Bounds and Algorithms for Frameproof Codes and Related Combinatorial Structures
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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 × n matrix having entries in {0,1,…, q-1} and with the property that for any column 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 𝐜. In this paper, we show the existence of q-ary (k,n)-frameproof codes of length t = O(k^2/qlog n) for q ≤ k, using the Lovász Local Lemma, and of length t = O(k/log(q/k)log(n/k)) for q > k using the expurgation method. Remarkably, for the practical case of q ≤ k our findings give codes whose length almost matches the lower bound Ω(k^2/qlog klog 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.
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