Filtering by
- All Subjects: Elliptic Curves
- All Subjects: Polynomials
- Creators: Childress, Nancy
Many current cryptographic algorithms will eventually become easily broken by Shor's Algorithm once quantum computers become more powerful. A number of new algorithms have been proposed which are not compromised by quantum computers, one of which is the Supersingular Isogeny Diffie-Hellman Key Exchange Protocol (SIDH). SIDH works by having both parties perform random walks between supersingular elliptic curves on isogeny graphs of prime degree and eventually end at the same location, a shared secret.<br/><br/>This thesis seeks to explore some of the theory and concepts underlying the security of SIDH, especially as it relates to finding supersingular elliptic curves, generating isogeny graphs, and implementing SIDH. As elliptic curves and SIDH may be an unfamiliar topic to many readers, the paper begins by providing a brief introduction to elliptic curves, isogenies, and the SIDH Protocol. Next, the paper investigates more efficient methods of generating supersingular elliptic curves, which are important for visualizing the isogeny graphs in the algorithm and the setup of the protocol. Afterwards, the paper focuses on isogeny maps of various degrees, attempting to visualize isogeny maps similar to those used in SIDH. Finally, the paper looks at an implementation of SIDH in PARI/GP and work is done to see the effects of using isogenies of degree greater than 2 and 3 on the security, runtime, and practicality of the algorithm.
factors into linear factors over $K$, as do all of its derivatives. Such a polynomial
is said to be {\it proper} if
its roots are distinct. An unresolved question in the literature is
whether or not there exists a proper $\Q$-derived polynomial of degree 4. Some examples
are known of proper $K$-derived quartics for a quadratic number field $K$, although other
than $\Q(\sqrt{3})$, these fields have quite large discriminant. (The second known field
is $\Q(\sqrt{3441})$.) I will describe a search for quadratic fields $K$
over which there exist proper $K$-derived quartics. The search finds examples for
$K=\Q(\sqrt{D})$ with $D=...,-95,-41,-19,21,31,89,...$.\\
For the second topic, by Krasner's lemma there exist a finite number of degree $n$ extensions of $\Q_p$. Jones and Roberts have developed a database recording invariants of $p$-adic extensions for low degree $n$. I will contribute data to this database by computing the Galois slope content, inertia subgroup, and Galois mean slope for a variety of wildly ramified extensions of composite degree using the idea of \emph{global splitting models}.