Two Number-Theoretic Problems That Illustrate the Power and Limitations of Randomness

Andrew Shallue, Illinois Wesleyan University

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Dissertation for Doctor of Philosophy (Mathematics) at the University of Wisconsin – Madison.

Abstract

This thesis contains work on two problems in algorithmic number theory. The first problem is to give an algorithm that constructs a rational point on an elliptic curve over a finite field. A fast and easy randomized algorithm has existed for some time. We prove that in the case where the finite field has characteristic 2, there is a deterministic algorithm with the same asymptotic running time as the existing randomized algorithm.