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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

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.

ContributorsSteele, Aaron J (Author) / Jones, John (Thesis director) / Childress, Nancy (Committee member) / Computer Science and Engineering Program (Contributor, Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2021-05
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Let $E$ be an elliptic curve defined over a number field $K$, $p$ a rational prime, and $\Lambda(\Gamma)$ the Iwasawa module of the cyclotomic extension of $K$. A famous conjecture by Mazur states that the $p$-primary component of the Selmer group of $E$ is $\Lambda(\Gamma)$-cotorsion when $E$ has good ordinary

Let $E$ be an elliptic curve defined over a number field $K$, $p$ a rational prime, and $\Lambda(\Gamma)$ the Iwasawa module of the cyclotomic extension of $K$. A famous conjecture by Mazur states that the $p$-primary component of the Selmer group of $E$ is $\Lambda(\Gamma)$-cotorsion when $E$ has good ordinary reduction at all primes of $K$ lying over $p$. The conjecture was proven in the case that $K$ is the field of rationals by Kato, but is known to be false when $E$ has supersingular reduction type. To salvage this result, Kobayashi introduced the signed Selmer groups, which impose stronger local conditions than their classical counterparts. Part of the construction of the signed Selmer groups involves using Honda's theory of commutative formal groups to define a canonical system of points. In this paper I offer an alternate construction that appeals to the Functional Equation Lemma, and explore a possible way of generalizing this method to elliptic curves defined over $p$-adic fields by passing from formal group laws to formal modules.
ContributorsReamy, Alexander (Author) / Sprung, Florian (Thesis advisor) / Childress, Nancy (Thesis advisor) / Paupert, Julien (Committee member) / Montaño, Jonathan (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2023
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Iwasawa theory is a branch of number theory that studies the behavior of certain objects associated to a $\mathbb{Z}_p$-extension. We will focus our attention to the cyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields for varying primes p, and will give some conditions for when the corresponding lambda-invariants are greater than

Iwasawa theory is a branch of number theory that studies the behavior of certain objects associated to a $\mathbb{Z}_p$-extension. We will focus our attention to the cyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields for varying primes p, and will give some conditions for when the corresponding lambda-invariants are greater than 1.
ContributorsStokes, Christopher Mathewson (Author) / Childress, Nancy (Thesis advisor) / Sprung, Florian (Committee member) / Montaño, Johnathan (Committee member) / Paupert, Julian (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2023