The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation x → g x mod n. Analysis of the security of many cryptographic algorithms depends on the assumption that it is statistically impossible to distinguish the use of this map from the use of a randomly chosen map with similar characteristics. This paper focuses on a prime modulus, p, for which it is shown that the basic structure of the functional graph produced by this map is largely dependent on an interaction between g and p − 1. We deal with two of the possible structures, permutations and binary functional graphs. Estimates exist for the shape of a random permutation, but similar estimates must be created for the binary functional graphs. Experimental data suggest that both the permutations and binary functional graphs correspond well to the theoretical predictions.

Ludwig Van Beethoven's Orchestration of the Trombone Daniel R. CloutierLudwig Van Beethoven is considered one of the first composers to fully realize the wideranging performance potential of the trombone. Although the instrument had already been scored in sacred and operatic works by major composers (and symphonies by lesser-known composers), it was Beethoven who was considered to be the first to truly emancipate the trombone from its sacred and operatic boundaries and write for it in various secular, choral, and even chamber music genres. In this paper, every composition of Beethoven's that includes trombones is examined. The historical context of each individual work will be established, along with any general trombone-related observations from the work in question.Musical examples from each work will be examined in detail, with the view of elucidating compositional techniques such as melodic and solo usage, doubling, range, dynamic range, percentage of measures actually played, and placement within orchestration. Comparisons between genres and time periods will also be made, with the objective being to establish patterns of musical evolution-if any-across time or between genres.iii Dedication This Research Project is dedicated to the memory of my mother, Pauline J. Cloutier, who died unexpectedly during the last stages of this Project. She lived a full and active life, right up to her final days. She had been looking forward to this Research Project's completion, and it is my hope that her love of writing is reflected in the words found on these pages. iv

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