## 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_{I}_{P}_{P}_{C}(N, q, t) and M_{T}_{A}(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_{I}_{P}_{P}_{C}(N, q, t) ≤ rq^{⌈}^{N}^{/}^{(}^{v}^{-}^{1}^{)}^{⌉}+ (v- 1 - r) q^{⌊}^{N}^{/}^{(}^{v}^{-}^{1}^{)}^{⌋}, where v= ⌊ (t/ 2 + 1) ^{2}⌋ , 0 ≤ r≤ v- 2 and N≡rmod(v-1). This new bound improves two previously known bounds of Blackburn, and Alon and Stav. On the other hand, M_{T}_{A}(N, q, t) is still not known for almost all t. In 2010, Blackburn, Etzion and Ng asked whether MTA(N,q,t)≤cq⌈N/t2⌉ 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.

Original language | English |
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Pages (from-to) | 1727-1737 |

Number of pages | 11 |

Journal | Designs, Codes, and Cryptography |

Volume | 86 |

Issue number | 8 |

DOIs | |

State | Published - 1 Aug 2018 |

Externally published | Yes |

## Keywords

- Parent-identifying codes
- Traceability codes
- Traitor tracing schemes