3. Electronic Theses and Dissertations (ETDs) - All submissions

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The algebra and geometry of using continued fractions for approximating real and complex numbers
(2022) Mennen, Carminda Margaretha
Through geometric analysis we forge a new interpretation of the link between nearest integer continued fractions and the Farey tessellation of hyperbolic space. Instead of just truncating the continued fraction to generate approximations, we focus on • how to parse the product of maps derived from the nearest integer continued fraction into a product of parabolic and elliptic Mobius maps and ¨ • on the collection of points on which these maps act. It turns out that we need to set apart the elliptic maps that permute a collection of six vertices, three values in R, namely ∞, 0 and 1, and three values in C, namely i, 1 + i and 1+i 2 . The action of the remaining parabolic maps on the same six vertices results in the creation of a sequence of nested Farey quadrilaterals, containing the target, whose boundaries are based in the Schmidt arrangement formed by the Farey sets and dual Farey sets of Schmidt.
Continued fractions and cryptographic applications
(2022) Sookraj, Priyanka
In this dissertation, Michael J. Wiener’s proposed attack on short secret exponents used in the RSA cryptosystem is studied and thus an application of continued fractions in the cryptanalysis is highlighted. Furthermore, some variants and improvements to the attack proposed by A. Dujella, M. Bunder and J. Tonien are studied.