# Probabilistic Existence Results for Parent-Identifying Schemes

## TL;DR

This paper establishes probabilistic existence bounds for parent-identifying schemes, linking combinatorial structures to forbidden configurations, and provides asymptotically optimal bounds for certain parameters in digital fingerprinting and group testing.

## Contribution

It introduces a novel relationship between parent-identifying schemes and forbidden configurations, deriving probabilistic bounds and analyzing structures for improved scheme design.

## Key findings

- Probabilistic lower bounds for $t$-IPPS and $t$-MIPPC.
- Asymptotically optimal order of magnitude for $t$-IPPS size.
- Best known asymptotic code rate for $t$-MIPPC when $t
eq 2$.

## Abstract

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 combinatorial structures for parent-identifying schemes. First, we establish an equivalent relationship between parent-identifying schemes and forbidden configurations. Based on this relationship, we derive 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 $t\geq 3$. Furthermore, we analyze the structure of $2$-IPPS and prove some bounds for certain cases.