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.

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.

In this paper we outline a method of producing reduced elliptic curves with many integral points and provide the results of the outlined computations, including several curves with hundreds of integral points. The first three sections give back-ground and describe our work with integral points on elliptic curves. The last section is unrelated to elliptic curves and provides a complete classification of self-descriptive numbers.

A one-way function (OWF) is a function that is computationally feasible to compute in one direction, but infeasible to invert. Many current cryptosystems make use of properties of OWFs to provide ways to send secure messages. This paper reviews some simple OWFs and examines their use in contemporary cryptosystems and other cryptographic applications. This paper also discusses the broader implications of OWF-based cryptography, including its relevance to fields such as complexity theory and quantum computing, and considers the importance of OWFs in future cryptographic development