New upper bounds for parent-identifying codes and traceability codes

Abstract

In the last two decades, parent-identifying codes and traceability codes are introduced to prevent copyrighted digital data from unauthorized use. They have important applications in the scenarios like digital fingerprinting and broadcast encryption schemes. A major open problem in this research area is to determine the upper bounds for the cardinalities of these codes. In this paper we will focus on this theme. Consider a code of length N which is defined over an alphabet of size q. Let $M_{IPPC}(N,q,t)$ and $M_{TA}(N,q,t)$ denote the maximal cardinalities of t-parent-identifying codes and t-traceability codes, respectively, where t is known as the strength of the codes. We show $M_{IPPC}(N,q,t)\le rq^{\lceil N/(v-1)\rceil }+(v-1-r)q^{\lfloor N/(v-1)\rfloor }$, where $v=\lfloor (t/2+1)^2\rfloor$, $0\le r\le v-2$ and $N\equiv r \mod (v-1)$. This new bound improves two previously known bounds of Blackburn, and Alon and Stav. On the other hand, $M_{TA}(N,q,t)$ is still not known for almost all t. In 2010, Blackburn, Etzion and Ng asked whether $M_{TA}(N,q,t)\le cq^{\lceil N/t^2\rceil }$ or not, where c is a constant depending only on N, and they have shown the only known validity of this bound for $t=2$. By using some complicated combinatorial counting arguments, we prove this bound for $t=3$. This is the first non-trivial upper bound in the literature for traceability codes with strength three.