New Bounds on 2-Frameproof Codes of Length 4

Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. These digital fingerprints are generally denoted as codewords in Qn, where Q is an alphabet of size q and n is a positive integer. A 2-frameproof code is a code C such that any 2 codewords in C cannot form a new codeword under a particular rule. Thus, no pair of users can frame a user who is not a member of the coalition. This paper concentrates on the upper bound for the size of a q-ary 2-frameproof code of length 4. Our new upper bound shows that C≤2q2−2q+1 when q is odd and q>10.

Secure frameproof codes through biclique covers

Graph Theory International audience For a binary code Γ of length v, a v-word w produces by a set of codewords {w1,...,wr}⊆Γ if for all i=1,...,v, we have wi∈{w1i,...,wri} . We call a code r-secure frameproof of size t if |Γ|=t and for any v-word that is produced by two sets C1 and C2 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≥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 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.