Probabilistic existence results for parent-identifying schemes

Parent-identifying schemes provide a way to identify causes from effects for some information systems, such as digital fingerprinting and group testing. In this paper, we consider the combinatorial structures for parent-identifying schemes. First, we establish an equivalent relationship between the parent-identifying schemes and forbidden configurations. Based on this relationship, we derive the probabilistic existence lower bounds for two related combinatorial structures, that is, $t$ -parent-identifying set systems ( $t$ -IPPS) and $t$ -multimedia parent-identifying codes ( $t$ -MIPPC), which are used in broadcast encryption and multimedia fingerprinting, respectively. The probabilistic lower bound for the maximum size of a $t$ -IPPS has the asymptotically optimal order of magnitude in many cases, and that for $t$ -MIPPC provides the asymptotically optimal code rate when $t=2$ and the best known asymptotic code rate when $t\ge 3$ . Furthermore, we analyze the structure of 2-IPPS and prove some bounds for certain cases.