abstract: Finding the optimal solution to a problem with an enormous search space can be challenging. Unless a combinatorial construction technique is found that also guarantees the optimality of the resulting solution, this could be an infeasible task. If such a technique is unavailable, different heuristic methods are generally used to improve the upper bound on the size of the optimal solution. This dissertation presents an alternative method which can be used to improve a solution to a problem rather than construct a solution from scratch. Necessity analysis, which is the key to this approach, is the process of analyzing the necessity of each element in a solution. The post-optimization algorithm presented here utilizes the result of the necessity analysis to improve the quality of the solution by eliminating unnecessary objects from the solution. While this technique could potentially be applied to different domains, this dissertation focuses on k-restriction problems, where a solution to the problem can be presented as an array. A scalable post-optimization algorithm for covering arrays is described, which starts from a valid solution and performs necessity analysis to iteratively improve the quality of the solution. It is shown that not only can this technique improve upon the previously best known results, it can also be added as a refinement step to any construction technique and in most cases further improvements are expected. The post-optimization algorithm is then modified to accommodate every k-restriction problem; and this generic algorithm can be used as a starting point to create a reasonable sized solution for any such problem. This generic algorithm is then further refined for hash family problems, by adding a conflict graph analysis to the necessity analysis phase. By recoloring the conflict graphs a new degree of flexibility is explored, which can further improve the quality of the solution. / Dissertation/Thesis / Ph.D. Computer Science 2011

abstract: Broadcast Encryption is the task of cryptographically securing communication in a broadcast environment so that only a dynamically specified subset of subscribers, called the privileged subset, may decrypt the communication. In practical applications, it is desirable for a Broadcast Encryption Scheme (BES) to demonstrate resilience against attacks by colluding, unprivileged subscribers. Minimal Perfect Hash Families (PHFs) have been shown to provide a basis for the construction of memory-efficient t-resilient Key Pre-distribution Schemes (KPSs) from multiple instances of 1-resilient KPSs. Using this technique, the task of constructing a large t-resilient BES is reduced to finding a near-minimal PHF of appropriate parameters. While combinatorial and probabilistic constructions exist for minimal PHFs with certain parameters, the complexity of constructing them in general is currently unknown. This thesis introduces a new type of hash family, called a Scattering Hash Family (ScHF), which is designed to allow for the scalable and ingredient-independent design of memory-efficient BESs for large parameters, specifically resilience and total number of subscribers. A general BES construction using ScHFs is shown, which constructs t-resilient KPSs from other KPSs of any resilience ≤w≤t. In addition to demonstrating how ScHFs can be used to produce BESs , this thesis explores several ScHF construction techniques. The initial technique demonstrates a probabilistic, non-constructive proof of existence for ScHFs . This construction is then derandomized into a direct, polynomial time construction of near-minimal ScHFs using the method of conditional expectations. As an alternative approach to direct construction, representing ScHFs as a k-restriction problem allows for the indirect construction of ScHFs via randomized post-optimization. Using the methods defined, ScHFs are constructed and the parameters' effects on solution size are analyzed. For large strengths, constructive techniques lose significant performance, and as such, asymptotic analysis is performed using the non-constructive existential results. This work concludes with an analysis of the benefits and disadvantages of BESs based on the constructed ScHFs. Due to the novel nature of ScHFs, the results of this analysis are used as the foundation for an empirical comparison between ScHF-based and PHF-based BESs . The primary bases of comparison are construction efficiency, key material requirements, and message transmission overhead. / Dissertation/Thesis / M.S. Computer Science 2012

abstract: The construction of many families of combinatorial objects remains a challenging problem. A t-restriction is an array where a predicate is satisfied for every t columns; an example is a perfect hash family (PHF). The composition of a PHF and any t-restriction satisfying predicate P yields another t-restriction also satisfying P with more columns than the original t-restriction had. This thesis concerns three approaches in determining the smallest size of PHFs. Firstly, hash families in which the associated property is satisfied at least some number lambda times are considered, called higher-index, which guarantees redundancy when constructing t-restrictions. Some direct and optimal constructions of hash families of higher index are given. A new recursive construction is established that generalizes previous results and generates higher-index PHFs with more columns. Probabilistic methods are employed to obtain an upper bound on the optimal size of higher-index PHFs when the number of columns is large. A new deterministic algorithm is developed that generates such PHFs meeting this bound, and computational results are reported. Secondly, a restriction on the structure of PHFs is introduced, called fractal, a method from Blackburn. His method is extended in several ways; from homogeneous hash families (every row has the same number of symbols) to heterogeneous ones; and to distributing hash families, a relaxation of the predicate for PHFs. Recursive constructions with fractal hash families as ingredients are given, and improve upon on the best-known sizes of many PHFs. Thirdly, a method of Colbourn and Lanus is extended in which they horizontally copied a given hash family and greedily applied transformations to each copy. Transformations of existential t-restrictions are introduced, which allow for the method to be applicable to any t-restriction having structure like those of hash families. A genetic algorithm is employed for finding the "best" such transformations. Computational results of the GA are reported using PHFs, as the number of transformations permitted is large compared to the number of symbols. Finally, an analysis is given of what trade-offs exist between computation time and the number of t-sets left not satisfying the predicate. / Dissertation/Thesis / Doctoral Dissertation Computer Science 2019