Separating hash families

In der vorliegenden Dissertation wird angestrebt, offene Probleme im Zusammenhang mit sogenannten "separating hash families" zu diskutieren und zu lösen. Separating hash families (SHF) sind interessante kombinatorische Strukturen, die verschiedene bekannte Objekte als Spezialfälle einschließen, wie z.B. perfect hash families (PHF), frameproof codes, secure frameproof codes und codes with identifiable parent property. Ferner finden SHFs zahlreiche kryptographische Anwendungen, z.B. in key distribution patterns, broadcast encryption, secret sharing schemes, visual cryptography und in den Codes für den Urheberrechtsschutz. In dieser Dissertation konzentrieren wir uns auf die Herleitung oberer Schranken für die Anzahl der Spalten einer SHF. Zuerst werden spezifische Typen von SHFs untersucht und einige ihrer Eigenschaften bewiesen. Basierend darauf erzielen wir neue obere Schranken für die maximale Anzahl der Spalten bzw. untere Schranken für die minimale Anzahl der Zeilen einer SHF. Für bestimmte Parameter geben wir Konstruktionen von SHFs, so dass die erzielten Schranken mit Gleichheit erfüllt sind. Damit sind die Schranken im oberen Fall optimal. Anschließend untersuchen wir generelle SHFs und stellen drei neue obere Schranken vor, die schärfer als alle bisher bekannten Schranken sind

Separating hash families with large universe

Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of universe for a $t$-perfect hash family of length $N$ over an alphabet of size $q$. In this paper, we show that $q^{2-o(1)}<p_t(t, q)=o(q^2)$ for all $t\geq 3$, which answers an open problem about separating hash families raised by Blackburn et al. in 2008 for certain parameters. Previously, this result was known only for $t=3, 4$. Our proof is obtained by establishing the existence of a large set of integers avoiding nontrivial solutions to a set of correlated linear equations.Comment: 17 pages, no figur

Fingerprinting Codes and Separating Hash Families

The thesis examines two related combinatorial objects, namely fingerprinting codes and separating hash families. Fingerprinting codes are combinatorial objects that have been studied for more than 15 years due to their applications in digital data copyright protection and their combinatorial interest. Four well-known types of fingerprinting codes are studied in this thesis; traceability, identifiable parent property, secure frameproof and frameproof. Each type of code is named after the security properties it guarantees. However, the power of these four types of fingerprinting codes is limited by a certain condition. The first known attempt to go beyond that came out in the concept of two-level traceability codes, introduced by Anthapadmanabhan and Barg (2009). This thesis extends their work to the other three types of fingerprinting codes, so in this thesis four types of two-level fingerprinting codes are defined. In addition, the relationships between the different types of codes are studied. We propose some first explicit non-trivial constructions for two-level fingerprinting codes and provide some bounds on the size of these codes. Separating hash families were introduced by Stinson, van Trung, and Wei as a tool for creating an explicit construction for frameproof codes in 1998. In this thesis, we state a new definition of separating hash families, and mainly focus on improving previously known bounds for separating hash families in some special cases that related to fingerprinting codes. We improve upper bounds on the size of frameproof and secure frameproof codes under the language of separating hash families

Linear time Constructions of some dd-Restriction Problems

We give new linear time globally explicit constructions for perfect hash families, cover-free families and separating hash functions

Improved Constructions of Frameproof Codes

Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let $M_{c,l}(q)$ be the largest cardinality of a $q$-ary $c$-frameproof code of length $l$ and $R_{c,l}=\lim_{q\rightarrow \infty}M_{c,l}(q)/q^{\lceil l/c\rceil}$. It has been determined by Blackburn that $R_{c,l}=1$ when $l\equiv 1\ (\bmod\ c)$, $R_{c,l}=2$ when $c=2$ and $l$ is even, and $R_{3,5}=5/3$. In this paper, we give a recursive construction for $c$-frameproof codes of length $l$ with respect to the alphabet size $q$. As applications of this construction, we establish the existence results for $q$-ary $c$-frameproof codes of length $c+2$ and size $\frac{c+2}{c}(q-1)^2+1$ for all odd $q$ when $c=2$ and for all $q\equiv 4\pmod{6}$ when $c=3$. Furthermore, we show that $R_{c,c+2}=(c+2)/c$ meeting the upper bound given by Blackburn, for all integers $c$ such that $c+1$ is a prime power.Comment: 6 pages, to appear in Information Theory, IEEE Transactions o