Bounds and Algorithms for Frameproof Codes and Related Combinatorial Structures

In this paper, we study upper bounds on the minimum length of frameproof codes introduced by Boneh and Shaw to protect copyrighted materials. A $q$-ary $(k,n)$-frameproof code of length $t$ is a $t\times n$ matrix having entries in $\{0,1,\dots ,q-1\}$ and with the property that for any column $\mathbf{c}$ and any other $k$ columns, there exists a row where the symbols of the $k$ columns are all different from the corresponding symbol (in the same row) of the column $\mathbf{c}$. In this paper, we show the existence of $q$-ary $(k,n)$-frameproof codes of length $t=O(\frac{k^2}{q}\log n)$ for $q\le k$, using the Lovász Local Lemma, and of length $t=O(\frac{k}{\log (q\mathrm{/}k)}\log (n\mathrm{/}k))$ for $q>k$ using the expurgation method. Remarkably, for the practical case of $q\le k$ our findings give codes whose length almost matches the lower bound $\mathrm{\Omega }(\frac{k^2}{q\log k}\log n)$ on the length of any $q$-ary $(k,n)$-frameproof code and, more importantly, allow us to derive an algorithm of complexity $O(t{n}^2)$ for the construction of such codes.