Discrete and Complex Algorithms for Curves
This dissertation consists of two parts. The first part pertains to the Schottky problem, which asks to characterize Jacobians of curves amongst abelian varieties. This has a complete solution only in the first non-trivial case where the genus is four, and the solution is described in terms of the Riemann theta function. We first present a Julia package for numerical evaluations of the Riemann theta function. We then describe numerical approaches to the Schottky problem in genus four and five. We present a solution to a variant of the Schottky problem in genus five, for Jacobians with a vanishing theta null. Finally, we describe solutions to the tropical Schottky problem, and relate the tropical and classical solutions to the Schottky problem in genus four.
The second part of this dissertation relates to cryptography. We first study cycles of pairing-friendly elliptic curves, for an application in pairing-based cryptography. We next study the concrete security of the Learning With Errors problem in lattice-based cryptography, when sampling the secret from a non-uniform, small distribution.