Fingerprinting Codes and Related Combinatorial Structures

Fingerprinting codes were introduced by Boneh and Shaw in 1998 as a method of copyright control. The desired properties of a good fingerprinting code has been found to have deep connections to combinatorial structures such as error-correcting codes and cover-free families. The particular property that motivated our research is called "frameproof". This has been studied extensively when the alphabet size q is at least as large as the colluder size w. Much less is known about the case q < w, and we prove several interesting properties about the binary case q = 2 in this thesis. When the length of the code N is relatively small, we have shown that the number of codewords n cannot exceed N, which is a tight bound since the n = N case can be satisfied a trivial construction using permutation matrices. Furthermore, the only possible candidates are equivalent to this trivial construction. Generalization to a restricted parameter set of separating hash families is also given. As a consequence, the above result motivates the question of when a non-trivial construction can be found, and we give some definitive answers by considering combinatorial designs. In particular, we give a necessary and sufficient condition for a symmetric design to be a binary 3-frameproof code, and provide example classes of symmetric designs that satisfy or fail this condition. Finally, we apply our results to a problem of constructing short binary frameproof codes

Sequential and Dynamic Frameproof Codes

There are many schemes in the literature for protecting digital data from piracy by the use of digital fingerprinting, such as frameproof codes and traitor-tracing schemes. The concept of traitor tracing has been applied to a digital broadcast setting in the form of dynamic traitor-tracing schemes and sequential traitor-tracing schemes, which could be used tocombat piracy of pay-TV broadcasts, for example. In this paper we extend the properties of frameproof codes to this dynamic model, defining and constructing both l-sequential frameproof codes and l-dynamic-frameproof codes. We also give bounds on the number of users supported by such schemes

Construction of Almost Disjunct Matrices for Group Testing

In a \emph{group testing} scheme, a set of tests is designed to identify a small number $t$ of defective items among a large set (of size $N$) of items. In the non-adaptive scenario the set of tests has to be designed in one-shot. In this setting, designing a testing scheme is equivalent to the construction of a \emph{disjunct matrix}, an $M \times N$ matrix where the union of supports of any $t$ columns does not contain the support of any other column. In principle, one wants to have such a matrix with minimum possible number $M$ of rows (tests). One of the main ways of constructing disjunct matrices relies on \emph{constant weight error-correcting codes} and their \emph{minimum distance}. In this paper, we consider a relaxed definition of a disjunct matrix known as \emph{almost disjunct matrix}. This concept is also studied under the name of \emph{weakly separated design} in the literature. The relaxed definition allows one to come up with group testing schemes where a close-to-one fraction of all possible sets of defective items are identifiable. Our main contribution is twofold. First, we go beyond the minimum distance analysis and connect the \emph{average distance} of a constant weight code to the parameters of an almost disjunct matrix constructed from it. Our second contribution is to explicitly construct almost disjunct matrices based on our average distance analysis, that have much smaller number of rows than any previous explicit construction of disjunct matrices. The parameters of our construction can be varied to cover a large range of relations for $t$ and $N$.Comment: 15 Page