New results for traitor tracing schemes

In the last two decades, several classes of codes are introduced to protect the copyrighted digital data. They have important applications in the scenarios like digital fingerprinting and broadcast encryption schemes. In this paper we will discuss three important classes of such codes, namely, frameproof codes, parent-identifying codes and traceability codes. Firstly, suppose $N(t)$ is the minimal integer such that there exists a binary $t$-frameproof code of length $N$ with cardinality larger than $N$, we prove that $N(t)\ge\frac{15+\sqrt{33}}{24} (t-2)^2$, which is a great improvement of the previously known bound $N(t)\ge\binom{t+1}{2}$. Moreover, we find that the determination of $N(t)$ is closely related to a conjecture of Erd\H{o}s, Frankl and F\"uredi posed in the 1980's, which implies the conjectured value $N(t)=t^2+o(t^2)$. Secondly, we derive a new upper bound for parent-identifying codes, which is superior than all previously known bounds. Thirdly, we present an upper bound for 3-traceability codes, which shows that a $q$-ary 3-traceability code of length $N$ can have at most $cq^{\lceil N/9\rceil}$ codewords, where $c$ is a constant only related to the code length $N$. It is the first meaningful upper bound for 3-traceability codes and our result supports a conjecture of Blackburn et al. posed in 2010.