In this dissertation, we study two problems arising from arithmetic geometry.

Falting's theorem states that there are only finitely many rational points on curves of genus greater than 1. However, an explicit determination of all such points on a curve remains a hard problem. There are various approaches to computing rational points on higher genus curves and we use Coleman's theory of p-adic line integrals to study a particular class of curves with rich arithmetic origins, namely, the modular curves. In join work with Chen and Kedlaya, we implement a new algorithm that does not use the models of the modular curves and illustrate this method through the computation of several examples.

On the other hand, in anticipation of the development of powerful quantum computers in the next few decades, we study cryptosystems that rely on the hardness of certain number theoretical problems. In joint work with Arpin, Bilingsley, Hast, Perlner and Robinson, we investigate BIKE, a cryptosystem presented as one of the candidates for the National Institute of Standards and Technology Post-Quantum Cryptography Standardization Process. We identified several factors that affect the security of the code-based cryptosystem as a potential quantum-attack-resistant candidate for real world applications through extensive simulations.