On Small Degree Extension Fields in Cryptology

This thesis studies the implications of using public key cryptographic primitives that are based in, or map to, the multiplicative group of finite fields with small extension degree. A central observation is that the multiplicative group of extension fields essentially decomposes as a product of algebraic tori, whose properties allow for improved communication efficiency. Part I of this thesis is concerned with the constructive implications of this idea. Firstly, algorithms are developed for the efficient implementation of torus-based cryptosystems and their performance compared with previous work. It is then shown how to apply these methods to operations required in low characteristic pairing-based cryptography. Finally, practical schemes for high-dimensional tori are discussed. Highly optimised implementations and benchmark timings are provided for each of these systems. Part II addresses the security of the schemes presented in Part I, i.e., the hardness of the discrete logarithm problem. Firstly, an heuristic analysis of the effectiveness of the Function Field Sieve in small characteristic is given. Next presented is an implementation of this algorithm for characteristic three fields used in pairing-based cryptography. Finally, a new index calculus algorithm for solving the discrete logarithm problem on algebraic tori is described and analysed

COMPUTATIONAL MODELING OF GENE REGULATION, GAMETE FORMATION, AND EMBRYO IMPLANTATION

DNA located in genes is transcribed into RNA which is translated into protein. The regulation of transcription and translation is carried out by several factors including a gene’s primary sequence, cis-regulatory elements (CREs) in non-coding DNA regions, epigenetic marks on the histones which compact DNA, and trans-binding factors (or proteins). The differential expression of a gene is crucial for establishing lineage-specific cell identity and phenotypic variability. Mutation or dysregulation may lead to natural variation within a population or aberrant gene expression and disease; trait-associated variation is known to be enriched in putative CREs, supporting their role in the origins of disease. Understanding the mechanisms by which CREs interact with one another and their cellular environment to regulate transcription may inform knowledge of biological pathways and provide a crucial foundation for developing new treatments. Further, because all DNA is passed to an offspring from their parents, it is important to understand not just the outcomes on expression due to coding and non-coding variation, but also how genetic material is passed to future generations. These dissertation chapters apply modeling approaches to large amounts of genetic and gene expression data in order to 1) better understand how the sequence and epigenetic makeup of CREs impact gene expression within hematopoiesis; 2) scan for selfish genetic elements which are preferentially passed to offspring within human sperm samples; and 3) predict implantation success for euploid embryos given gene expression profiles. Our models within Chapters 2-4 describe the impact of CREs within the blood cell lineage, connecting CREs to putative target genes, and establishing that the hematopoietic CREs were enriched for blood-trait associated genetic variation. Within Chapter 5, we find no compelling evidence of selfish genetic elements within a large sample of human sperm. Finally, within Chapter 6, we identify some genes which seem to impact the success of IVF embryo implantation by acting through regulation of translation