On the Computation and Composition of Belyiˇ Maps and Dessins D'enfants
This dissertation centers on computing with dessins d'enfants, in the form of constellations, and on the monodromy group of compositions of Belyiˇ maps. To begin, a discussion of known effective and efficient algorithms for computing constellations, Belyiˇ Maps, and dessins d'enfants from one another is presented. Following this is an analysis of how to use double cosets in an optimal way to count equivalence classes of constellations. In addition, class multiplication coefficients are used to count trees with certain passports, culminating in a new proof of a result of Mednykh. The method given by Wood for computing the constellation of a composition of Belyiˇ maps is further developed and extended to allow Belyiˇ maps which are defined over the complex numbers. By utilizing the fact that the monodromy group of the composition of Belyiˇ maps β о γ is a subgroup of a wreath product, generators of the monodromy group of β о γ are found by a simple algorithm. Additionally, a group is determined from β alone which allows one to find the monodromy group of β о γ , for any γ, simply by applying the monodromy representation of γ. Finally, using the previous results, a cryptographic protocol utilizing compositions of Belyiˇ maps is proposed. A probabilistic method for efficiently deciding if the monodromy group of a Belyiˇ map is either the alternating or symmetric group is discussed. Although the protocol, in its current form, is not efficient enough for practical use, it demonstrates the ability to design a cryptographic protocol around the problem of computing Belyiˇ maps.
Goins, Purdue University.
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