There are several ways to write the number 5 as a sum of positive integers, disregarding order. A quick calculation shows that this can be done in 7 ways: 5 = 5, 5 = 4 + 1, 5 = 3 + 2, 5 = 3 + 1 + 1, 5 = 2 + 2 + 1, 5 = 2 + 1 + 1 + 1 and 5 = 1 + 1 + 1 + 1 + 1. In the same way, we can count the number of ways for each integer n. Although this calculation is trivial, a closed form for this function is not as easily obtained as one for combinations, for example. This work formulates and proves Rademacher's formula for these partition numbers. It also tries to uncover some key ideas behind the proof, the supporting theory and other inspirations.

Digitalization of sensitive information such as bankaccounts enforces privacy to be ensured. Mathematical objects called elliptic curves have ensured exchange of this type of information due to their algebraic structure. The security relies on so called ECDLP and solving these types of problems, but there has yet to be found an efficient algorithm to solve these problems. The aim of the thesis is to study the applications of elliptic curves in cryptography, in which Diffie-Hellman and Digital Signature Algorithm will be discussed. Furthermore, the safety of ECC will be compared to the generally more well-known RSA cryptosystem. Additionally, specific types of elliptic curves have certain properties which cause their ECDLP to be significantly uncomplicated to solve and some of these will be dealt with as well.