covers of Kneser graphs, providing new insights and bounds for their construction and properties.

Contribution

It characterizes the existence of secure frameproof codes via biclique covers of Kneser graphs and explores their relation to cover-free families, offering new bounds and theoretical insights.

Findings

Existence of secure frameproof codes characterized by 1-biclique covers of Kneser graphs.

Connection established between biclique covers and cover-free families.

An upper bound for the 1-biclique covering number of Kneser graphs provided.

Abstract

For a binary code $\Gamma$ of length $v$, a $v$-word $w$ produces by a set of codewords $\{w^1,...,w^r\} \subseteq \Gamma$ if for all $i=1,...,v$, we have $w_i\in \{w_i^1, ..., w_i^r\}$ . We call a code $r$-secure frameproof of size $t$ if $|\Gamma|=t$ and for any $v$-word that is produced by two sets $C_1$ and $C_2$ of size at most $r$ then the intersection of these sets is nonempty. A $d$-biclique cover of size $v$ of a graph $G$ is a collection of $v$-complete bipartite subgraphs of $G$ such that each edge of $G$ belongs to at least $d$ of these complete bipartite subgraphs. In this paper, we show that for $t\geq 2r$, an $r$-secure frameproof code of size $t$ and length $v$ exists if and only if there exists a 1-biclique cover of size $v$ for the Kneser graph ${\rm KG}(t,r)$ whose vertices are all $r$-subsets of a $t$-element set and two $r$-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of $d$-biclique covers of Kneser graphs and cover-free families, where an $(r,w; d)$ cover-free family 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. Also, we present an upper bound for 1-biclique covering number of Kneser graphs.