防帧码及相关组合结构的界和算法
Bounds and Algorithms for Frameproof Codes and Related Combinatorial Structures
摘要：
本文研究了Boneh和Shaw提出的保护版权材料的防帧码的最小长度的上界。长度为PDMATHABM 0的$q$元$(k,n)$防帧码是在$\{0,1,\ldots, q-1\}$中具有条目的$t \times n$矩阵，并且具有这样的性质：对于任何列$\mathbf{c}$和任何其它$k$列，存在一行，其中$k$列的符号都与列$\mathbf{c}$的对应符号（在同一行中）不同.本文利用Lovász局部引理证明了$q$元$(k,n)$ -防帧码的存在性，对于$q \leq k$，码长为$t = O(\frac{k^2}{q} \log n)$;对于$q > k$，码长为$t = O(\frac{k}{\log(q/k)}\log(n/k))$.值得注意的是，对于$q \leq k$的实际情况下，我们的研究结果给出的代码的长度几乎匹配的下限$Ω(\frac{k^2}{q\log k} \log n)$上的任何$q$元$(k,n)$防帧代码的长度，更重要的是，让我们得到一个算法的复杂性$O(t n^2)$的建设这样的代码。
英文摘要：
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 $\{0,1,\ldots, q-1\}$ 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 $q \leq k$, using the Lovász Local Lemma, and of length $t = O(\frac{k}{\log(q/k)}\log(n/k))$ for $q > k$ using the expurgation method. Remarkably, for the practical case of $q \leq k$ our findings give codes whose length almost matches the lower bound $Ω(\frac{k^2}{q\log k} \log n)$ 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.
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