Date of Award

5-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Edray Herber Goins

Committee Member 1

Donu V.B. Arapura

Committee Member 2

David Ben McReynolds

Committee Member 3

Samuel S. Wagstaff, Jr.

Abstract

This dissertation centers on computing with dessins d’enfants, in the form of constellations, and on the monodromy group of compositions of Bely˘ı maps. To begin, a discussion of known e ective and eÿcient algorithms for computing constellations, Bely˘ı 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 coeÿcients 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 Bely˘ı maps is further developed and extended to allow Bely˘ı maps which are defned over the complex numbers. By utilizing the fact that the monodromy group of the composition of Bely˘ı maps is a subgroup of a wreath product, generators of the monodromy group of are found by a simple algorithm. Additionally, a group is determined from alone which allows one to fnd the monodromy group of , for any , simply by applying the monodromy representation of . Finally, using the previous results, a cryptographic protocol utilizing compositions of Bely˘ı maps is proposed. A probabilistic method for eÿciently deciding if the monodromy group of a Bely˘i map is either the alternating or symmetric group is discussed. Although the protocol, in its current form, is not eÿcient enough for practical use, it demonstrates the ability to design a cryptographic protocol around the problem of computing Bely˘ı maps.

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